need help finding Probabilities...?

Question

suppose the random variable X has a geometric distribution with mean of 2.5. determine the following probabilities.

a) P(X=1)
b) P(X=5)

Answer 1

mean = 2.5
mean = 1/p
2.5= 1/p
p = 1/2.5
p = .4

P(X = n) = (1-p)^n-1*p
P(x=1) = (1-.4)^(1-1)*.4
=(.6^0)*.4
=1*.4
=.4

P(x=5) = (1-.4)^(5-1)*.4
=(.6^4)*.4
=.1296*.4
=.05184

Answer 2

the mass function for the geometric distribution with success probability p is:

P(X = x) = p * (1 - p)^(x-1) for x = 0, 1, 2, 3, 4, 5, ....
P(X = x) = 0 otherwise

think of this a one success after x -1 failures

the mean of the geometric is 1/p.

as such the success probability is 2.5 = 1/p; p = 0.4

P(X =1 ) = 0.4 * (1 - 0.4) ^ 0 = 0.4
P(X = 5) = 0.4 * (1 - 0.4) ^ 4 = 0.05184