Lecture9.pdf - ESE 520 Probability and Stochastic Processes Lecture 9 ”Expectation(continued Covariance and correlation.” Some more examples Example

# Lecture9.pdf - ESE 520 Probability and Stochastic Processes...

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ESE 520 Probability and Stochastic Processes Lecture 9 ”Expectation (continued). Covariance and correlation.” Some more examples. Example 1. Let X be a random variable with N ( μ, σ 2 ) probability dis- tribution. Then V ar ( X ) = σ 2 . Indeed: V ar ( X ) = Z -∞ ( x - μ ) 2 1 2 πσ e - 1 2 ( x - μ σ ) 2 dx = [ z = x - μ σ ] = Z -∞ σ 2 z 2 1 2 π e - 1 2 z 2 dz. More generally, for m = 1 , 2 , ... , we have E ( X - μ ) 2 m = .... = Z -∞ σ 2 m z 2 m 1 2 π e - 1 2 z 2 dz = σ 2 m Z -∞ z 2 m - 1 z 2 π e - 1 2 z 2 dz = - we use integration by parts here - σ 2 m [ z 2 m - 1 ( - 1 2 π e - 1 2 z 2 ) | -∞ + Z -∞ (2 m - 1) z 2 m - 2 1 2 π e - 1 2 z 2 dz ] σ 2 m [0 + Z -∞ (2 m - 1) z 2 m - 2 1 2 π e - 1 2 z 2 dz ] . As the result, we obtain the following formula: E ( X - μ ) 2 m = (2 m - 1)(2 m - 3) .... 3 × 1 σ 2 m = (2 m - 1)!! σ 2 m In particular, it follows then that V ar ( X ) = E ( X - μ ) 2 = σ 2 . Also, if Z has N (0 , 1) distribution, then E ( Z 2 m ) = (2 m - 1)!! 1

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Example 2. (”maximum and minimum statistics”) Let X i , i = 1 , 2 , ..., n be a sequence of independent and identically distributed (i.i.d.) random variables with the cdf F ( x ) (respectively with the pdf f ( x )). Define Y := max { X 1 , X 2 , ..., X n } , and Z := min { X 1 , X 2 , ..., X n } . Find F Y ( y ) and F Z ( z ) (respectively f Y ( y ) and f Z ( z )).
• Spring '14
• Arthur
• Probability distribution, Probability theory, 2m, dθ, 1 2m

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