This preview shows pages 1–4. 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 Document
Unformatted text preview: 2. Mean of a Random Variable 3 Definition: The mean (Expected Value or Expectation) of the discrete random variable X (with probability mass function p ( x )) is defined as E( X ) = μ = X { all x } xp ( x ) That is a measure of central tendency for the distribution of X . Example 2: Suppose the probability distribution of X is given as follows: x1 1 2 p ( x ) 0.1 0.4 0.2 0.3 Then, its mean is E( X ) = μ = 2 X x = 1 xp ( x ) = 1(0 . 1) + 0(0 . 4) + 1(0 . 2) + 2(0 . 3) = 0 . 7 Interpretation: If you collect a big sample from the distribution p ( x ) (with replace ment), then your bar chart of relative frequencies should be very similar to the one of p ( x ) and the sample mean should be very close to μ (”The Law of Large Numbers”). 2. Mean of a Random Variable 4 • Note that the concept of expectation can be generalized to: E[ g ( X )] = X { all x } g ( x ) p ( x ) where g is any function. • Using Example 2: E(3 X + 2) = 2 X x = 1 (3 x + 2) p ( x ) = 3 2 X x = 1 xp ( x ) + 2 2 X x = 1 p ( x ) = 3E( X ) + 2 = 3(0 . 7) + 2 = 2 . 3 and E( X 2 ) = 2 X x = 1 x 2 p ( x ) = ( 1) 2 (0 . 1) + 0 2 (0 . 4) + 1 2 (0 . 2) + 2 2 (0 . 3) = 1 . 5 • Following the idea of computing E(3 X + 2) on this page, E( aX + b ) = a E( X ) + b where a and b are nonrandom numbers. 3. Variance of a Random Variable 5 Definition: The variance of the discrete random variable X is defined as: Var( X ) = σ 2 = X { all x } ( x μ ) 2 p ( x ) = E( X μ ) 2 and its standard deviation is the positive square root of its variance....
View
Full
Document
This note was uploaded on 01/01/2010 for the course ISOM ISOM111 taught by Professor Anthonychan during the Spring '09 term at HKUST.
 Spring '09
 AnthonyChan

Click to edit the document details