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Let the two dimensional discrete random variable (X,Y) have the following joint

probability distribution.
X
Y
1 2 3
0 2p 0.1 0.25
1 0.08 0.1 0.08
2 0.05 p 0.13
a. Find the value of p that makes the distribution a legitimate distribution
b. Find the marginal probability distributions of X and Y
c. Are X and Y are independent? Why?
a. Find the marginal distribution of X for a given Y=2 i.e๐‘“(๐‘‹|๐‘Œ = 2)๏ฟฝ

To find the value of p that makes the distribution a legitimate distribution, we need to ensure that all the probabilities in the distribution sum up to 1.

We can start by summing up all the probabilities in the joint probability distribution:

2p + 0.1 + 0.25 + 0.08 + 0.1 + 0.08 + 0.05 + p + 0.13 = 1

Simplifying the equation, we get:

3p + 0.59 = 1

3p = 0.41

p = 0.41/3

p โ‰ˆ 0.1367

Therefore, the value of p that makes the distribution a legitimate distribution is approximately 0.1367.

To find the marginal probability distribution of X, we can sum up the probabilities for each value of X:

For X = 0: 2p + 0.1 + 0.25 = 2p + 0.35
For X = 1: 0.08 + 0.1 + 0.08 = 0.26
For X = 2: 0.05 + p + 0.13 = p + 0.18

So the marginal probability distribution of X is:

P(X = 0) = 2p + 0.35
P(X = 1) = 0.26
P(X = 2) = p + 0.18

To find the marginal probability distribution of Y, we can sum up the probabilities for each value of Y:

For Y = 0: 0.25 + 0.1 + 0 + 0.08 = 0.43
For Y = 1: 0.1 + 0.08 + p + 0.13 = p + 0.31
For Y = 2: 2p + 0.1 + 0.08 + 0.05 = 2p + 0.23

So the marginal probability distribution of Y is:

P(Y = 0) = 0.43
P(Y = 1) = p + 0.31
P(Y = 2) = 2p + 0.23

To determine whether X and Y are independent, we need to check if the joint probability distribution can be factored into the product of the marginal probability distributions of X and Y.

For independence, we need:

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

We can compare the calculated joint probabilities with the product of the corresponding marginal probabilities:

For example, let's check for X = 0 and Y = 2:

P(X = 0 and Y = 2) = 2p = 2 * 0.1367 โ‰ˆ 0.2734

P(X = 0) * P(Y = 2) = (2p + 0.35) * (2p + 0.23) = (2 * 0.1367 + 0.35) * (2 * 0.1367 + 0.23) โ‰ˆ 0.2734

Since the equation holds true for all values of X and Y, we can conclude that X and Y are independent.

Note: The marginal distribution of X for a given Y=2 is obtained by summing up the probabilities where Y=2 for each value of X:

P(X = 0 | Y = 2) = (2p + 0.1) / (2p + 0.23)
P(X = 1 | Y = 2) = 0.08 / (2p + 0.23)
P(X = 2 | Y = 2) = (0.05) / (2p + 0.23)