Question

Define a new random variable by y = 2px. show that, as p l 0, the mgf of y converges to that of a chi squared random variable with 2r degrees of freedom by showing that

Answer 1

The moment generating function (mgf) of y is given by

$M_y(t)=E(e^{ty})=E(e^{2pxt})=M_x(2pt)=p(1-e^{2pt}(1-p))^{-1}$

Since the momoent generating function of x is given by $M_x(t)=p(1-e^{t}(1-p))^{-1}$

When p tends to 0, we have

$\lim_{p \to 0} M_y(t)= \lim_{p \to 0} \frac{p}{1-e^{2pt}(1-p)} = \frac{0}{0}$

Applying L'Hopital's rule we have:

$\lim_{p \to 0} M_y(t)=\lim_{p \to 0} \frac{1}{e^{2pt}+2te^{2pt}+2pte^{2pt}} = \frac{1}{1+2t} , \ \ \ t\ \textless \ \frac{1}{2}$

This shows that y converges to a chi squared random variable with 2r degrees of freedom.