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Random Variable Random Variables STATISTICS – Lecture no. 3 Jiřı́ Neubauer Department of Econometrics FEM UO Brno office 69a, tel. 973 442029 email:[email protected] 13. 10. 2009 Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Random Variable Many random experiments have numerical outcomes. Definition A random variable is a real-valued function X (ω) defined on the sample space Ω. The set of possible values of the random variable X is called the range of X . M = {x; X (ω) = x}. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Random Variable We denote random variables by capital letters X , Y , . . . (eventually X1 , X2 , . . . ) and their particular values by small letters x, y , . . . . Using random variables we can describe random events, for example X = x, X ≤ x, x1 < X < x2 etc. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Random Variable We denote random variables by capital letters X , Y , . . . (eventually X1 , X2 , . . . ) and their particular values by small letters x, y , . . . . Using random variables we can describe random events, for example X = x, X ≤ x, x1 < X < x2 etc. Examples of random variables: the number of dots when a die is rolled, the range is M = {1, 2, . . . 6} the number of rolls of a die until the first 6 appears, the range is M = {1, 2, }˙ the lifetime of the lightbulb, the range is M = {x; x ≥ 0}, Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Random Variable According to the range M we separate random variables to discrete . . . M is finite or countable, Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Random Variable According to the range M we separate random variables to discrete . . . M is finite or countable, continuous . . . M is a closed or open interval. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Random Variable Examples of discrete random variables: the number of cars sold at a dealership during a given month, M = {0, 1, 2, . . . } the number of houses in a certain block, M = {1, 2, . . . } the number of fish caught on a fishing trip, M = {0, 1, 2, . . . } the number of heads obtained in three tosses of a coin, M = {0, 1, 2, 3} Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Random Variable Examples of discrete random variables: the number of cars sold at a dealership during a given month, M = {0, 1, 2, . . . } the number of houses in a certain block, M = {1, 2, . . . } the number of fish caught on a fishing trip, M = {0, 1, 2, . . . } the number of heads obtained in three tosses of a coin, M = {0, 1, 2, 3} Examples of continuous random variables: the height of a person, M = (0, ∞) the time taken to complete an examination, M = (0, ∞) the amount of milk in a bottle, M = (0, ∞) Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Random Variable For the description of random variables we will use some functions: a (cumulative) distribution function F (x), a probability function p(x) – only for discrete random variables, a probability density function f (x) – only for continuous random variables. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Random Variable For the description of random variables we will use some functions: a (cumulative) distribution function F (x), a probability function p(x) – only for discrete random variables, a probability density function f (x) – only for continuous random variables. and some measures: measures of location, measures of dispersion, measures of concentration. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Distribution Function Definition Let X be any random variable. The distribution function F (x) of the random variable X is defined as F (x) = P(X ≤ x), x ∈ R. Note: distribution function = cumulative distribution function Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Distribution Function We mention some important properties of F (x): for every real x: 0 ≤ F (x) ≤ 1, Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Distribution Function We mention some important properties of F (x): for every real x: 0 ≤ F (x) ≤ 1, F (x) is a non-decreasing, right-continuous function, Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Distribution Function We mention some important properties of F (x): for every real x: 0 ≤ F (x) ≤ 1, F (x) is a non-decreasing, right-continuous function, it has limits lim F (x) = 0, lim F (x) = 1, x→−∞ x→∞ if range of X is M = {x; x ∈ (a, bi} then F (a) = 0 a F (b) = 1, Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Distribution Function We mention some important properties of F (x): for every real x: 0 ≤ F (x) ≤ 1, F (x) is a non-decreasing, right-continuous function, it has limits lim F (x) = 0, lim F (x) = 1, x→−∞ x→∞ if range of X is M = {x; x ∈ (a, bi} then F (a) = 0 a F (b) = 1, for every real numbers x1 and x2 : P(x1 < X ≤ x2 ) = F (x2 ) − F (x1 ). Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Discrete Random Variable For a discrete random variable X , we are interested in computing probabilities of the type P(X = xk ) for various values xk in range of X . Definition Let X be a discrete random variable with range {x1 , x2 , . . . } (finite or countably infinite). The function p(x) = P(X = x) is called the probability function of X . Note: probability function = probability mass function Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Discrete Random Variable We mention some important properties of p(x) for every real number x, 0 ≤ p(x) ≤ 1, Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Discrete Random Variable We mention some important properties of p(x) for every real number x, 0 ≤ p(x) ≤ 1, X p(x) = 1 x∈M Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Discrete Random Variable We mention some important properties of p(x) for every real number x, 0 ≤ p(x) ≤ 1, X p(x) = 1 x∈M for every two real numbers xk and xl (xk ≤ xl ): P(xk ≤ X ≤ xl ) = xl X p(xi ). xi =xk Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Probability Function The probability function p(x) can be described by the table, X p(x) x1 p(x1 ) x2 p(x2 ) Jiřı́ Neubauer ... ... xi p(xi ) Random Variables ... ... P 1 Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Probability Function the graph [x, p(x)], Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Probability Function the formula, for example π(1 − π)x p(x) = 0 x = 0, 1, 2, . . . , otherwise, where π is a given probability. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example The shooter has 3 bullets and shoots at the target until the first hit or until the last bullet. The probability that the shooter hits the target after one shot is 0.6. The random variable X is the number of the fired bullets. Find the probability and the distribution function of the given random variable. What is the probability that the number of the fired bullets will not be larger then 2? Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example Random variable X is discrete with the range M = {1, 2, 3}. The probability function is: p(1) = P(X = 1) = 0.6, Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example Random variable X is discrete with the range M = {1, 2, 3}. The probability function is: p(1) = P(X = 1) = 0.6, p(2) = P(X = 2) = 0.4 · 0.6 = 0.24, Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example Random variable X is discrete with the range M = {1, 2, 3}. The probability function is: p(1) = P(X = 1) = 0.6, p(2) = P(X = 2) = 0.4 · 0.6 = 0.24, p(3) = P(X = 3) = 0.4·0.4·0.6+0.4·0.4·0.4 = 0.4·0.4 = 0.16. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example All results are summarized in the table x p(x) 1 0.6 2 0.24 3 0.16 P 1 The probability function can be described by the formula 0.6 · 0.4x−1 x = 1, 2, 0.42 x = 3, p(x) = 0 otherwise. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example We can calculate some values of the distribution function F (x): F (0) = P(X ≤ 0) = 0, Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example We can calculate some values of the distribution function F (x): F (0) = P(X ≤ 0) = 0, F (1) = P(X ≤ 1) = p(1) = 0.6, Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example We can calculate some values of the distribution function F (x): F (0) = P(X ≤ 0) = 0, F (1) = P(X ≤ 1) = p(1) = 0.6, F (1.5) = P(X ≤ 1.5) = P(X ≤ 1) = p(1) = 0.6, Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example We can calculate some values of the distribution function F (x): F (0) = P(X ≤ 0) = 0, F (1) = P(X ≤ 1) = p(1) = 0.6, F (1.5) = P(X ≤ 1.5) = P(X ≤ 1) = p(1) = 0.6, F (2) = P(X ≤ 2) = p(1) + p(2) = 0.84, Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example We can calculate some values of the distribution function F (x): F (0) = P(X ≤ 0) = 0, F (1) = P(X ≤ 1) = p(1) = 0.6, F (1.5) = P(X ≤ 1.5) = P(X ≤ 1) = p(1) = 0.6, F (2) = P(X ≤ 2) = p(1) + p(2) = 0.84, F (3) = P(X ≤ 3) = p(1) + p(2) + p(3) = 1, Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example We can calculate some values of the distribution function F (x): F (0) = P(X ≤ 0) = 0, F (1) = P(X ≤ 1) = p(1) = 0.6, F (1.5) = P(X ≤ 1.5) = P(X ≤ 1) = p(1) = 0.6, F (2) = P(X ≤ 2) = p(1) + p(2) = 0.84, F (3) = P(X ≤ 3) = p(1) + p(2) + p(3) = 1, F (4) = P(X ≤ 4) = p(1) + p(2) + p(3) = 1. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example We can write 0 x < 1, 0.6 1 ≤ x < 2, F (x) = 0.84 2 ≤ x < 3, 1 x ≥ 3. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example Figure: The probability and the distribution function Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example What is the probability that the number of the fired bullets will not be larger then 2? P(X ≤ 2) = P(X = 1) + P(X = 2) = p(1) + p(2) = F (2) = = 0.6 + 0.24 = 0.84. Jiřı́ Neubauer Random Variables Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Random Variable Probability Density Function If the cumulative distribution function is a continuous function, then X is said to be a continuous random variable. Definition The probability density function of the random variable X is a non-negative function f (x) such that Zx f (t)dt, x ∈ R. F (x) = −∞ Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Probability Density Function Some properties of f (x): R R∞ f (x)dx = f (x)dx = 1, −∞ M Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Probability Density Function Some properties of f (x): R R∞ f (x)dx = f (x)dx = 1, −∞ f (x) = M dF (x) dx = F 0 (x), where the derivative exists, Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Probability Density Function Some properties of f (x): R R∞ f (x)dx = f (x)dx = 1, −∞ f (x) = M dF (x) dx = F 0 (x), where the derivative exists, P(x1 ≤ X ≤ x2 ) = P(x1 < X < x2 ) = P(x1 < X ≤ x2 ) = Rx2 P(x1 ≤ X < x2 ) = F (x2 ) − F (x1 ) = f (x)dx x1 Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Probability Density Function Some properties of f (x): R R∞ f (x)dx = f (x)dx = 1, −∞ f (x) = M dF (x) dx = F 0 (x), where the derivative exists, P(x1 ≤ X ≤ x2 ) = P(x1 < X < x2 ) = P(x1 < X ≤ x2 ) = Rx2 P(x1 ≤ X < x2 ) = F (x2 ) − F (x1 ) = f (x)dx x1 If X is a continuous random variable, then P(X = x) = 0. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Probability Density Function The function f (x) we can describe by a formula or a graph, for example ( 1 − x−2 5 pro x > 2, 5e f (x) = 0 pro x ≤ 2. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example The random variable X has the probability density function 2 cx (1 − x) 0 < x < 1, f (x) = 0 otherwise. Determine a constant c in order that f (x) is a probability density function. Find a distribution function of the random variable X . Calculate the probability P(0.2 < X < 0.8). Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example The probability density function has to fulfil Z f (x)dx = 1. M R1 0 h 3 R1 cx 2 (1 − x)dx = c (x 2 − x 3 )dx = c x3 − 0 c = c 13 − 14 = 12 = 1, we get c = 12. Jiřı́ Neubauer Random Variables i1 x4 4 0 = Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example The distribution function can be calculated by the definition of the probability density function. We can write for 0 < x < 1 h 3 Rx 12t 2 (1 − t)dt = 12 (t 2 − t 3 )dt = 12 t3 − 0 h 0 i x3 x4 3 = 12 3 − 4 = 4x − 3x 4 . F (x) = Rx F (x) = x 3 (4 0 x ≤ 0, − 3x) 0 < x < 1, 1 otherwise. Jiřı́ Neubauer Random Variables ix t4 4 0 = Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Figure: The probability density function and the distribution function Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example Using the probability density function we can calculate Z0.8 0.8 P(0.2 < X < 0.8) = 12x 2 (1 − x)dx = 4x 3 − 3x 4 0.2 = 0.792. 0.2 Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example Using the probability density function we can calculate Z0.8 0.8 P(0.2 < X < 0.8) = 12x 2 (1 − x)dx = 4x 3 − 3x 4 0.2 = 0.792. 