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

Question

. 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.e𝑓(𝑋|𝑌 = 2)�

Answer 1

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

Summing up all the probabilities in the table, we have:

0.2 + 0.1 + 0.25 + 0.08 + 0.1 + 0.08 + 0.05 + p + 0.13 = 1

Simplifying the equation, we get:

p + 0.99 = 1

Subtracting 0.99 from both sides, we have:

p = 0.01

So, the value of p that makes the distribution a legitimate distribution is p = 0.01.

b. To find the marginal probability distributions of X and Y, we need to sum up the probabilities for each value of X and Y, respectively.

The marginal probability distribution of X can be found by summing up the probabilities for each value of X:

X = 0: 0.2
X = 1: 0.1 + 0.08 + 0.1 = 0.28
X = 2: 0.05 + p + 0.13 = 0.18 + p

So, the marginal probability distribution of X is:

P(X = 0) = 0.2
P(X = 1) = 0.28
P(X = 2) = 0.18 + p

Similarly, the marginal probability distribution of Y can be found by summing up the probabilities for each value of Y:

Y = 0: 0.2 + 0.08 + 0.05 = 0.33
Y = 1: 0.1 + 0.1 + p = 0.2 + p
Y = 2: 0.1 + 0.08 + 0.1 + p + 0.13 = 0.31 + p

So, the marginal probability distribution of Y is:

P(Y = 0) = 0.33
P(Y = 1) = 0.2 + p
P(Y = 2) = 0.31 + p

c. To check if X and Y are independent, we need to compare the joint probability distribution with the product of the marginal probability distributions.

If X and Y are independent, then 𝑓(𝑋,𝑌) = 𝑓(𝑋) × 𝑓(𝑌) for all values of X and Y.

Let's compare the joint probabilities with the product of the marginal probabilities:

0.2 ≠ (0.33)(0.2) = 0.066

0.1 ≠ (0.33)(0.2 + 0.01) = 0.0726

0.25 ≠ (0.33)(0.31 + 0.01) = 0.111

0.08 ≠ (0.28)(0.2) = 0.056

0.1 ≠ (0.28)(0.2 + 0.01) = 0.0624

0.08 ≠ (0.28)(0.31 + 0.01) = 0.0996

0.05 ≠ (0.18 + 0.01)(0.2) = 0.038

p ≠ (0.18 + 0.01)(0.2 + 0.01) = 0.0418

0.13 ≠ (0.18 + 0.01)(0.31 + 0.01) = 0.0646

Since the joint probabilities are not equal to the product of the marginal probabilities for all values of X and Y, we can conclude that X and Y are not independent.