Let X have a Poisson distribution with parameter ?. to

Chapter 7, Problem 100SE

(choose chapter or problem)

Suppose F is defined as in Definition 7.3.

a If \(W_{2}\) is fixed at \(W_{2}\), then \(F=W_{1} / c\), where \(c=w_{2} v_{1} / v_{2}\). Find the conditional density of

𝐹 for fixed \(W_{2}=w_{2}\).

b Find the joint density of 𝐹 and \(W_{2}\)

c Integrate over \(w_{2}\) to show that the probability density function of 𝐹 —say, 𝘨 (𝘺)—is given by

\(g(y)=\frac{\Gamma\left[\left(v_{1}+v_{2}\right) / 2\right]\left(v_{1} / v_{2}\right)^{v_{1} / 2}}{\Gamma\left(v_{1} / 2\right) \Gamma\left(v_{2} / 2\right)} y^{\left(v_{1} / 2\right)-1}\left(1+\frac{v_{1} y}{v_{2}}\right)^{-\left(\nu_{1}+v_{2}\right) / 2}, \quad 0<y<\infty\)

Equation Transcription:

     

Text Transcription:

W2

W2

F=W1/c      

c=w2v1/v2

W2=w2

W2

w2

\(g(y)=\Gamma [(v_1+v_2\) / 2\](v_1 / v_{2)^v_1 / 2\Gamma(v_{1} / 2) \Gamma(v_2 / 2) y^(v_1 / 2)-1(1+v_1 y v_2)^-(\nu_1+v_2) / 2, \quad 0<y<\infty