Question

Let X1, X2, . . . be a sequence of independent and identically distributed random variables where the distribution is given by the so-called zero-truncated Poisson distribution with probability mass function; P(X = x) = λ^x/ (x!(eλ − 1)), x = 1, 2, 3...

Let N ∼ Binomial(n, 1−e^−λ ) be another random variable that is independent of the Xi ’s.

1) Show that Y = X1 +X₂ + ... + X_N has a Poisson distribution with mean nλ.

Answer 1

Let X1, X2, . . .XN be a sequence of independent and identically distributed random variables where the distribution is given by the so-called zero-truncated Poisson distribution with probability mass function

Let N ∼ Binomial(n, 1−e^−λ ) be another random variable that is independent of the Xi ’s. The probability mass function is

1) Show that Y = X1 +X₂ + ... + X_N has a Poisson distribution with mean nλ. Using the moment generating function(mgf), we found out the distribution of Y. Using the compound distribution function, If the primary distribution N has mgf M_N(t) and the secondary distribution X has mgf M_X(t), then the mgf of Y is

In terms of probability generating function(pgf), it is define as

So, we use pgf to get the distribution.

pgf of zero-truncated Poisson distribution is

pgf of binomial distribution is

Then, using the define equation

This is the probability generating function of Poisson distribution with mean nλ.