This preview shows pages 1–12. 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 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 DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Continuous Uniform Distribution; Exponential Distribution Andrew Liu October 5, 2011 Andrew Liu Textbook section: 45, 48 Review; Mean and Variance of Continuous Random Variables Mean and variance of functions of random variables Suppose that Y = h ( X ). (For example, Y = 3 X 2 4.) Let f ( x ) denote the density function of X . Andrew Liu Textbook section: 45, 48 Review; Mean and Variance of Continuous Random Variables Mean and variance of functions of random variables Suppose that Y = h ( X ). (For example, Y = 3 X 2 4.) Let f ( x ) denote the density function of X . Discrete random variable Continuous random variable Mean E ( Y ) = X i h ( x i ) P ( X = x i ) E ( Y ) = Z  h ( x ) f ( x ) dx Var V ( Y ) = X i h ( x i ) 2 P ( X = x i ) [ E ( Y )] 2 V ( Y ) = Z  h 2 ( x ) f ( x ) dx [ E ( Y )] 2 Andrew Liu Textbook section: 45, 48 Mean and Variance Examples E.g.1 Suppose f ( x ) = 0 . 05 for 0 x 20. Determine the mean and variance of Y = X 2 . Andrew Liu Textbook section: 45, 48 Mean and Variance Examples E.g.1 Suppose f ( x ) = 0 . 05 for 0 x 20. Determine the mean and variance of Y = X 2 . Solution: E ( Y ) = Andrew Liu Textbook section: 45, 48 Mean and Variance Examples E.g.1 Suppose f ( x ) = 0 . 05 for 0 x 20. Determine the mean and variance of Y = X 2 . Solution: E ( Y ) = Z  h ( x ) f ( x ) dx = Andrew Liu Textbook section: 45, 48 Mean and Variance Examples E.g.1 Suppose f ( x ) = 0 . 05 for 0 x 20. Determine the mean and variance of Y = X 2 . Solution: E ( Y ) = Z  h ( x ) f ( x ) dx = Z 20 ( x 2 * . 05) dx = 0 . 05 * x 3 3 20 . Andrew Liu Textbook section: 45, 48 Mean and Variance Examples E.g.1 Suppose f ( x ) = 0 . 05 for 0 x 20. Determine the mean and variance of Y = X 2 . Solution: E ( Y ) = Z  h ( x ) f ( x ) dx = Z 20 ( x 2 * . 05) dx = 0 . 05 * x 3 3 20 . V ( Y ) = Z  h ( x ) 2 f ( x ) [ E ( Y )] 2 = Z 20 ( x 4 * . 05) dx [ E ( Y )] 2 = . 05 * x 5 5 20 [ E ( Y )] 2 . Andrew Liu Textbook section: 45, 48 Continuous Uniform Distribution The Simplest Continuous Distribution Uniform Distribution: Density Function Uniform Distribution: Cumulative Distribution Function Andrew Liu Textbook section: 45, 48 Continuous Uniform Distribution Density Function, Mean and Variance Density function A continuous random variable X is called a continuous uniform random variable if it has the the following probability density function. f ( x ) = 1 ( b a ) , a x b . Andrew Liu Textbook section: 45, 48 Continuous Uniform Distribution Density Function, Mean and Variance Density function A continuous random variable X is called a continuous uniform random variable if it has the the following probability density function....
View
Full
Document
 Winter '08
 Xangi

Click to edit the document details