Question

Suppose you roll two 6-sided dice, letting X be the sum of the numbers shown on the dice and Y be the number of dice that show an odd number.

a) Find the joint pmf of <X,Y>

b) Find the maringals pmf's for both variables.

c) Are X and Y independent?

Answer 1

Answer to part a)
All possible 36 outcomes are as shown below

.

The pmf for it is shown below:

Like for sum 2: both the dice got odd numbers, so value of y = 2, and since there is only 1 outcome out of 36 , its probability is 1/36 = 0.0278

Like this all the probability values for different values of X and Y are obtained as shown below:

.

Answer to part b)

The marginal value of X are :

Add all probability values rowwise to get the marginal value for each value of X

.

The marginal value of Y :

Add all the probability value scolumn wise to get the marginal value of each value of Y

.

Part c)

To test independent

P(X and Y) = P(X)*P(Y)

Let us take X = 6, and Y = 2

P(X=6) = 0.1389

P(Y = 2) = 0.2500

P(X =6 and Y=2) = 0.0833

[we got these values from the pmf table above and the marginal tables]

.

Now P(X=6) * P(Y=2) = 0.1389 *0.2500 = 0.0347 which is not equal to P(x=6 AND Y=2) = 0.0833

Hence since the equation P(X)*P(Y) = P(X and Y) is not satisfied for these two variable they are not independent