a) here as we can see that X depends on a random variable U ; therefore X is a random variable cause randomness in U will be imposed on value of X.
b)
as we know that mean of U =E(U) =(a+b)/2 =(0+1)/2 =0.5
hence E(X)=E(2U+1)=2E(U)+1 =2*0.5+1
E(X)=2
(20 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 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) =
(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) =
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
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.
(c) (20 pts.) Let X have a uniform distribution U(0, 2) and let the considiton; distribution of Y given X = x be U(0, x3) i. Determine f (x, y). Make sure to describe the support of f. ii. Calculate fy (y) iii. Find E(Y).
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 +...
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.
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 [0, U]. Calculate P(T> 1/2)
2. Assume the random variable y has the continuous uniform distribution defined on the interval a to b, that is, f(y) = 1/6 - a), a sy<b. For this problem let a = 0 and b = 2. (a) Find P(Y < 1). (Hint: Use a picture.) (b) Find u and o2 for the distribution.