0.2 If the distribution function is known, we can do simpler calculation P(0.2 < X < 0.8) = F (0.8) − F (0.2) = = 0.83 (4 − 3 · 0.8) − 0.23 (4 − 3 · 0.2) = 0.792. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example A random variable X is described by the distribution function 0 x ≤ 0, F (x) = −x 1−e x > 0. Find a probability density function. Jiřı́ Neubauer Random Variables Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Random Variable Example Using mentioned formula f (x) = and the fact that d dx (1 dF (x) dx − e −x ) = e −x we get f (x) = Jiřı́ Neubauer 0 e −x x ≤ 0, x > 0. Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Measures of Location The distribution function (the probability function or the probability density function) gives us the complete information about the random variable. Sometimes is useful to know some simpler and concentrated formulation of this information such as measures of location, dispersion and concentration. The best known measures of location are a mean (an expected value), quantiles (a median, an upper and a lower quartile, . . . ) and a mode. Jiřı́ Neubauer Random Variables Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Random Variable Expected Value Definition The mean (the expected value) E (X ) of the random variable X (sometimes denoted as µ) is the value that is expected to occur per repetition, if an experiment is repeated a large number of times. For the discrete random variable is defined as X E (X ) = xi p(xi ), M for the continuous random variable as Z E (X ) = xf (x)dx M if the given sequence or integral absolutely converges. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Expected Value We mention some properties of the mean: the mean of the constant c is equal to this constant E (c) = c, Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Expected Value We mention some properties of the mean: the mean of the constant c is equal to this constant E (c) = c, the mean of the product of the constant c and the random variable X is equal to the product of the given constant c and the mean of X E (cX ) = cE (X ), Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Expected Value the mean of the sum of random variables X1 , X2 , . . . , Xn is equal to the sum of the mean of the given random variables, E (X1 + X2 + · · · + Xn ) = E (X1 ) + E (X2 ) + · · · + E (Xn ), Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Expected Value the mean of the sum of random variables X1 , X2 , . . . , Xn is equal to the sum of the mean of the given random variables, E (X1 + X2 + · · · + Xn ) = E (X1 ) + E (X2 ) + · · · + E (Xn ), if X1 , X2 , . . . , Xn are independent, then the mean of their product is equal to the product of their means E (X1 X2 . . . Xn ) = E (X1 )E (X2 ) . . . E (Xn ). Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Random Variables and Independency The random variables X1 , X2 , . . . , Xn are independent if and only if for any numbers x1 , x2 , . . . , xn ∈ R is P(X1 ≤ x1 , X2 ≤ x2 , . . . , Xn ≤ xn ) = = P(X1 ≤ x1 ) · P(X2 ≤ x2 ) · · · P(Xn ≤ xn ). Let us have the random vector X = (X1 , X2 , . . . , Xn ), whose components X1 , X2 , . . . , Xn are the random variables. F (x) = F (x1 , x2 , . . . , xn ) = P(X1 ≤ x1 , X2 ≤ x2 , . . . , Xn ≤ xn ) is the distribution function of the vector X and F (x1 ), F (x2 ), . . . , F (xn ) are the distribution functions of the random variables X1 , X2 , . . . , Xn . The random variables X1 , X2 , . . . , Xn are independent if and only if F (x1 , x2 , . . . , xn ) = F (x1 ) · F (x2 ) · · · F (xn ). Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Random Variables and Independency If X is the random vector whose components are the discrete random variables, the function p(x) = p(x1 , x2 , . . . , xn ) = P(X1 = x1 , X2 = x2 , . . . , Xn = xn ) is the probability function of the vector X, p(x1 ), p(x2 ), . . . , p(xn ) are the probability functions of X1 , X2 , . . . , Xn , then: X1 , X2 , . . . , Xn are independent if and only if p(x1 , x2 , . . . , xn ) = p(x1 ) · p(x2 ) · · · p(xn ). If X is the random vector whose components are the continuous random variables, the function f (x) = f (x1 , x2 , . . . , xn ) is the probability density function of the vector X, f (x1 ), f (x2 ), . . . , f (xn ) are the probability density functions of X1 , X2 , . . . , Xn , then: X1 , X2 , . . . , Xn are independent if and only if f (x1 , x2 , . . . , xn ) = f (x1 ) · f (x2 ) · · · f (xn ). Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Quantile Definition 100P% quantile xP of the random variable with the increasing distribution function is such value of the random variable that P(X ≤ xP ) = F (xP ) = P, Jiřı́ Neubauer 0 < P < 1. Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Quantile The quantile x0.50 we call median Me(X ), it fulfils P(X ≤ Me(X )) = P(X ≥ Me(X )) = 0.50. The quantile x0.25 is called the lower quartile, the quantile x0.75 is called the upper quartile. The selected quantiles of some important distributions are tabulated. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Mode Definition The mode Mo(X ) is the value of the random variable with the highest probability (for the discrete random variable), or the value, where the function f (x) has the maximum (for the continuous random variable ). Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example Find the mean (the expected value) and the mode of the random variable defined as the number of fired bullets (see the example before). The probability function is 0.6 · 0.4x−1 x = 1, 2, 0.42 x = 3, p(x) = 0 otherwise. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example The mean (the expected value) we get using the formula from the definition of E (X ) E (x) = 3 X xi p(xi ) = 1 · 0.6 · 0.40 + 2 · 0.6 · 0.41 + 3 · 0.42 = 1.56. i=1 Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example The mean (the expected value) we get using the formula from the definition of E (X ) E (x) = 3 X xi p(xi ) = 1 · 0.6 · 0.40 + 2 · 0.6 · 0.41 + 3 · 0.42 = 1.56. i=1 The mode is the value of the given random variable with the highest probability which is Mo(X ) = 1, because p(1) = 0.6. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example The random variable X is described by the probability density function 12x 2 (1 − x) 0 < x < 1, f (x) = 0 otherwise. Find the mean (the expected value) and the mode. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example We calculate the mean using the definition formula Z1 E (X ) = Z1 xf (x)dx = 0 x4 x5 x·12x (1−x) = 12 − 4 5 2 0 Jiřı́ Neubauer Random Variables 1 = 0 3 = 0.6. 5 Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example We calculate the mean using the definition formula Z1 E (X ) = Z1 xf (x)dx = 0 x4 x5 x·12x (1−x) = 12 − 4 5 2 0 1 = 0 3 = 0.6. 5 The mode is the maximum of the probability density function. We have to 2find themaximum of f 2(x) on the interval 0 < x < 1, d dx 12x (1 − x) = 12(2x − 3x ) = 0, x(2 − 3x) = 0, we get x = 0 or x = 2/3. The maximum of f (x) is in x = 2/3 ⇒ Mo(X ) = 2/3. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example Find the median,the upper and the lower quartile of the random variable X with the distribution function 1 − x13 x > 1, F (x) = 0 x ≤ 1. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example The quantile is defined by the formula F (xP ) = P. 1− 1 = P, xP3 xP = √ 3 Jiřı́ Neubauer 1 . 1−P Random Variables Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Random Variable Example The quantile is defined by the formula F (xP ) = P. 1− 1 = P, xP3 xP = √ 3 median x0.50 = lower quartile x0.25 = upper quartile x0.75 = 1 √ 3 1−0.50 1 √ 3 1−0.25 1 √ 3 1−0.75 Jiřı́ Neubauer 1 . 1−P = 1.260, = 1.101, = 1.587. Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Measures of Dispersion The elementary and widely-used measures of dispersion are the variance and the standard deviation. