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 Document
Unformatted text preview: WEEK 4, Monday , Sept 20 Today: Cover Sections 31 through 33 . Discuss discrete and continuous random variables and their distributions as probability models. Outcomes are on the real number line, often the events of interest are intervals. Discrete – Probability mass function . Properties are p ( x ) where p ( x ) ≥ 0 (31) and 1 ) ( = ∑ x all x p . (32) Cumulative distribution function . ∑ ≤ = ≤ = x i i p x X P x F ) ( ) ( ) ( (33) Expectation . ) ( ) ( all ∑ = = x x xp X E μ (34) is called the mean of X or simply the expected value of X . For any function h and Y = h ( X ), Y is a random variable with expectation ∑ = = x all x p x h X h E Y E ) ( ) ( )) ( ( ) ( . (35) Variance and Standard Deviation . )] ( [ ) ( ] ) [( ) ( 2 2 2 2 2 2 μ μ μ σ ∑ = = = = x p x X E X E X V (37) and (38) ) ( ) ( X V X SD = = σ (39) 1 Please note . The units of measure for E ( X ) and SD ( X ) are the same as the units of measure for X . If X is measured in feet , then E ( X ) and SD ( X ) are in units of feet . If X is in units of ($1,000), then E ( X ) and SD ( X ) are in units of ($1,000). Example 1. Total on the roll of a pair of fair dice . x 2 3 4 5 6 7 8 9 10 11 12 p ( x ) 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36 Probability Mass Function p and Cumulative Distribution Function F x p (x) F ( x ) 2 1/36 1/36 3 2/36 3/36 4 3/36 6/36 5 4/36 10/36 6 5/36 15/36 7 6/36 21/36 8 5/36 26/36 9 4/36 30/36 10 3/36 33/36 11 2/36 35/36 12 1/36 36/36 2 Prob Dist of X = Total on Two Dice 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 2 3 4 5 6 7 8 9 10 11 12 x P(x) Prob Dist of X = Total on Two Dice 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 2 3 4 5 6 7 8 9 10 11 12 x P(x) Way to Calculate Mean, Variance, and Standard Deviation of a Discrete Random Variable 3 7 ) ( ) ( = = = ∑ x xp X E μ 8333 . 5 36 / 210...
View
Full Document
 Spring '08
 Anderson
 Standard Deviation, Probability distribution

Click to edit the document details