Lecture Notes Chapter 7.pdf - CHAPTER 7 MOMENTS AND MOMENT-GENERATING FUNCTIONS Moments The rth moment about the origin of the random variable X is

# Lecture Notes Chapter 7.pdf - CHAPTER 7 MOMENTS AND...

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CHAPTER 7 MOMENTS AND MOMENT-GENERATING FUNCTIONS Moments The r th moment about the origin of the random variable X is given by μ 0 r = E ( X r ) = x x r f ( x ) if X is discrete Z - x r f ( x ) dx if X is continuous N OTE . The first moment μ 0 = E ( X ) is simply the mean number or the mathematical expectation of X . The second moment μ 0 2 = E ( X 2 ) is usually calculated in terms of Var ( X )+[ E ( X )] 2 . E XAMPLE 7.1. Find each of the following moments of the random variable X . (a) The first moment of X , where X has the PDF f ( x ) = ( 3 x 2 0 < x < 1 0 otherwise . (b) The second moment of X , where X is the binomial random variable with parameters n = 10 and p = 0 . 25. (c) The third moment of X , where X is the exponential random variable with parameter β = 2. (d) The forth moment of X , where X is the gamma random variable with parameters α = 2 and β = 1. (e) The fifth moment of X , where X is the normal random variable with mean 0 and variance 4. (f) The ( 2 n - 1 ) th moment of Z , where Z is the standard normal random variable and n Z + .

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