Poisson and Geometric Random Variable questions
Can anyone tell me if I got the correct answers for the below two questions? Any help is appreciated.
Q:Let X have a Poisson distribution with parameter λ. If we know that P(X = 1|X ≤ 1) = 0.8, then what is the expectation and variance of X?
A:So, (X=0|X≤ 1)=0.2, which equals e^-λ=0.2. Taking the log of both sides and multiplying by -1 we get λ =1.60944=E(X)=VAR(X). Is this right?
Q:A random variable X is a sum of three independent geometric random variables with mean 4. Find the mean, variance, and moment generating function of X/4.
A: Each of the three geometric r.v.'s has E(X)=(1-p)/p=4. Solving for p we get 0.2. Since these are three independent geometric r.v.'s then their sum is a negative binomial r.v. with parameters r=3 and p=0.2. E(X) of a negative binomial r.v. is (r*(1-p))/p, which gives us 12 and VAR(X) (r*(1-p))/p^2=60. Are these correct?
I am not sure how to get the moment generating function of X/4. The moment generating function is defined as E(e^(tx)). In my notes I have that E(e^(atx)) gives us the moment generating function Mx(at). The moment generating function for neg. binomial r.v. is (p/(1-(1-p)e^t))^r. Since the constant a=1/4 would the moment generating function just be (p/(1-(1-p)e^t/4))^r?