Problem 3. Let X1, . . . , Xn be independent Poisson(λ) random variables. Find a BUE of e−2λ . Hint: Compute Pλ(X1 = 0, X2 = 0)
Problem 3. Let X1, . . . , Xn be independent Poisson(λ) random variables. Find a...
Let X1, X2, X3 be independent random variables with E(X1) = 1, E(X2) = 2 and E(X3) = 3. Let Y = 3X1 − 2X2 + X3. Find E(Y ), Var(Y ) in the following examples. X1, X2, X3 are Poisson. [Recall that the variance of Poisson(λ) is λ.] X1, X2, X3 are normal, with respective variances σ12 = 1, σ2 = 3, σ32 = 5. Find P(0 ≤ Y ≤ 5). [Recall that any linear combination of independent normal...
Let λ >0 and suppose that X1,X2,...,Xn be i.i.d. random variables with Xi∼Exp(λ). Find the PDF of X1+···+Xn. Use convolution formula and prove by induction
Let X1, . . . , Xn be independent Poisson(θ) random variables with parameter θ > 0. (1) Find the Bayes estimator of θ for a Gamma(α, β) prior. (2) Find the MSE of the Bayes estimator.
Let X1, . . . , Xn be independent and identically distributed random variables with Xi ∼ Poisson(λ) for i = 1, . . . , n. It could be useful to recall that Σ Xi ∼ Poisson(nλ). Find the form the rejection region for a most powerful test of H0 : λ = λ0 vs HA : λ = λA For λA < λ0
Let X1,... , Xn be independent random variables, each following an exponential distri- bution with rate λ. Let Y = min(X1, .. . , Xn). Find the cd.f. and pdf. of Y. HINT:
Let Xi and X2 be independent Poisson random variables with means λ! and λ2. (a) Find the distribution of x, + x, (b) Compute the conditional distribution of Xi given that X + X.-n
3. (25 pts.) Let X1, X2, X3 be independent random variables such that Xi~ Poisson (A), i 1,2,3. Let N = X1 + X2+X3. (a) What is the distribution of N? (b) Find the conditional distribution of (X1, X2, X3) | N. (c) Now let N, X1, X2, X3, be random variables such that N~ Poisson(A), (X1, X2, X3) | N Trinomial(N; pi,p2.ps) where pi+p2+p3 = 1. Find the unconditional distribution of (X1, X2, X3).
3. (25 pts.) Let X1,...
Let X1, X2, ..., Xr be independent exponential random variables with parameter λ. a. Find the moment-generating function of Y = X1 + X2 + ... + Xr. b. What is the distribution of the random variable Y?
Let Ņ, X1. X2, . . . random variables over a probability space It is assumed that N takes nonnegative inteqer values. Let Zmax [X1, -. .XN! and W-min\X1,... ,XN Find the distribution function of Z and W, if it suppose N, X1, X2, are independent random variables and X,, have the same distribution function, F, and a) N-1 is a geometric random variable with parameter p (P(N-k), (k 1,2,.)) b) V - 1 is a Poisson random variable with...
Let X1,X2 be two independent
exponential random variables with λ=1, compute the
P(X1+X2<t) using the joint density function. And let Z be gamma
random variable with parameters (2,1). Compute the probability that
P(Z < t). And what you can find by comparing P(X1+X2<t) and
P(Z < t)? And compare P(X1+X2+X3<t) Xi iid
(independent and identically distributed) ~Exp(1) and P(Z < t)
Z~Gamma(3,1) (You don’t have to compute)
(Hint: You can use the fact that Γ(2)=1,
Γ(3)=2)
Problem 2[10 points] Let...