Suppose U has a uniform distribution on the interval (0₁, 02). Then, the moment generating function of U is derived as follows. mu(t)= E = = 02 01 702 J0₁ du
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- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.If a random variable X has the moment generating function Mx (t)= 2 - ť Determine the variance of X.Let the random variable X have the moment generating function M(t) = e³t 1-t² = -1 < t < 1 What are the mean and the variance of X, respectively?
- Let X1, X2,... , Xn be independent Exp(A) random variables. Let Y = X(1)min{X1, X2, ... , Xn}. Show that Y follows Exp(nA) dis- tribution. Hint: Find the pdf of YB) Let the random variable X have the moment generating function e3t M(t) for -1(ii) Suppose that X₁ and X₂ have joint pdf 2, 10, Compute the joint pdf of random variables Y₁ = X₁ and Y₂ = X2. f(x1, x₂): for 0 < x1 < x₂ < 1 otherwise =The moment generating function M (t) for random variables y are as follow. Examine and identify the distribution of the random variables and state the parameter involves. 1. M(t) = e +2 3 1 2. M()-(3-28) = 3. M(t)= -(-) 4. M(t)= 5 5-tIf the pdf of a random variable X is f(x) = { 0.15-0.15(x-0.5), x ≥ 0.5 otherwise Find the moment generating function and use it to find the mean and variance of X.The p.d.f. of a random variable X' is as shown in the figure. The pdf is zero for X 5. Calculate (i) the maximum value of p.d.f. (ii) expectation of X, E(X) (iii) variance of X. fx (x) k10 If 0₁ <0₂, derive the moment-generating function of a random variable that has a uniform distribution on the interval (0₁, 02). Suppose U has a uniform distribution on the interval (0₁, 02). Then, the moment generating function of U is derived as follows. mu(t) = E = II II .02 102 du2 Let X (t) be a random process with mean 3 and auto correlation R(t, t2) = 9 + 4 e-0.2 ,-t, Determine the mean, variance and covariance of the random variables Z =X (5) and wX (8).