X, be equally likely to be any of the n! permutations of 72. Let X1, ....
Let XXn Be equally likely to be any of the n! permutations of (1, 2,., ).,Argue that 7 jx, 2 m(n + 1)2 Let XXn Be equally likely to be any of the n! permutations of (1, 2,., ).,Argue that 7 jx, 2 m(n + 1)2
We roll two dice. Assume all 36 possibilities are equally likely. Let X1 and X2 be the result of the first and second die, respectively. Let S be the sum of the scores, that is S = X1 + X2. Calculate the following: (a) P(S = k), for k 2,3,... 12. (b) P(X1 = 2 S = k) for k = 2,3,... 12. (c) P(X1 = 6|S = k) for k = 2, 3, ... 12.
Additional Problem 5. Suppose that X is equally likely to take any of the values 1, 2, 3, and 4. Let Y sin Ysin㈜ oampate EV). ). Compute E(Y). sin
Let X be a continuous random variable with density, and let X1, X2 be two independent draws from X. Then, not usually is it the case that the random variable 2X is distributed as X1 + X2. However, the Cauchy density, which is given by the form , possesses the following property; X1+X2 has the same distribution as the random variable 2X. a. Let X be a binomial. Argue, based on the properties of the binomial distribution, that X1 +...
Let x[n] ←→ X(z) and let x[n] = αnu[n]. Let x1[n] ←→X(z3). Find x1[n] without computing X(z) or using properties of the z-transform.
id 3. Let X1, X2, ..., X 1 N(0,03) and Y1, 72,..., Ym N(02,03) independently. Denote 0 = (01,02,0z)" (a) Write down the expression for the log-likelihood function (0). (b) Find the maximum likelihood estimator Ô of . (You do not need to perform the second derivative test.) (e) Find the Fisher information. (d) Consider using –2 log(LR) to test H. : 0 = 0against H, : 01 + 02. Find the maximum likelihood estimator of O under H, and...
Any help would be appreciated! Problem 4 Let (X, Y)~ N and Z = X1(XY > 0}-X1(XY < 0} (1) Find the distribution of Z (2) Show that the joint distribution of Y and Z is not bivariate normal.
3. For n 2 2, let X have n-dimensional normal distribution MN(i, V). For any 1 3 m < n, let X1 denote the vector consisting of the last n - m coordinates of X < n, let 1 (a). Find the mean vector and the variance-covariance matrix of X1. (b). Show that Xi is a (n- m)-dimensional normal random vector.
(1) Let f : [n] [n] be a permutation. A fixed point of f is an element x e [n] such that f(x) - x. Now consider random permutations of [n] and let X be the random variable which represents the number of fixed points of a given permutation. (a) What is the probability that X 0? (b) What is the probability that X-n -2? (c) What is the probability that X-n-1? (d) What is the expectation of X? (Hint:...
Let f [n]n] be a permutation. A fixed point of f is an element x e [n] such that f(x)-x. Now consider random permutations of [n] and let X be the random variable which represents the number of fixed points of a given permutation. (a) What is the probability that X 0? (b) What is the probability that X 2? (c) What is the probability that X--1? (d) What is the expectation of X? (Hint: As usual, express X as...