. 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?
d. Find the marginal distribution of X for a given Y=2 i.e๐(๐|๐ = 2)๏ฟฝ
a. To find the value of p that makes the distribution a legitimate distribution, we need to check if the sum of all the probabilities is equal to 1.
Sum of all probabilities = 2p + 0.1 + 0.25 + 0.08 + 0.1 + 0.08 + 0.05 + p + 0.13 = 2p + 0.69 + 2p + 0.26
Now, equating this sum to 1:
2p + 0.69 + 2p + 0.26 = 1
4p + 0.95 = 1
4p = 0.05
p = 0.05/4
p = 0.0125
Therefore, the value of p that makes the distribution a legitimate distribution is p = 0.0125.
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.
Marginal probability distribution of X:
f(X=0) = 2p + 0.1 + 0.25 = 2(0.0125) + 0.1 + 0.25 = 0.035 + 0.1 + 0.25 = 0.385
f(X=1) = 0.08 + 0.1 + 0.08 = 0.26
f(X=2) = 0.05 + p + 0.13 = 0.05 + 0.0125 + 0.13 = 0.1925
Therefore, the marginal probability distribution of X is:
X | f(X)
--------------
0 | 0.385
1 | 0.26
2 | 0.1925
Marginal probability distribution of Y:
f(Y=0) = 0.1 + 0.08 + 0.05 = 0.23
f(Y=1) = 0.25 + 0.1 + p = 0.25 + 0.1 + 0.0125 = 0.3625
f(Y=2) = 2p + 0.08 + 0.13 = 2(0.0125) + 0.08 + 0.13 = 0.025 + 0.08 + 0.13 = 0.235
Therefore, the marginal probability distribution of Y is:
Y | f(Y)
--------------
0 | 0.23
1 | 0.3625
2 | 0.235
c. To determine if X and Y are independent, we need to check if the joint probability distribution can be factorized into the product of the marginal probability distributions.
Multiplying the marginal probability distributions of X and Y, we get:
f(X) * f(Y) =
(0.385 * 0.23) (0.385 * 0.3625) (0.385 * 0.235)
(0.26 * 0.23) (0.26 * 0.3625) (0.26 * 0.235)
(0.1925 * 0.23) (0.1925 * 0.3625) (0.1925 * 0.235)
We can compare each element of the joint probability distribution with the corresponding element in the product of the marginal probability distributions. If they are equal, then X and Y are independent.
Comparing elements:
0.1 = 0.08855
0.1 = 0.09445
0.08 = 0.0749
0.08 = 0.08065
Since the elements do not match, X and Y are not independent.
d. To find the marginal distribution of X for a given Y=2, i.e., f(X|Y=2), we need to normalize the probabilities for each value of X when Y=2.
f(X=0|Y=2) = 0.25 / (0.25 + 0.08 + 0.13) = 0.25 / 0.46 โ 0.5435
f(X=1|Y=2) = 0.08 / (0.25 + 0.08 + 0.13) = 0.08 / 0.46 โ 0.1739
f(X=2|Y=2) = 0.13 / (0.25 + 0.08 + 0.13) = 0.13 / 0.46 โ 0.2826
Therefore, the marginal distribution of X for a given Y=2 is:
X | f(X|Y=2)
-----------------
0 | 0.5435
1 | 0.1739
2 | 0.2826