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Mathematical Statistics HW#4-992

Mathematical Statistics HW#4-992 - (2 Suppose that the...

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Mathematical Statistics-992 Homework #4 Due 4/7/2011 1. From textbook: Sec. 4.2 3, 8, 15 Sec. 4.5 22, 30, 34, 37, 40 Sec. 4.8 43, 46, 47, 53, 57 2. 課堂上曾用一階動差、二階動差求機率的上界(或下界),也可以用 mgf 求機 率的上界(或下界)。 Let M X ሺtሻ be the moment generating function of random variable X . (1) Use Markov inequality to show that PሺX ൒ kሻ ൑ e ି୲୩ M X ሺtሻ holds for all positive t and PሺX ൑ kሻ ൑ e ି୲୩ M X ሺtሻ holds for all negative t . ( 因對所有 t 都成立,可以挑選合適的 t 得到理想的結果
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Unformatted text preview: ) (2) Suppose that the moment generating function is M X ሺtሻ ൌ e ౪ మ మ . Show that Pሺx ൒ kሻ ൑ e ି ౡ మ మ . (Hint: use (a) and let t ൌ k ) 3. Let ሺx, Yሻ be a random vector such that the variance of Y is finite. Show that (1) EሾEሺY|xሻሿ ൌ EሺYሻ . (2) VሾEሺY|xሻሿ ൅ EሾVሺY|xሻሿ ൌ VሺYሻ , and therefore VሾEሺY|xሻሿ ൑ VሺYሻ . (3) With reference to exercise 3.74 and 4.58 to verify parts (1) and (2)....
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