(a)
When a fair coin has been flipped the possible outcomes are head and tail with equal probability. So we have
P(X=1) = 0.5, P(X=0) = 0.5
When head came, urn H selected so we have
P(Y=1 | X=1) = 0.60, P(Y=0 | X=1) = 0.40
When tail came, urn T selected so we have
P(Y=1 | X=0) = 0.40, P(Y=0 | X=0) = 0.60
Now the joint probabilities are
P(Y=1 and X=1) = P(Y=1 | X=1)P(X=1) = 0.60 *0.50 = 0.30
P(Y=0 and X=1) = P(Y=0 | X=1)P(X=1) = 0.40 *0.50 = 0.20
P(Y=1 and X=0) = P(Y=1 | X=0)P(X=0) = 0.40 *0.50 = 0.20
P(Y=0 and X=0) = P(Y=0 | X=0)P(X=0) = 0.60 *0.50 = 0.30
Following table shows the joint pdf:
X | ||||
0 | 1 | P(Y=y) | ||
Y | 0 | 0.3 | 0.2 | 0.5 |
1 | 0.2 | 0.3 | 0.5 | |
P(X=x) | 0.5 | 0.5 | 1 |
(b)
Following table shows the calculations for E(Y):
Y | P(Y=y) | yP(Y=y) |
0 | 0.5 | 0 |
1 | 0.5 | 0.5 |
Total | 0.5 |
So,
The probability that ball is green is
P(Y=1) = 0.5
(c)
Following table shows the calculations for Var(Y):
So,
---------------
Following table shows the conditional pdf of Y given X=0 and calculations for Var(Y|X=0):
So,
Following table shows the conditional pdf of Y given X=1 and calculations for Var(Y|X=1):
So,
(d)
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