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 (x, y).

(b) Sketch the set of integers (x, y) for which f (x, y) > 0. From the shape of this region, can X and Y be independent?Why or why not?

(c) Find P(X = 10,Y = 4).

(d) Give the marginal pmf of X.

(e) Find P(X ≤ 11).

Answer :

Step 1 of 1 :

Let X denote the number of good items and

Let y denote the number of seconds and 1-X-Y defective.

Here 15 items are selected at random from the production line.

So n = 15.

We know that

The probabilities of defective item is

6/10 = 0.6,

3/10 = 0.3

1/10 = 0.1and

Here and

pmf(x,y) = Prob {X good items, Y seconds defective}

Our goal is to find

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

b). Sketch the set of integer (x,y) for f(x,y)>0. From the shape of this region, can X and Y be

independent ? why or why not?

c). Find P(X=10,Y=4)

d). Find the marginal pmf of X.

e). Find

a).

Now we have to determine the joint pmf of X and Y.

Then the joint pmf X and Y is

Hence the the joint pmf of X and Y is above.