Page 149 - DMTH404_STATISTICS

Unit 10: Approximate Expressions for Expectations and Variance

Similarly, the probability distribution of Y is Notes

Y 1 2 3 4 5 6 Total
1 1 1 1 1 1
P j 1
6 6 6 6 6 6

1 1 1 1 1 1 21
and ( ) 1.E Y = + 2. + 3. + 4. + 5. + 6. = = 3.5
6 6 6 6 6 6 6
The conditional distribution of X when Y = 5 is

X 1 2 3 Total
1 6 1 1 6 1 1 6 1
/Y = 5 ´ = ´ = ´ = 1
P
i
18 1 3 18 1 3 18 1 3
1
\ ( /E X Y = 5) = (1 2 3+ + ) 2=
3
The conditional distribution of Y when X = 2 is

Y 1 2 3 4 5 6 Total
1 1 1 1 1 1
P
/X = 2 1
j
6 6 6 6 6 6
1
\ ( /E Y X = 2) = (1 2 3 4 5 6+ + + + + ) 3.5=
6
Since the conditional distribution of X is same as its marginal distribution (or equivalently the
conditional distribution of Y is same as its marginal distribution), X and Y are independent
random variables.

Example 10: Two unbiased coins are tossed. Let X be a random variable which denotes
the total number of heads obtained on a toss and Y be a random variable which takes a value 1
if head occurs on first coin and takes a value 0 if tail occurs on it. Construct the joint probability
distribution of X and Y. Find the conditional distribution of X when Y = 0. Are X and Y independent
random variables?
Solution.

There are 4 elements in the sample space of the random experiment. The possible values that X
can take are 0, 1 and 2 and the possible values of Y are 0 and 1. The joint probability distribution
of X and Y can be written in a tabular form as follows:

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