Let X and Y be IID N(0, 1) random variables. Let Z~ Exp (1) be an...

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Let X and Y be IID N(0, 1) random variables. Let Z~ Exp (1) be an exponentially distributed random variable with X = (1/2). (a) Find E(ex²) and E(erZ), i.e., the MGF of X2 and Z. Find out the region of convergence or [r_(X²), r+(X²)] and [r_(Z), r+(Z)]. Identify whether gx² (r) and gz (r) converges at the boundary points or not. (b) It can be shown that if the MGF of a random variable V is identical to that of W in a small neighborhood rc (8,8) with 8 > 0, then V and W have the same distribution. Usc part (a) to find the distribution of X2 + y². Let X and Y be IID N(0, 1) random variables. Let Z~ Exp (1) be an exponentially distributed random variable with X = (1/2). (a) Find E(ex²) and E(erZ), i.e., the MGF of X2 and Z. Find out the region of convergence or [r_(X²), r+(X²)] and [r_(Z), r+(Z)]. Identify whether gx² (r) and gz (r) converges at the boundary points or not. (b) It can be shown that if the MGF of a random variable V is identical to that of W in a small neighborhood rc (8,8) with 8 > 0, then V and W have the same distribution. Usc part (a) to find the distribution of X2 + y².

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