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Continuous Probability Distributions Previously we have been discussing the probabilities associated with discrete random variables – where a rv can only assume a select number of values. Now, we’ll extend this concept to continuous random variables – where a rv can assume any value within a specified range. Definition Continuous Random Variable: a rv with a set of possible values in an entire interval of numbers, A and B, where A ≤ B. Examples: height of tree, time waiting for service, age, etc. PDF of Continuous Rv The probability distribution of probability density function (pdf) of X is a function f(x) such that for any two numbers a and b, a ≤ b: b P ( a ≤ X ≤ b ) = ∫ f ( x ) dx a Conditions f ( x) ≥ 0 , ∀ x +∞ ∫−∞ f ( x ) dx = area under the entire curve f ( x ) = 1 The probability that X takes on a value in the interval [a,b] is the area under the graph of the density function: f(x) a b x Figure 1. Continuous Distribution CDF for Continuous RV The cumulative distribution function (cdf), F(x), is defined for x as the area under the curve to the left of x: x F ( X ) = P ( X ≤ x) = ∫ f ( y ) dy −∞ Expected Value and Variance of Continuous Distributions The mean of a continuous random variable X with a pdf f(x) is: µx = E ( x ) = ∫ ∞ −∞ xf ( x )dx If X is a continuous rv with pdf f(x) and h(x) is any function of X, then: E ⎡⎣h ( x ) ⎤⎦ = ∫ ∞ −∞ h ( x ) f ( x )dx The variance of a continuous rv X with a pdf f(x) and mean value µ is: ∞ 2 2⎤ 2 ⎡ σx =V x = x − µ f x dx = E X − µ ⎢⎣ ⎥⎦ −∞ ( ) ∫ ( ) ( ) ( ) − ⎡⎣E ( X )⎤⎦ 2 =E X 2 ( ) The probability of X equally any number c, P(X=c) = 0 and for any two numbers a and b with a < b: P ( a ≤ X ≤ b) = P ( a < X ≤ b) = P ( a ≤ X < b) = P ( a < X < b) Uniform Distribution A distribution that has constant probability. A continuous rv is said to have uniform distribution on the interval [A, B] if the pdf of X is: ⎧ 1 A≤ x≤ B ⎪ f ( x; A, B ) = ⎨ B − A ⎪⎩ 0 otherwise Uniform Example Suppose every 10 minutes a bus arrives at your stop. Due to the variation in the time you leave your house, you don’t always get to the bus stop at the same time. Therefore, the time spent waiting for the bus, X, is a continuous random variable. What is the probability that you wait between 1 and 3 minutes? Graphically 1 10 0 10 0 1 3 10 ⎧ 1 ⎪ f ( x;0,10 ) = ⎨10 − 0 ⎪⎩ 0 0 ≤ x ≤ 10 otherwise 1 P (1 ≤ X ≤ 3 ) = ∫ f ( x )dx = ∫ dx 1 1 10 3 3 x =3 x 3 1 2 = = − = 10 x =1 10 10 10 1 = = 0.20 5 Normal Distribution The most well known distribution is the normal distribution. Also referred to as the Gaussian distribution, after Carl Friedrich Gauss, or the “bell curve”, after the shape of the distribution: PDF and CDF of Normal Distribution The normal density is given by: 1 −( x − µ ) f ( x) = e 2πσ 2 2σ 2 , −∞ < x < ∞ where, π and e are the constants ~ 3.14 and 2.72; and µ and σ are the mean and standard deviation, respectively Characteristics of the Normal Distribution the distribution is symmetric about the mean the mean = median = mode large standard deviations result in flatter curves smaller standard deviations result in taller curves µ ± σ ~68% of the distribution area µ ± 2σ ~95% of the distribution area µ ± 3σ ~99.7% of the distribution area the mean and standard deviation completely define the distribution: X ~ N(µ, σ) Standard Normal ~N(0,1) The standard normal distribution, is a normal distribution with a mean = 0 and standard deviation = 1: 1 −( z ) f ( z) = e 2π 2 x − µ) ( z= σ 2 , −∞ < z < ∞ Graphically 0.08 0.05 0.03 -3.0 -1.0 .0 1.0 2.0 3.0 4.0 Probability 0.10 CDF Normal Cum Prob 0.9 0.7 0.5 0.3 0.1 -4 -3 -2 -1 0 1 Column 1 2 3 4 Of note: Data generated from a N(µ, σ) will not necessarily have a mean = µ and a standard deviation = σ Though the data was generated using a normal distribution with mean = 0 and standard deviation = 1, the estimated mean = -0.02 and the estimated standard deviation = 0.999. Probability of X The entire area under the curve of a standard normal distribution is 1 unit. Therefore, the relationship between the normal distribution and the standard normal distribution can be used to determine the probability of z being ≤, < , ≥, >, or = to any zO. Other Distributions There are numerous other continuous distributions: – Gamma – Exponential – Chi-Squared