Suppose that X and Y are independent discrete random variables and each assumes the values 0,1, and 2 with probability of 1/3 each. Find the frequency function of X+Y.
To find the frequency function of X + Y, we need to determine the probability of each possible value of X + Y.
Since X and Y are independent random variables, we can use the fact that the probability of the sum of independent events is the product of their individual probabilities.
Let's consider the possible values of X and Y, which are 0, 1, and 2.
For X = 0:
- The only way for X and Y to add up to 0 is if both X and Y are 0. So, the probability is P(X=0) * P(Y=0) = (1/3) * (1/3) = 1/9.
For X = 1:
- X + Y can be 1 if X=1 and Y=0, or if X=0 and Y=1. So, the probability is P(X=1) * P(Y=0) + P(X=0) * P(Y=1) = (1/3) * (1/3) + (1/3) * (1/3) = 2/9.
For X = 2:
- The only way for X and Y to add up to 2 is if both X and Y are 2. So, the probability is P(X=2) * P(Y=2) = (1/3) * (1/3) = 1/9.
Therefore, the frequency function of X + Y is given by:
P(X + Y = 0) = 1/9
P(X + Y = 1) = 2/9
P(X + Y = 2) = 1/9
Note: The sum of the probabilities should equal 1, which is the case here:
1/9 + 2/9 + 1/9 = 4/9 + 4/9 = 8/9 + 1/9 = 9/9 = 1.
So, the frequency function of X + Y is {0: 1/9, 1: 2/9, 2: 1/9}.