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w5_ExtraNote_Cov - 2. Mean of a Random Variable 3...

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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: x-1 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 non-random 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....
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This note was uploaded on 01/01/2010 for the course ISOM ISOM111 taught by Professor Anthonychan during the Spring '09 term at HKUST.

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w5_ExtraNote_Cov - 2. Mean of a Random Variable 3...

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