Recall that the sum of a xed number of independent Gaussian random variables is

Question

Recall that the sum of a xed number of independent Gaussian random variables is Gaussian. Recall also that X is a Gaussian random variable having zero mean if and only if its MGF has the form MX(s) = e((2s2)/2) , where 2 is the variance of X. This exercise investigates whether a random sum of independent Gaussian random variables may also be Gaussian. Suppose that N is a random variable taking values 1 and 3, each with probability 1/2 , and Xi, i = 1,2,..., are independent and identically distributed Gaussian random variables with mean 0 and variance 1, and also independent from N. Let WN = from i = 1 to N of Xi. Is WN a Gaussian random variable? Try to nd a discrete random variable N taking positive integer values such that WN is Gaussian. Note then that we must have E[e((Ns2)/2)] = eas^2 for some positive constant a. If this equation holds, what should the value of the constant a be? What kind of random variables N can match this?

Explanation / Answer

Here, use normal random variable to calculate constant values.

Related Questions

Navigate

• Browse (All)
• Browse R
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.