Let Z1, Z2, and Z3 have independent standard normal

QUESTION:

Let \(Z_{1}, Z_{2}\) and \(Z_{3}\) have independent standard normal distributions,\(N(0,1)\).

(a) Find the distribution of

                         \(W=\frac{Z_{1}}{\sqrt{\left(Z_{2}^{2}+Z_{3}^{2}\right) / 2}}\)

(b) Show that

                          \(V=\frac{Z_{1}}{\sqrt{\left(Z_{1}^{2}+Z_{2}^{2}\right) / 2}}\)

has pdf \(f(v)=1 /\left(\pi \sqrt{2-v^{2}}\right),-\sqrt{2}<v<\sqrt{2}\)

(c) Find the mean of \(V\).

(d) Find the standard deviation of \(V\).

(e) Why are the distributions of \(W\) and \(V\) so different?

Equation Transcription:

Z1, Z2,

Z3 

Text Transcription:

Z_1, Z_2,

Z_3

W=Z_1/sqrt(Z_2^2+Z_3^2)/2

V=Z_1/sqrt(Z_2^2+Z_3^2)/2

F(v)=1/(pi sqrt 2-v^2), -sqrt 2<v<sqrt 2

V

W