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Unformatted text preview: M ± ሺtሻ, െh ൏ ݐ ൏ 0 . (3) Prove the statement: if VarሺXሻ ൌ 0 , then PሺX ൌ µሻ ൌ 1 . 4. Let ሺX ଵ , X ଶ ሻ be a random vector such that the variance of X ଶ is finite. Prove that (1) EሾEሺX ଶ |X ଵ ሻሿ ൌ EሺX ଶ ሻ (2) VሾEሺX ଶ |X ଵ ሻሿ VሺX ଶ ሻ . 5. Let X and Y be independent with moment generating functions M ± ሺtሻ ൌ e ୲ మ and M ± ሺtሻ ൌ e ଶ୲ మ ା୲ . (1) Find the mean and variance of X . (2) Find the moment generating function of X 2Y 3 . 6. Let T ൌ ∑ X ୧ N ୧ୀଵ , where ² is a random variable with a finite expectation and the X ୧ are the random variables that are independent of ² and have finite mean EሺXሻ . Show that (1) EሺTሻ ൌ Eሺ²ሻEሺXሻ . (2) VሺTሻ ൌ ሾEሺXሻሿ ଶ Vሺ²ሻ Eሺ²ሻVሺXሻ ....
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This note was uploaded on 05/09/2011 for the course DIF 1486 taught by Professor Yow-jenjou during the Spring '11 term at National Chiao Tung University.
- Spring '11