02. Some review material from Statistics 100A

02 Some review - University of California Los Angeles Department of Statistics Statistics 100B Instructor Nicolas Christou Continuous probability

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Unformatted text preview: University of California, Los Angeles Department of Statistics Statistics 100B 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 • Cumulative distribution function (cdf): F ( x ) = P ( X ≤ x ) = Z x-∞ f ( x ) dx • Therefore f ( x ) = F ( x ) • 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 • 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 • 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 > ,x ≥ . 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 ∞ x α-1 e-x dx....
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This note was uploaded on 01/14/2011 for the course STATS 100B 100B taught by Professor Christou during the Winter '10 term at UCLA.

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02 Some review - University of California Los Angeles Department of Statistics Statistics 100B Instructor Nicolas Christou Continuous probability

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