(5 pts) Let U be a random variable following a uniform distribution on the interval [0,...
(5 pts) Let U be a random variable following a uniform distribution on the interval [0,1 Let Calculate analytically the variance of X. (HINT: E g(x)f(x)dx, and the p.d.f. 10SzSI 0 o.t.w. f(x) of a uniform distribution is f(x) =
(20 pts) Let U be a random variable following a uniform distribution on the interval [0 Let X=2U + 1 (a) Is X a random variable? Why or why not? (b) Calculate E[X] analytically
Please show your works (3) (5 pts) Let U be a random variable following a uniform distribution on the interval [0, 1 Let X-2U1 Calculate analytically the variance of X. (HINT: E[J(x)-「 f(x) of a uniform distribution is f(x) g(x)f(x)dz, and the pdf. 0 o.t.w.
Please show your works (2) (20 pts) Let U be a random variable following a uniform distribution on the interval [0,1 Let (a) Is X a random variable? Why or why not? (b) Calculate EX] analytically
Other answers show 12 in the denominator when solving for the final answer, please explain why. (3) (5 pts) Let U be a random variable following a uniform distribution on the interval [0, 1]. Let X-2U1 Calculate analytically the variance of X. (HINT: El( f(x) of a uniform distribution is f(x) g) f(x)dx, and the p.d.f 0 о.t.u.
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.
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 +...
Suppose that U is a random variable with a uniform distribution on (0,1). Now suppose that f is the PDF of some continuous random variable of interest, that F is the corresponding CDF, and assume that F is invertible (so that the function F-1 exists and gives a unique value). Show that the random variable X = F-1(U) has PDF f(x)—that is, that X has the desired PDF. Hint: use results on transformations of random variables. This cute result allows...
7. Suppose the random variable U has uniform distribution on [0,1]. Then a second random variable T is chosen to have uniform distribution on [O, U] Calculate P(T > 1/2)
.Let U be the uniform random variable over range [0, 1]. Let Z be defined as follows: Z = tan(Ur-05) (i). Find the pdf fz(z)i. Find the mean EZ