This preview shows pages 1–7. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 components, 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 ( Xμ ) 2 = Z ∞∞ ( xμ ) 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 ba , 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 distributions 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 α1 ex 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 α1 ex dx....
View
Full
Document
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.
 Winter '10
 Christou

Click to edit the document details