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RideAnS [48]
4 years ago
6

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 wi

th 2r degrees of freedom by showing that
Mathematics
1 answer:
Mandarinka [93]4 years ago
5 0
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 <span>a chi squared random variable with 2r degrees of freedom</span>.
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