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Variance Definition The variance D(X ) of the random variable X (sometimes denoted as σ 2 ) is defined by the formula D(X ) = E [X − E (X )]2 . The variance of the discrete random variable is given by X D(X ) = [xi − E (X )]2 p(xi ), M the variance of the continuous random variable is Z D(X ) = [x − E (X )]2 f (x)dx. M Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Variance Some properties of the variance: D(k) = 0, where k is a constant, Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Variance Some properties of the variance: D(k) = 0, where k is a constant, D(kX ) = k 2 D(X ), Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Variance Some properties of the variance: D(k) = 0, where k is a constant, D(kX ) = k 2 D(X ), D(X + Y ) = D(X ) + D(Y ), if X and Y are independent, Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Variance Some properties of the variance: D(k) = 0, where k is a constant, D(kX ) = k 2 D(X ), D(X + Y ) = D(X ) + D(Y ), if X and Y are independent, D(X ) ≥ 0 for every random variable, Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Variance D(X ) = E (X 2 ) − E (X )2 , D(X ) = E [X − E (X )]2 = E [X 2 − 2XE (X ) + E (X )2 ] = = E (X 2 ) − E [2XE (X )] + E [E (X )2 ] = = E (X 2 ) − 2E (X )E (X ) + E (X )2 = E (X 2 ) − E (X )2 For the discrete random variable X D(X ) = xi2 p(xi ) − E (X )2 , M for the continuous random variable Z D(X ) = x 2 f (x)dx − E (X )2 . M Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Standard Deviation Definition The standard deviation σ(X ) of the random variable X is defined as the square root of the variance p σ(X ) = D(X ). The standard deviation has the same unit as the random variable X. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Figure: Relation between the mean and the standard deviation Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example Find the variance and the standard deviation of the random variable defined as the number of fired bullets (see the example before). Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example The mean is E (X ) = 1.56 (see the previous example). For the purpose of calculation of the variance we use the formula D(X ) = E (X 2 ) − E (X )2 Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example The mean is E (X ) = 1.56 (see the previous example). For the purpose of calculation of the variance we use the formula D(X ) = E (X 2 ) − E (X )2 2 E (X ) = X M xi2 p(xi ) = 3 X xi2 p(xi ) = 12 ·0.6+22 ·0.24+32 ·0.16 = 3, i=1 then D(X ) = 3 − 1.562 = 0.566. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example The mean is E (X ) = 1.56 (see the previous example). For the purpose of calculation of the variance we use the formula D(X ) = E (X 2 ) − E (X )2 2 E (X ) = X M xi2 p(xi ) = 3 X xi2 p(xi ) = 12 ·0.6+22 ·0.24+32 ·0.16 = 3, i=1 then D(X ) = 3 − 1.562 = 0.566. The standard deviation is the square root of the variance p σ(X ) = D(X ) = 0.753. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example Find the variance and the standard deviation of the random variable X with the probability density function 12x 2 (1 − x) 0 < x < 1, f (x) = 0 otherwise. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example The mean is E (X ) = 3/5 (see the previous example). D(X ) = E (X 2 ) − E (X )2 E (X 2 ) = R M x 2 f (x)dx = R1 x 2 · 12x 2 (1 − x)dx = 12 0 = 25 = 0.4, Jiřı́ Neubauer Random Variables h x5 5 − i1 x6 6 0 Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example The mean is E (X ) = 3/5 (see the previous example). D(X ) = E (X 2 ) − E (X )2 E (X 2 ) = R M x 2 f (x)dx = R1 x 2 · 12x 2 (1 − x)dx = 12 0 = 25 = 0.4, 2 3 1 2 = = 0.04. D(X ) = − 5 5 25 The standard deviation is σ(X ) = p 1 D(X ) = = 0.2. 5 Jiřı́ Neubauer Random Variables h x5 5 − i1 x6 6 0 Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Measures of Concentration We will focus on the measures describing the shape of random variables distribution (skewness and kurtosis). These measures are defined by moments. Jiřı́ Neubauer Random Variables Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Random Variable Measures of Concentration Definition The r th moment µ0r of the random variable X is defined by the formula µ0r (X ) = E (X r ) for r = 1, 2, . . . . The r th moment of the discrete random variable is given by X µ0r (X ) = xir p(xi ), M r th moment of the continuous random variable is Z 0 µr (X ) = x r f (x)dx. M Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Measures of Concentration Definition The r th central moment µr of the random variable X is defined by the formula µr (X ) = E [X − E (X )]r for r = 2, 3, . . . . The r th central moment of the discrete random variable is given by X µr (X ) = [xi − E (X )]r p(xi ), M the r th central moment of the continuous random variable is Z µr (X ) = [x − E (X )]r f (x)dx. M Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Skewness Definition The skewness α3 (X ) is defined by the formula α3 (X ) = Jiřı́ Neubauer µ3 (X ) . σ(X )3 Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Skewness Definition The skewness α3 (X ) is defined by the formula α3 (X ) = µ3 (X ) . σ(X )3 According to the values of the skewness we can tell whether distribution is symmetric or asymmetric. α3 = 0, distribution is symmetric, α3 < 0, distribution is skewed to the right, α3 > 0, distribution is skewed to the left. