How $P[X=x]= \frac2{3^x}$ can give an even value for $x =1,2,3,\dots$

Question

I have a question that says “Let $X$ be a discrete random variable with probability function $P[X=x] = \frac2{3^x}$ for $x = 1,2,3,\dots$ What is the probability that $X$ is even? The thing is I don’t understand how this function can give $2,4,6,8$ etc..... if $x$ is from the natural numbers.

Answer 1

Because even numbers such as $2,4,6$ are natural number.

For example, even though the dice can gives value from $1$ to $6$, we can ask for the probability that it is even itsn't it.

To solve the problem, compute

$$\sum_{i=1}^\infty \frac{2}{3^{2x}}$$

Answer 2

Probability of $X$ being even implies: $$\sum_{n=1}^{\infty}P(X=2n)=\sum_{n=1}^{\infty} \frac{2}{3^{2n}}=2\cdot \frac{\frac19}{1-\frac19}=\frac14.$$

Answer 3

"What is the probability that $X$ is even?"

That is the question!

Not: "for what $x$ is the probability $P(X=x)$ even?"