Question

Let X1,X2,X3, and X4 be independent random variables. Assume that X2,X,,and X each are Poisson distributed with mean 5, and suppose that Y X1 X2 X3 X4 POI(25). (a) What is the distribution of X1?

Answer 1

Given that X2, X3, and X4 have a Poisson distribution with a mean of 5 each, and the sum of X1 through X4 is Poisson with a mean of 25, the distribution of X1 is Poisson with a mean of 10.

Step-by-step explanation:

The question is about determining the distribution of X1 given that X1, X2, X3, and X4 are independent random variables with X2, X3, and X4 being Poisson distributed with a mean of 5, and Y which is the sum of X1, X2, X3, and X4 is also Poisson distributed with a mean of 25.

Since the mean of Y is the sum of the means of X1, X2, X3, and X4, and X2 to X4 each has a mean of 5, this sums up to 15.

Therefore, X1 must be distributed such that its mean is the remaining value, which is 10. So, X1 is Poisson distributed with a mean of 10 (X1~Poisson(10)).

The properties of Poisson distribution allow for this because the sum of independent Poisson random variables is also Poisson distributed with the mean equal to the sum of their means.