0

Given random variable X and N so that, N ∼ Poisson(λ), and X|N ∼ Bin(N,p) where p is a constant (Assume that X = 0 when N = 0 and 0 < p < 1). Note that the moment generating function of a Bernoulli random variable with parameter p is 1 − p + etp, and the moment generating function for Poisson(λ) distribution is exp[λ(et − 1)].

Show that X/λ → p, as λ → ∞

I believe I found the pmf of X not conditioned on N below

$ P(x=x) = \frac{e^{-\lambda p}(\lambda p)^x}{x!} $

Then used $ E(e^{tx}) $ to find the mgf

$ M_x(t) = e^{\lambda p(e^t-1) } $

But I am no unsure of how to show convergence in distribution.

Greyson21
  • 3

1 Answers1

0
  • You've shown $M_X(t) = e^{\lambda p (e^t-1)}$
  • Show that consequently, $M_{X/\lambda}(t) = e^{\lambda p (e^{t/\lambda} - 1)}$.
  • Note that $\lim_{\lambda \to \infty} \frac{e^{t/\lambda} - 1}{t/\lambda} = 1$ to conclude that $\lim_{\lambda \to \infty} M_{X/\lambda}(t) = e^{pt}$
angryavian
  • 89,882