Value of Constant for Legitimate P.M.F.

A discrete random variable X has the probability mass function

$\displaystyle P(X=x)=\left{c(0.90)^{x} \quad \text{if x = 0,2,4,6,8,...}$

and

$\displaystyle P(X=x) = 0 \quad \text{otherwise}$

What must be the value of the constant c in order for this to be a legitimate p.m.f.?

I would think that,

$\displaystyle \sum_{x=0}^{\infty}c(0.9)^{x} = 1, \quad \text{Where x is only even values}$

But how does one solve for c?

Does

$\displaystyle \sum(0.9)^{x}$

decompose into a more convient form somehow?

Thanks again!

Re: Value of Constant for Legitimate P.M.F.

Hey jegues.

Hint: r^2x = (r^2)x and use the fact that a geometric series ar + ar^2 + .. + ar^n + .... = a/(1-r) if the series converges if |r| < 1.

Re: Value of Constant for Legitimate P.M.F.

jegues,

To further explain what chiro said, think about expanding the summation and setting up an equation.

In this situation:

c(0.9)^0 + c(0.9)^2 + c(0.9)^4 + c(0.9)^6 + ... = 1

(This equation is equal to 1 since all pmfs should add up to 1)

This should make it easier to see how to solve for c.

Don't forget that geometric series converge. Additionally, you may want to look up Benford's Law; that should help clarify your thinking.