Question

Suppose that Y takes on the values {0, 1, 2, 3}, each with probability 1/4. Take a random sample of 5 numbers from this distribution. In how many possible samples can the sum of the five numbers (Y1 + Y2 + Y3 + Y4 + Y5) equal to 1? [Give the number of samples, not the proportion of possible samples]

Answer 1

The total number of samples that give this outcome is 5.

Explanation:

Since Y takes values in {0,1,2,3}, For us to have that
`Y_1+Y_2+Y_3+Y_4+Y_5=1` implies that all of them are zero but one. The one that is non-zero necessarily is equal to 1. To calculate the number of samples that give this outcome is equivalent to counting the total number of ways in which we can pick the i-index such that
`Y_i=1`. Note that in this case we can either choose Y1 to be 1, Y2 to be 1 and so on. So, the total number of samples that give this outcome is 5.