07 Continuous probability distributions

# 07 Continuous probability distributions - University of...

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University of California, Los Angeles Department of Statistics Statistics 100A Instructor: Nicolas Christou Continuous probability distributions Let X be a continuous random variable, -∞ < X < f ( x ) is the so called probability density function (pdf) if Z -∞ f ( x ) dx = 1 Area under the pdf is equal to 1. How do we compute probabilities? Let X be a continuous r.v. with pdf f ( x ). Then P ( X > a ) = Z a f ( x ) dx P ( X < a ) = Z a -∞ f ( x ) dx P ( a < X < b ) = Z b a f ( x ) dx Note that in continuous r.v. the following is true: P ( X a ) = P ( X > a ) This is NOT true for discrete r.v. 1

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Cumulative distribution function (cdf): F ( x ) = P ( X x ) = Z x -∞ f ( x ) dx Therefore f ( x ) = F ( x ) 0 Compute probabilities using cdf: P ( a < X < b ) = P ( X b ) - P ( X a ) = F ( b ) - F ( a ) Example: Let the lifetime X of an electronic component in months be a continuous r.v. with f ( x ) = 10 x 2 ,x > 10. a. Find P ( X > 20). b. Find the cdf. c. Use the cdf to compute P ( X > 20). d. Find the 75 th percentile of the distribution of X . e. Compute the probability that among 6 such electronic com- ponents, at least two will survive more than 15 months. 2
Mean of a continuous r.v. μ = E ( X ) = Z -∞ xf ( x ) dx Mean of a function of a continuous r.v. E [ g ( X )] = Z -∞ g ( x ) f ( x ) dx Variance of continuous r.v. σ 2 = E ( X - μ ) 2 = Z -∞ ( x - μ ) 2 f ( x ) dx Or σ 2 = Z -∞ x 2 f ( x ) dx - [ E ( X )] 2 Some properties: Let a,b constants and X , Y r.v. E ( X + a ) = a + E ( X ) E ( X + Y ) = E ( X ) + E ( Y ) var ( X + a ) = var ( X ) var ( aX + b ) = a 2 var ( X ) If X,Y are independent then var ( X + Y ) = var ( X ) + var ( Y ) 3

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Example: Let X be a continuous r.v. with f ( x ) = ax + bx 2 , and 0 < x < 1. a. If E ( X ) = 0 . 6 ﬁnd a,b . b. Find var ( X ). 4
Uniform probability distribution: A continuous r.v. X follows the uniform probability distribution on the interval a,b if its pdf function is given by f ( x ) = 1 b - a , a x b Find cdf of the uniform distribution. Find the mean of the uniform distribution. Find the variance of the uniform distribution. 5

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The gamma distribution The gamma distribution is useful in modeling skewed distribu- tions for variables that are not negative. A random variable X is said to have a gamma distribution with parameters α,β if its probability density function is given by f ( x ) = x α - 1 e - x β β α Γ( α ) , α,β > 0 ,x 0 . E ( X ) = αβ and σ 2 = αβ 2 . A brief note on the gamma function: The quantity Γ( α ) is known as the gamma function and it is equal to: Γ( α ) = Z 0 x α - 1 e - x dx. If α = 1, Γ(1) = R 0 e - x dx = 1. With integration by parts we get Γ( α + 1) = α Γ( α ) as follows: Γ( α + 1) = Z 0 x α e - x dx = Let, v = x α dv dx = αx α - 1 du dx = e - x u = - e - x Therefore, Γ( α + 1) = Z 0 x α e - x dx = - e - x x α | 0 - Z 0 - e - x αx α - 1 dx = α Z 0 x α - 1 e - x dx. Or, Γ( α + 1) = α Γ( a ). 6
Similarly, using integration by parts it can be shown that, Γ( α + 2) = ( α + 1)Γ( α + 1) = ( α + 1) α Γ( α ), and, Γ( α + 3) = ( α + 2)( α + 1) α Γ( α ).

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07 Continuous probability distributions - University of...

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