Let X be a random variable having a pmf given by PX 0 1P P

Let X be a random variable having a pmf given by P(X = 0) = 1-P, P(X = l) = p (you can think of this as modeling an event that happens with a certain probability p) Part a: What is the mean and variance of X, in terms of p? Part b: What happens to the variance as p rightarrow 0 and as p rightarrow 1? Does this make sense? Part c: Find the mgf of X.

Solution

a)

x * P(X)

0 * (1-p) + 1*p

mean = p

variance

x^2 * P(X)

0^2 * (1-p) + 1^2 * p

p

variance = p - p^2

part b)

when p is almost 0 variance is 0

when p is almost 1 variance is 0

this make sence because the probability just have 2 option, or is 0 or 1

so if the value is more closer to one of them the variance is going to fall

c)

I can gladly help you but you should post it on a new question