Chapter 4.3, Problem 3E

Let W equal the weight of laundry soap in a 1-kilogram box that is distributed in Southeast Asia. Suppose that $P\left(W<1\right)=0.02$ and $P\left(W>1.072\right)=0.08$.

Call a box of soap light, good, or heavy, depending on whether $\left\{W<1\right\},\left\{1\le W\le 1.072\right\}$, or $\left\{W>1.072\right\}$, respectively. In $n=50$ independent observations of these boxes, let X equal the number of light boxes and V the number of good boxes.

(a) What is the joint pmf of X and Y?

(b) Give the name of the distribution of V along with the values of the parameters of this distribution.

(c) Given that $X=3$, how is V distributed conditionally?

(d) Determine $E\left(Y|X=3\right)$.

(c) Find $\rho$, the correlation coefficient of X and Y.