Given
. Consider the transformation
. Then
i) The PDF of
is
ii) The expected value is
Since
, when
is odd.
.Let U be the uniform random variable over range [0, 1]. Let Z be defined as...
(5 pts) Let U be a random variable following a uniform distribution on the interval [0, 1]. Let X=2U + 1 Calculate analytically the variance of X. (HINT : Elg(z)- g(z)f(x)dr, and the pdf. 0 < z < 1 0 o.t.w. f(x) of a uniform distribution is f(x) =
Consider the random variable Z defined as follows: for z=c for z 1 2K fz(z) for zc otherwise 0 for some constant c> 1 (a) Determine the value of K. Z2 (b) Determine the distribution of the random variable W (c) Find the moment generating function of Z when c 2 (d) Using the moment generating function, determine the mean and the variance of Z when c 2
Let F be a continuous distribution function and let U be a uniform (0, 1) random variable (a) If X F-(U), show that X has distribution function F. Show that -log(U) is an exponential random variable with mean 1.
Let X be a uniform(0, 1) random variable and let Y be uniform(1,2) with X and Y being independent. Let U = X/Y and V = X. (a) Find the joint distribution of U and V . (b) Find the marginal distributions of U.
Let X1, ..., X, bei.i.d. random variable with pdf fe defined as follows: fo (2) = 0x0-11(0<x< 1) where 0 is some positive number. Using the MLE Ô, find the shortest confidence interval for 6 with asymptotic level 85% using the plug-in method. To avoid double jeopardy, you may use V for the appropriate estimator of the asymptotic variance V (Ô), and/or I for the Fisher information I (Ô) evaluated at ê, or you may enter your answer without using...
3. (Bpoints) Let X, Y and Z be independent uniform random variables on the interval (0, 2), Let W min(X, y.z a) Find pdf of W Find E(1-11 b)
3. (Bpoints) Let X, Y and Z be independent uniform random variables on the interval (0, 2), Let W min(X, y.z a) Find pdf of W Find E(1-11 b)
1. Consider the uniform distribution X defined over the interval [0, 2pi]. Now let Y = sin(X) (a) Calculate the CDF FY(y) of Y. (b) Calculate the PDF f(y) of Y. In particular, in what interval [a, b] is Y defined? (this mean f(y) = 0 for y < a and for y > b). (c) Verify that f(y) is a PDF.
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 +...
2. Let Xbe a random variable with a continuous uniform density between -1 and 1, i.e, X U(-1,1) Random variable Y is defined by the following transformation: (1) Var(Y)-?
2. Let Xbe a random variable with a continuous uniform density between -1 and 1, i.e, X U(-1,1) Random variable Y is defined by the following transformation: (1) Var(Y)-?
2. Let Xbe a random variable with a continuous uniform density between -1 and 1, i.e, X U(-1,1) Random variable Y is defined by the following transformation: (1) Var(Y)-?
2. Let Xbe a random variable with a continuous uniform density between -1 and 1, i.e, X U(-1,1) Random variable Y is defined by the following transformation: (1) Var(Y)-?