Question

The moment generating function (MGF) for a random variable X is: Mx (t) = E[e'X]. Onc useful property of moment generating functions is that they make it relatively casy to compute weighted sums of independent random variables: Z=aX+BY M26) - Mx(at)My (Bt). (A) Derive the MGF for a Poisson random variable X with parameter 1. (B) Let X be a Poisson random variable with parameter 1, as above, and let y be a Poisson random variable with parameter y. X and Y are independent. Use the MGF to show that their sum is also Poisson. What is the parameter of the resulting distribution? (C) Derive the MGF for an exponential random variable X with parameter 1. For what values of t does the MGF MX(t) exist? (D) Let X be an exponential random variable with parameter 1, and let a > 0 be a constant. Show that the random variable aX is also exponentially distributed. What is its parameter? (E) Imagine that you have k i.i.d. random variables X1, X2, ..., Xk, each of which is exponentially distributed with parameter 2. Show that this sum is gamma distributed and compute its parameters. This will require finding the MGF of the gamma distribution. Use the following parameterization of the gamma density: f(z; C,B) = 24-12-Bz

Answer 1

The Moment Generating function uniquely defines the distribution of a random variable.

So, for any unknown distribution, we calculate the moment generating function and compare them to a known Moment generating function(of known distributions).

If the calculated Moment generating function of the unknown distribution has the same form as that of the known MGF of some Known distribution, then the unknown distribution has the same probability distribution as that of the known distribution.

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