Exercise 2.5. Of never married, and two are divorced. Three of the executives ar
ID: 3054110 • Letter: E
Question
Exercise 2.5. Of never married, and two are divorced. Three of the executives are to be selected for promotion. Let }i denote the number of married executives and Y2, the number of never- married executives among the three selected for promotion. (a) Assuming that the three are randomly selected from the nine available, find the joint probability distribution of Y and Y2. Name it if possible. nine executives in a certain business firm, four are married, three have (b) Find the marginal probability distribution of Y, the number of married executives among the three selected for promotion. (c) Find PYi 112 -2). (d) If we let Y3 denote the number of divorced executives among the three selected for promotion, then Y3 3-Y -Y2. Find P(Y 1|Y2)Explanation / Answer
Of nine executive in certain business firm 4 are married, 3 are non married and 2 are divorced.
Now we have to select 3 executives for promotion.
Let Y1 = number of married executives.
Y2 = number of non married executives.
Y1 takes values 0,1,2 and 3.
Y2 takes values 0,1,2 and 3.
a) Joint probability distribution of Y1 and Y2 is,
The joint pmf of Y1 and Y2 is,
P(y1,y2) = (4 C y1) * (3 C y2) * (2 C 3-y1-y2) / (9 C 3)
0 <= y1 <= 3
0 <= y2 <= 3
1 <= y1+y2 <= 3
By using this pmf we can find joint probability.
b) Marginal distribution of Y1 :
c) Find P(Y1 = 1 / Y2=2)
By using conditional probability,
P(Y1 = 1 / Y2 = 2) = P(Y1 = 1 and Y2 = 2) / P(Y2 = 2)
= 0.2143 / 0.3571 = 0.6
e) Y1 and Y2 are independent iff
f(y1,y2) = f(y1)*f(y2)
We have to check each pair iff result fails in atleast case then the variables are dependent.
For the pair (0,0) :
0 = 0.1190*0.2381 = 0.0283
Both sides are not same.
In first case result fail so Y1 and Y2 are dependent.
f) E(Y1) = sum (y1 * P(y1)) = 1
h) E(0.2Y1 + 0.8Y2)
First find E(Y2)
E(Y2) = 1.3332
E(0.2Y1 + 0.8Y2) = 0.2E(Y1) + 0.8*E(Y2)
= 0.2*1 + 0.8*1.3332
= 1.27
unmarried married Y2 Y1 0 1 2 3 total 0 0 0.047619 0.142857 0.047619 0.238095 1 0.035714 0.285714 0.214286 0 0.535714 2 0.071429 0.142857 0 0 0.214286 3 0.011905 0 0 0 0.011905 0.119048 0.47619 0.357143 0.047619 1Related Questions
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