CS代考程序代写 University of California, Los Angeles Department of Statistics

University of California, Los Angeles Department of Statistics
Instructor: Nicolas Christou
Quiz 8
Answer the following questions:
a. Suppose X1, . . . , Xn are i.i.d. Poisson(λ). Let S = 􏰃ni=1 Xi. Show that (X1 = x1, . . . , Xn = xn) conditioned
on S = N follows the multinomial distribution with parameters S and ( 1 , . . . , 1 ). ′􏰃nn′
Hint 1: Find the joint pmf of Xis. Given that S = N, i Xi = N. Use this result in the joint pmf of the Xis.
Hint 2: Continue by expressing the conditional pmf as the ratio of the joint and the marginal pmf’s.
Note on the multinomial probability distribution:
A sequence of n independent experiments is performed and each experiment can result in one of r possible
outcomes with probabilities p1, p2, . . . , pr with 􏰃ri=1 pi = 1. Let Xi be the number of the n experiments that
resultinoutcomei,i=1,2,…,r. Then,P(X1 =x1,X2 =x2,…,Xr =xr)= n! px1px2 ···pxr. n1!n2!···nr! 1 2 r
b. Refer to question (a). Suppose we know that E[X(X −1)] = λ2 and var[X(X −1)] = 2λ2 +4λ3. Please explain how you would verify these two results. Let T1 = 1 􏰃n Xi(Xi − 1). Show that T1 is unbiased estimator for
n i=1
λ2 and compute its variance.
c. Refer to question (a). Let T2 = E[T1|S]. Show that T2 = S(S−1) . Note that E[Xi|S] = S 1 and var(Xi|S) =
S 1 􏰁1 − 1 􏰂. Is T2 unbiased estimator of λ2? Find var(T2). nn
d. Let X1,…,Xn be i.i.d. random variables with f(x) = αxα−1 ,α > 0,θ > 0,0 ≤ x ≤ θ. Assume that θ is θα
known. Use the factorization theorem to show that Πni=1Xi is a sufficient statistics for α.
Statistics 100B
n2 n

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