Given the joint probability distribution of two random variables X and Y:
X\Y 1 2 3
1 0.1 0.2 0.1
2 0.2 0.1 0.1
Calculate the marginal probability distributions P(X) and P(Y).
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To calculate the marginal probability distribution P(X), we sum up the probabilities of X for each value of Y.
P(X=1) = P(X=1,Y=1) + P(X=1,Y=2) + P(X=1,Y=3) = 0.1 + 0.2 + 0.1 = 0.4
P(X=2) = P(X=2,Y=1) + P(X=2,Y=2) + P(X=2,Y=3) = 0.2 + 0.1 + 0.1 = 0.4
So, the marginal probability distribution P(X) is:
X P(X)
1 0.4
2 0.4
To calculate the marginal probability distribution P(Y), we sum up the probabilities of Y for each value of X.
P(Y=1) = P(X=1,Y=1) + P(X=2,Y=1) = 0.1 + 0.2 = 0.3
P(Y=2) = P(X=1,Y=2) + P(X=2,Y=2) = 0.2 + 0.1 = 0.3
P(Y=3) = P(X=1,Y=3) + P(X=2,Y=3) = 0.1 + 0.1 = 0.2
So, the marginal probability distribution P(Y) is:
Y P(Y)
1 0.3
2 0.3
3 0.2
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