In: Statistics and Probability
A fair 4-sided die is rolled, let X denote the outcome. After that, if X = x, then x fair coins are tossed, let Y denote the number of Tails observed. a) Find P( X >= 3 | Y = 0 ). b) Find E( X | Y = 2 ). “Hint”: Construct the joint probability distribution for ( X, Y ) first. Write it in the form of a rectangular array with x = 1, 2, 3, 4 and y = 0, 1, 2, 3, 4.
a) The value of Y given different values of X here is computed as:
P(Y = 0 | X = 1) = 0.5
P(Y = 0 | X = 2) = 0.52 = 0.25
P(Y = 0 | X = 3) = 0.53 = 0.125
P(Y = 0 | X = 4) = 0.54 = 0.0625
Also for a fair 4 sided die, we get:
P(X = 1) = P(X = 2) = P(X = 3) = P(X = 4) = 0.25
Therefore, using law of total probability, we get here:
P(Y = 0) = P(Y = 0 |X = 1)P(X = 1) + P(Y = 0 | X = 2)P(X = 2) + P(Y
= 0 | X = 3)P(X = 3) + P(Y = 0 | X = 4)P(X = 4)
P(Y = 0) = 0.25*(0.5 + 0.25 + 0.125 + 0.0625) = 0.234375
Now P(X >= 3 | Y = 0) is computed using Bayes theorem here
as:
P(X >= 3 | Y = 0) = P(Y = 0 | X = 3)P(X = 3) + P(Y = 0 | X =
4)P(X = 4) / P(Y = 0)
= 0.25*(0.125 + 0.0625) / 0.234375
= 0.2
Therefore 0.2 is the required probability here
b) The value of Y given different values of X here is computed as:
P(Y = 2 | X = 1) = 0
P(Y = 2 | X = 2) = 0.52 = 0.25
P(Y = 2 | X = 3) = 3*0.53 = 0.375
P(Y = 2 | X = 4) = (4c2)*0.54 = 0.375
The expected value here is computed as:
E(X | Y = 2) = 0*P(Y = 2 | X = 1) + 2*P(Y = 2 | X = 2) + 3*P(Y = 2 | X = 3) + 4*P(Y = 2 | X = 4)
= 0*1 + 0.25*2 + 0.375*3 + 0.375*4
= 3.225
Therefore 3.225 is the expected value of X given Y = 2 here.