Automobile engines and transmissions are produced on

QUESTION:

Automobile engines and transmissions are produced on assembly lines, and are inspected for defects after they come off their assembly lines. Those with defects are repaired. Let X represent the number of engines, and Y the number of transmissions that require repairs in a one-hour time interval. The joint probability mass function of X and Y is as follows:

\(\begin{array}{ccccc} \hline & & \ \ \ \ \ {\boldsymbol{y}} \\ \boldsymbol{x} & \mathbf{0} & \mathbf{1} & \mathbf{2} & \mathbf{3} \\ \hline \mathbf{0} & 0.13 & 0.10 & 0.07 & 0.03 \\ \mathbf{1} & 0.12 & 0.16 & 0.08 & 0.04 \\ \mathbf{2} & 0.02 & 0.06 & 0.08 & 0.04 \\ \mathbf{3} & 0.01 & 0.02 & 0.02 & 0.02 \\ \hline \end{array}\)

a. Find the marginal probability mass function \(p_{X}(x)\).

b. Find the marginal probability mass function \(p_{Y}(y)\).

c. Find \(\mu_{X}\).

d. Find \(\mu_{Y}\).

e. Find \(\sigma_{X}\).

f. Find \(\sigma_{Y}\).

g. Find \(\rm{Cov}(X, Y)\).

h. Find \(\rho_{X, Y}\).