2 years ago

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Mathematics

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Anastaziya [24]2 years ago

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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

Mandarinka [93]

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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3 years ago

Work out the circumference of this circle give your answer in terms of pi and state it’s units the diameter is 14cm

zalisa [80]

Answer:14pi and 43.98 (2dp)

Step-by-step explanation:

circumference =2xpixr

radius = 14 divide by2 =7

2x pi x 7=14pi

14pi =43.98229715

=43.98 (2dp)

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