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A manufactured item is classified as good, a “second,” or
Chapter 4, Problem 9E(choose chapter or problem)
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).
Questions & Answers
QUESTION:
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:
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.