If c is any constant and Y is a random variable such that E(Y) exists, show...
Problem 4 Let X be the following discrete random variable: Let Y = X2. Show that cov(X·Y) = 0, but X and Y are not independent random variable.
Q. 5. Let X be any random variable, with moment generating function M(S) = E[es], and assume M(s) < o for all s E R. The cumulant generating function of X is defined as A(s) = log Ele**] = log M(s), SER Show the following identities: (1) A'(0) = E[X]. (2) A”(0) = Var(X). (3) A"(0) = E[(X - E[X]))). Using the inversion theorem for MGFs, argue the following: (4) If A'(s) = 0 for all s ER, then P(X=...
2.6.11. Adegenerate random variable is a random variable taking a constant value Let X= c. Show that E(X)=c, and Var(X)=0. Also find the cumulative distribution function of the degenerate distribution of X.
1) Let X and Y be random variables. Show that Cov( X + Y, X-Y) Var(X)--Var(Y) without appealing to the general formulas for the covariance of the linear combinations of sets of random variables; use the basic identity Cov(Z1,22)-E[Z1Z2]- E[Z1 E[Z2, valid for any two random variables, and the properties of the expected value 2) Let X be the normal random variable with zero mean and standard deviation Let ?(t) be the distribution function of the standard normal random variable....
1. Let U C IRt be open, UR be a function, a U and 0 A v E R" such that Dof(a) exists. Show that DAvf(a) exists for every 0 λ E R, and DAwf(a-λDuf(a). 3 marks 1. Let U C IRt be open, UR be a function, a U and 0 A v E R" such that Dof(a) exists. Show that DAvf(a) exists for every 0 λ E R, and DAwf(a-λDuf(a). 3 marks
Problem 1. Let X be a normal random variable with mean 0 and variance 1 and let Y be uniform(0.1) with X and Y being independent. Let U-X + Y and V = X-Y. For this problem recall the density for a normal random variable is 2πσ2 (a) Find the joint distribution of U and V (b) Find the marginal distributions of U and V (c) Find Cov(U, V).
B is a connected ball of finite radius 2, Let f : U → Rm be Ci and let B be a compact connected subset of U Show that there exists a constant M such that for all a, y e B. (Hint: use the mean value theorem). Find an example which shows that the assumption that B was compact is essential 2, Let f : U → Rm be Ci and let B be a compact connected subset of...
Let X and Y be two independent random variables. Show that Cov (X, XY) = E(Y) Var(X).
1. Let U be a random variable that is uniformly distributed on the interval (0,1) (a) Show that V 1 - U is also a uniformly distributed random variable on the interval (0,1) (b) Show that X-In(U) is an exponential random variable and find its associated parameter (c) Let W be another random variable that is uformly distributed on (0,1). Assume that U and W are independent. Show that a probability density function of Y-U+W is y, if y E...
Let the random variable X and Y have the joint probability density function. fxy(x,y) lo, 3. Let the random variables X and Y have the joint probability density function fxy(x, y) = 0<y<1, 0<x<y otherwise (a) Compute the joint expectation E(XY). (b) Compute the marginal expectations E(X) and E(Y). (c) Compute the covariance Cov(X,Y).