Unformatted text preview: < γ ε := Z A ε p ( x ) dx = P { X 1 ∈ A ε } → , as ε → . Deﬁne, for each N , ˆ γ ε N = 1 N ∑ N i =1 1 { X i ∈ A ε } (1 B is the indicator function of set B ). (i) Find E (ˆ γ ε N ) and Var(ˆ γ ε N ); (ii) Suppose N is large enough. For each ε > 0 and N > 0, using the attached Normal Table to determine z ε N such that the probability that ˆ γ ε N ∈ ( γ εz ε N ,γ ε + z ε N ) is (approximately) 0 . 99. (iii) Show that lim ε → z ε N ˆ γ ε N = ∞ , a.s., no matter how large N is. 5. Let ξ be a random variable with positive density function p ( x ). Suppose that p is twice diﬀerentiable and satisﬁes the identity p ( x + y ) p ( x + y ) + p ( xy ) p ( xy ) = 2 p ( x ) p ( x ) , ∀ x,y ∈ (∞ , ∞ ) . Show that ξ must be a normal random variable. 1...
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 Spring '09
 Math, Normal Distribution, Variance, Probability theory, probability density function, Randomness

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