Question

Find the moment generating function of a Poisson random variable and hence determine its mean and variance.

Answer 1

Answer:

The exponential distribution is one of the distributions relating to a Poisson process. In a Poisson process, there is a certain rate λ of events occurring per unit time that is the same for any time interval. The time X to the first event is given by an exponential distribution with parameter λ.

The density function for an exponential distribution is

f(x)=λe−λx

for x∈[0,∞).

Its moment generating function turns out to be m(t)=(1−t/λ)−1.

Its mean and its standard deviation are both equal to 1/λ.

The moment generating function of a random variable X

is a function m(t) of a new variable t defined as the expectation of etX

m(t)=E(etX).

Let’s compute the moment generating function for the exponential distribution.

m(t)=====∫∞−∞etxf(x)dx∫∞0etxλe−λxdxλ∫∞0e(t−λ)xdxλe(t−λ)xt−λ∣∣∣∞0(limx→∞λe(t−λ)xt−λ)−λe0t−λ

Now if t<λ

, then the limit in the last line is 0, so in that case

m(t)=λλ−t=11−t/λ.

Step-by-step explanation: