A car dealer sells X cars each day and always tries to

QUESTION:

A car dealer sells \(X\) cars each day and always tries to sell an extended warranty on each of these cars. (In our opinion, most of these warranties are not good deals.) Let \(Y\) be the number of extended warranties sold; then \(Y \leq X\). The joint pmf of \(X\) and \(Y\) is given by

\(\begin{aligned}f(x, y) &=c(x+1)(4-x)(y+1)(3-y), \\x &=0,1,2,3, \quad y=0,1,2, \text { with } y \leq x .\end{aligned}\)

(a) Find the value of \(c\).

(b) Sketch the support of \(X\) and \(Y\).

(c) Record the marginal pmfs \(f_{X}(x)\) and \(f_{Y}(y)\) in the "margins."

(d) Are \(X\) and \(Y\) independent?

(e) Compute \(\mu_{X}\) and \(\sigma_{X}^{2}\).

(f) Compute \(\mu_{Y}\) and \(\sigma_{Y}^{2}\).

(g) Compute  \(\operatorname{Cov}(X, Y)\).

(h) Determine ρ, the correlation coefficient.

(i) Find the best-fitting line and draw it on your figure.

Equation Transcription:

.

Text Transcription:

X  

Y  

Y < or = X  

f(x,y)  =c(x+1)(4-x)(y+1)(3-y), x  =0,1,2,3, y=0,1,2,  y < or = x  

c

f_X(x)  

f_Y(y)  

mu_X  

sigma_X^2.  

mu_Y  

sigma_Y^2

Cov⁡(X,Y)

rho