Given the random variable Y in Problem 3.4.1, let U-g(Y) Y2 (a) Find Pu(u) (b) Find...
Let Y be a random variable with p(y) given in the accompanying table. Find E(Y), E(1/Y), E(Y2-1), and V(Y). y 1 2 3 4 p(y) .4 .3 .2 .1
Let y be a continuous uniform random variable, Y - Gumbel(B).for ß>0. That is, Y has cumulative density function PIY <y)=Fly)=e for YER. Showing all of your working, find the probability density function of Show that the inverse of the cumulative density function is given by F (y)=u-Bin(–In(y)). for YER. Given realisations {u,, uz,...,Ug} = {0.710,0.119,0.358,0.883,0.504} of a U[0, 1] variable, generate five realisations {y, Y2,..., Ys} of Y-Gumbel(5, 10). Clearly explain your method and any calculations required.
3. Suppose that Y is a continuous random variable with pdf су otherwise (a) Find the value of c E R so that fy(y) is a valid pdf. (b) If U-Y3, find the pdf fu(u) of U.
Question 4 Let Yı: Y2, .... Yn denote a random sample and let E(Y) = u and Var(Y) = o-y, i = 1, 2, ..., n. (b) Prove that the standard error of the sample mean Y SEⓇ) =
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 there be U, a random variable that is uniformly distributed over [0,1] . Find: 1) Density function of the random variable Y=min{U,1-U}. How is Y distributed? 2) Density function of 2Y 3)E(Y) and Var(Y) U Uni0,1
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).
Problem 1: Given the function g(x,y)-ke-xy-c)u(x-a)u(y-b) find the constant "k" in terms ofa, b, and c so that g(x,y) is a valid probability density function (15%). Are the random variables X and Y statistically independent (10%)? (Support your answer.)
3. (5 marks) Let U be a random variable which has the continuous uniform distribution on the interval I-1, 1]. Recall that this means the density function fu satisfies for(z-a: a.crwise. 1 u(z), -1ss1, a) Find thc cxpccted valuc and the variancc of U. We now consider estimators for the expected value of U which use a sample of size 2 Let Xi and X2 be independent random variables with the same distribution as U. Let X = (X1 +...
Problem The random variable X is exponential with parameter 1. Given the value r of X, the random variable Y is exponential with parameter equal to r (and mean 1/r) Note: Some useful integrals, for λ > 0: ar (a) Find the joint PDF of X and Y (b) Find the marginal PDF of Y (c) Find the conditional PDF of X, given that Y 2. (d) Find the conditional expectation of X, given that Y 2 (e) Find the...