simple probability function question?

Question

Let X be a random variable with probability function
f(x) = c / x! where x = 0, 1, 2, .....

note: x! means x factorial

Determine c.


The answer is 1/e. I'm not sure about the steps. Can someone show me step by step to the answer thanks!

Answer 1

Let me replace x by n in your probability function
as you should now; the sum of probability function from zero to infinity should be one so;

► Sigma (f(n))(n= 0 to infinity) = 1
so
► Sigma (c / n!)(x= 0 to infinity)=1


we know that (from macLaurin serie)
► Sigma ( x^n/ n!)(n= 0 to infinity)= exp(x)

from here it is clear that if we substitute x=1 in the above serie we'll have;

►Sigma ( 1^n/ n!)(n= 0 to infinity)= exp(1)=e

comparing this serie by serie of our probability function

►Sigma (c / n!)(x= 0 to infinity)=1
►Sigma ( 1/ n!)(n= 0 to infinity)= e => (1/e) Sigma ( 1/ n!)(n= 0 to infinity)= 1

So it's obvious that
►►c=1/e

Answer 2

For it to be a valid probability function then the sum of all probabilities must be 1. Therefore you need
c/0! + c/1! + c/2! + c/3! . . = 1. Factoring out the c gives
c(1/0/! + 1/1! + 1/2! + 1/3! . . . ) = 1. The bracket infinite sum is e so c must be 1/e to make the equation correct.

Answer 3

n
∑c/k! = ce
k=1
ce ≡ 1
c = 1/e