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COM2023 Mathematics Methods for Computing II Lecture 7& 8 Gianne Derks Department of Mathematics (36AA04) http://www.maths.surrey.ac.uk/Modules/COM2023 Autumn 2010 Use channel 04 on your EVS handset Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Discrete random variables Discrete random variable: example. . . Probability Mass Function . . . . . . . . Probability Mass Function: Examples . Biased die example revisited . . . . . . . The distribution function . . . . . . . . . Distribution function: Example 1 . . . . Distribution function: Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4 5 6 7 8 9 10 Continuous random variables 11 Probability density function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Probability density function: sketch 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Probability density function: sketch 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1 Overview ● Introduction to distributions: ◆ Discrete case ● ● probability mass function distribution function ◆ Continuous case (introduction) ● probability density function Discrete random variables (§4.1) Discrete random variable: example A biased die has the following probabilities for its outcomes: x 1 2 3 4 5 6 P (X = x) 2 12 2 12 1 12 5 12 1 12 1 12 The outcome is called a random variable, denoted by X, its value is usually denoted by x, where in this example x takes the values 1, . . . , 6, hence discrete values. An overview of the probabilities is given by the probability mass function, this is the full table above. Recall that the sum of all probabilities must be 1 and that all probabilities have to be between 0 and 1. 2 Probability Mass Function In general: The discrete random variable X takes discrete values x1 , x2 , . . . with probabilities p(x1 ), p(x2 ), . . ., i.e. p(xi ) = P ({X = xi }) for i = 1, 2, . . . Then p(x) is called the probability mass function or pmf of X. Two Properties of the pmf ● 0 ≤ p(x) ≤ 1; ● p(x1 ) + p(x2 ) + p(x3 ) + . . . = 1. Example A fair die is thrown and X is the outcome. What is the pmf of X? 1 2 3 4 5 6 x p(x) 1 6 1 6 1 6 1 6 1 6 1 6 Probability Mass Function: Examples ● ● ● What is the probability mass function (pmf) of the number of heads in three tosses of a fair coin? x 0 1 2 3 p(x) 1 8 3 8 3 8 1 8 A coin is tossed repeatedly and X is the number of tosses needed until the first ‘head’ is obtained. What is the pmf of X? x 1 2 3 4 5 ... p(x) 1 2 1 4 1 8 1 16 1 32 ... A pmf is specified by n ... n 1 ... 2 p(x) = cx, for x = 1, 2, 3, 4, where c is a constant. Find the value of c. Answer: Use that p(x1 ) + . . . + p(x4 ) = 1, then we get c = 3 1 10 . Biased die example revisited The random variable X associated with a biased die has the following pmf: x 1 2 3 4 5 6 p(x) 2 12 2 12 1 12 5 12 1 12 1 12 Now consider the events Ey to be the events that the random variable X takes a value less or equal to y, for y = 1, . . . , 6. What is the probability P (Ey ) for y = 1, . . . , 6? We get the following table y 1 2 3 4 5 6 P (Ey ) 2 12 4 12 5 12 10 12 11 12 12 12 Note that the values are between 0 and 1 and that they are increasing. The distribution function Given a discrete random variable X. To find the distribution function, consider the event Ey = {X = xi | xi ≤ y}, (i.e., the random variable X takes a value less or equal to y.) The cumulative probability distribution function, or for short distribution function is F (y) = P (Ey ) = P (X ≤ y) = X p(xj ) = p(x1 ) + . . . + p(xj0 ), xj ≤y where the index j0 is such that xj0 ≤ y and xj0 +1 > y. Properties of the distribution function ● 0 ≤ F (y) ≤ 1; ● F (−∞) = 0 and F (+∞) = 1; ● the graph of F is increasing, i.e., if y1 < y2 then F (y1 ) ≤ F (y2 ). 4 Distribution function: Example 1 Find the probability mass function and distribution function of a fair four-sided die. The pmf is: x 1 2 3 4 p(x) 1 4 1 4 1 4 1 4 The distribution function is y 1 2 3 4 F (y) 1 4 1 2 3 4 1 What are F (0.5), F (2.5), F (3.7), and F (5)? Answer: F (0.5) = 0, F (2.5) = F (2) = 12 , F (3.7) = F (3) = 43 , F (5) = 1. Distribution function: Example 2 The probability mass function (pmf) of a discrete random variable X is specified by p(x) = cx for x = 31 , 23 , 1, where c is a constant. Find the value of c and obtain the distribution function of X. ● Using that p(x1 ) + p(x2 ) + p(x3 ) = 1, we get c = 12 . ● The distribution function is y F (y) 1 3 1 6 Note: F (−1) = 0, F ( 73 ) = 16 , F ( 54 ) = 12 , F (2) = 1, etc. 5 2 3 1 2 1 1 Continuous random variables (§4.2) Probability density function A continuous random variable X takes values x in a range −∞ < x < ∞. We can only ask for the probability that x is in some interval, hence for P (a ≤ X ≤ b). The probability density function or pdf is a function f (x) such that ● the function is positive and all values are less or equal to 1: 0 ≤ f (x) ≤ 1 for all x; ● the total area under the function is 1:Z ∞ f (x)dx = 1; −∞ ● for any a ≤ b, the probability that X takes values between a and b is the area under the graph of f between a and b: Z b P (a ≤ X ≤ b) = f (x) dx. a Probability density function: sketch 1 The probability that X takes a value between 3.5 and 5.5 is the shaded area. 6 Probability density function: sketch 2 The probability that X takes a value less or equal 6 is the shaded area. 7