Let X and Y have joint probability mass function fX,Y (x, y) = (x + y)/30 for x = 0, 1, 2, 3 and y = 0,1,2. Find:
(a) Pr{X ≤ 2, Y = 1}(b) Pr{X > 2, Y ≤ 1}
(c) Pr{X +Y = 4}. (d) Pr{X > Y }.
(e) the marginal probability mass function of Y , and (f) E[XY].
(a)
(b)
(c)
(d)
(e)
Following table shows the joint and marginal pdfs:
X | ||||||
0 | 1 | 2 | 3 | P(Y=y) | ||
0 | 0 | 1/30 | 2/30 | 3/30 | 6/30 | |
Y | 1 | 1/30 | 2/30 | 3/30 | 4/30 | 10/30 |
2 | 2/30 | 3/30 | 4/30 | 5/30 | 14/30 | |
P(X=x) | 3/30 | 6/30 | 9/30 | 12/30 | 1 |
The marginal pdf of Y is
Y | P(Y=y) |
0 | 6/30 |
1 | 10/30 |
2 | 14/30 |
(f)
Following table shows the calculations for E(XY):
X | Y | P(X=x,Y=y) | xyP(X=x,Y=y) |
0 | 0 | 0 | 0 |
0 | 1 | 1/30 | 0 |
0 | 2 | 2/30 | 0 |
1 | 0 | 1/30 | 0 |
1 | 1 | 2/30 | 2/30 |
1 | 2 | 3/30 | 6/30 |
2 | 0 | 2/30 | 0 |
2 | 1 | 3/30 | 6/30 |
2 | 2 | 4/30 | 16/30 |
3 | 0 | 3/30 | 0 |
3 | 1 | 4/30 | 12/30 |
3 | 2 | 5/30 | 30/30 |
Total | 72/30 |
So,
Let X and Y have joint probability mass function fX,Y (x, y) = (x + y)/30...
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