7. Suppose X1,X2, ,X, are i.İ.d. from chisquare distribution with 3 degrees of freedom. Determine the...
Problem 3: Suppose X1, X2, is a sequence of i.i.d. random variables having the Poisson distribution with mean λ. Let A,-X, (a) Is λη an unbiased estimator of λ? Explain your answer. (b) Is in a consistent estimator of A? Explain your answer 72
7. Let X1, X2,.. be i.i.d. random variables, and let T(t)minn: X > t, t20. (a) Determine the distribution of T(t) (b) Show that, if p= P(X1> t)0 astoo, then pT(t)Exp(1) as to
7. Let X1, X2,.. be i.i.d. random variables, and let T(t)minn: X > t, t20. (a) Determine the distribution of T(t) (b) Show that, if p= P(X1> t)0 astoo, then pT(t)Exp(1) as to
1. Let X1, X2 be i.i.d with this distribution: f(x) = 3e cx, x ≥ 0 a. Find the value of c b. Recognize this as a famous distribution that we’ve learned in class. Using your knowledge of this distribution, find the t such that P(X1 > t) = 0.98. c. Let M = max(X1, X2). Find P(M < 10)
2. (10pts) Let X1, X2, , X20 be an i.i.d. sannple from a Normal distribution with mean μ and variance σ2, ie., Xi, X2, . . . , X20 ~ N(μ, σ2), with the density function Also let 20 20 10 20 -20 19 i-1 ー1 (a) (5pts) What are the distributions of Xi - X2 and (X1 - X2)2 respectively? Why? (b) (5pts) what are the distributions of Y20( and 201 ? Why? (X-μ)2
2. (10pts) Let X1, X2,...
Suppose X1, X2, . . . , Xn are i.i.d. Exp(µ) with the density f(x) = for x>0 (a) Use method of moments to find estimators for µ and µ^2 . (b) What is the log likelihood as a function of µ after observing X1 = x1, . . . , Xn = xn? (c) Find the MLEs for µ and µ^2 . Are they the same as those you find in part (a)? (d) According to the Central Limit...
3. Suppose that X1, X2, X3 be i.i.d. random variables with P(Xi 0) 2/5 and P(X 1) 3/5. Find the MGFof X, + X2 + X 3.
3. Suppose that X1, X2, X3 be i.i.d. random variables with P(Xi 0) 2/5 and P(X 1) 3/5. Find the MGFof X, + X2 + X 3.
Suppose X1,X2, .. ,X, is a random sample from a standard normal distribution and let Z be another standard normal variable that is independent of X1, X2, .., X,. 9 9 9 Determine the distribution of each of the variables X, U and V. (a) (b) Determine the distribution of the variable 3Z NU Determine the distribution of the variable W- (c) (d) Determine the distribution of the variable R -4y (where Y is the variable from (C)
7. (15 pts) Suppose X1, X2, ..., X, is a random sample from an exponential distribution with parameter 2. (Remember f(x;2) = ne-^x is the pdf for the exponential dista.) a) Find the likelihood function, L(X1, X2, Xn). b) Find the log-likelihood function, I = log L. c) Find d //d, set the result = 0 and solve for 2.
Let X1, ..., Xn be i.i.d. [Recall that i.i.d. stands for independent and identically distributed.] Since X1, ..., Xn all have the same distribution, they have the same expected value and variance. Let E(X1) = µ and V ar(X1) = σ 2 . Find the following in terms of µ and σ 2 . (a) E(X2 1 ). Note this is not µ 2 ! (b) E( Pn i=1 X2 i /n). (c) Now, define W by W = 1...
Suppose that X1,X2,... is a sequence of i.i.d. r.v.s having uniform distribution on [0,1]. Define Yn=n(1−max1≤i≤nXi) for n=1,2,.... Prove that Yn converges in distribution to an exponential distribution.