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Skewness Figure: The skewness Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Kurtosis Definition The kurtosis α4 (X ) is defined by the formula α4 (X ) = Jiřı́ Neubauer µ4 (X ) − 3. σ(X )4 Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Kurtosis Definition The kurtosis α4 (X ) is defined by the formula α4 (X ) = µ4 (X ) − 3. σ(X )4 Kurtosis is a measure of how outlier-prone a distribution is. The kurtosis of the normal distribution is 0. Distributions that are more outlier-prone than the normal distribution have kurtosis greater than 0; distributions that are less outlier-prone have kurtosis less than 0. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Kurtosis Figure: The kurtosis Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example Calculate the skewness and the kurtosis of the random variable defined as the number of fired bullets (see the previous examples). Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example First of all we have to calculate 3rd and 4th central moment. The mean of the given random variable is E (X ) = 1.56, the standard deviation is σ(X ) = 0.753. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example First of all we have to calculate 3rd and 4th central moment. The mean of the given random variable is E (X ) = 1.56, the standard deviation is σ(X ) = 0.753. µ3 = 3 P [xi − E (X )]3 p(xi ) = (1 − 1.56)3 · 0.6+ i=1 +(2 − 1.56)3 · 0.24 + (3 − 1.56)3 · 0.16 = 0.393 Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example First of all we have to calculate 3rd and 4th central moment. The mean of the given random variable is E (X ) = 1.56, the standard deviation is σ(X ) = 0.753. µ3 = 3 P [xi − E (X )]3 p(xi ) = (1 − 1.56)3 · 0.6+ i=1 +(2 − 1.56)3 · 0.24 + (3 − 1.56)3 · 0.16 = 0.393 µ4 = 3 P [xi − E (X )]4 p(xi ) = (1 − 1.56)4 · 0.6+ i=1 +(2 − 1.56)4 · 0.24 + (3 − 1.56)4 · 0.16 = 0.756 Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example The skewness is equal to α3 = µ3 = 0.922, σ3 Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example The skewness is equal to α3 = the kurtosis is α4 = µ3 = 0.922, σ3 µ4 − 3 = −0.644. σ4 Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example Calculate the skewness and the kurtosis of the random X with the probability density function 12x 2 (1 − x) 0 < x < 1, f (x) = 0 otherwise. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example The mean of the given random variable is E (X ) = 3/5. the standard deviation is 1/5. Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example The mean of the given random variable is E (X ) = 3/5. the standard deviation is 1/5. First of all we calculate 3rd and 4th central moment. Z1 µ3 = [x − 0.6]3 12x 2 (1 − x)dx = · · · = − 0 Jiřı́ Neubauer Random Variables 2 = −0.00229, 875 Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example The mean of the given random variable is E (X ) = 3/5. the standard deviation is 1/5. First of all we calculate 3rd and 4th central moment. Z1 µ3 = [x − 0.6]3 12x 2 (1 − x)dx = · · · = − 2 = −0.00229, 875 0 Z1 µ4 = [x − 0.6]4 12x 2 (1 − x)dx = · · · = 33 = 0.00377. 8750 0 Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example The skewness is equal to α3 = 2 µ3 = − = −0.286, 3 σ 7 Jiřı́ Neubauer Random Variables Random Variable Discrete Random Variable Continuous Random Variable Measures of Location Measures of Dispersion Measures of Concentration Example The skewness is equal to α3 = 2 µ3 = − = −0.286, 3 σ 7 the kurtosis is α4 = µ4 9 − 3 = − = −0.643. 4 σ 14 Jiřı́ Neubauer Random Variables
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