Chapter 4.1, Problem 9E

A manufactured item is classified as good, a “second.” or defective with probabilities 6/10, 3/10, and 1/10, respectively. Fifteen such items are selected at random from the production line. Let X denote the number of good items, Y the number of seconds, and $15-X-Y$ the number of defective items.

(a) Give the joint pmf of X and Y, $f\left(x,y\right)$.

(b) Sketch the set of points for which $f\left(x,y\right)>0$.

From the shape of this region, can X and Y be independent? Why or why not?

(c) Find $P\left(X=10,T=4\right)$.

(d) Give the marginal pmf of X.

(e) Find $P\left(X\le 11\right)$.