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Suppose that $X$ is a geometric random variable with parameter (probability of success) $p$.

Show that $\Pr(X > a+b \mid X>a) = \Pr(X>b)$

First I thought I'd start by calculating $\Pr(X>n)$ where $n=a+b$:

$$\Pr(X > n) = p_{n+1} + p_{n+2} + \cdots = ?\tag{1}$$

But I don't know how to determine the limit of equation (1). I know for an infinite geometric series starting at index zero:

$$\sum\limits_{n=0}^\infty ax^n=\cfrac{a}{1-x}\text{ for }|X|<1$$

But don't know what to do when index starts at $n$.

Next I thought I'd do:

$$\Pr(X > a+b) \mid X > a) = \cfrac{ \Pr[ (X > a+b) \cap (X > a)] }{\Pr(X > a)}$$ $$=\cfrac{\Pr(X > a+b)}{\Pr(X > a)}$$

and substitute my result from equation (1). Any help appreciated in advance. Thank you.

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2 Answers 2

up vote 1 down vote accepted

Here's how to handle an infinite geometric series when the index starts at $n$ instead of $0$: \begin{align} \sum_{k=n}^\infty ax^k & = ax^n + ax^{n+1} + ax^{n+2} + ax^{n+3}+\cdots \\[10pt] & = (ax^n) + (ax^n)x +(ax^n)x^2 + (ax^n)x^3+\cdots \\[10pt] & = b + bx + bx^2 + bx^3 + \cdots \end{align}

Now it starts at index $0$. And of course $b$ is $ax^n$.

It looks as if you can do the rest.

Second method: You mentioned that the probability of "success" is $p$. That means the probability of success on each trial is $p$.

If $X$ is defined as the number of trials needed to get one success, then the event $X>n$ is the same as the event of failure on all of the first $n$ trials, so that probability of that is $(1-p)^n$.

If $X$ is defined as the number of trials needed to get one failure, then the event $X>n$ is the same as the event of success on all of the first $n$ trials, so the probability of that is $p^n$.

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There are two ways to get $\Pr(X\gt n)$, the easy way and the hard way. Since there is something to be learned from both, we do it both ways.

The hard way: The probability that $X=n+1$ is the probability of $n$ failures in a row, then success. This is $(1-p)^np$.

Similarly, the probability that $X=n+2$ is the probability of $n+1$ failures in a row, then success. This has probability $(1-p)^{n+1}p$.

Continue, and add up. There is a $(1-p)^n p$ "in" each term. Take it out as a common factor. So we get $$(1-p)^n p\left(1+(1-p)+(1-p)^2 +\cdots\right).$$ The geometric series inside the brackets has sum $\frac{1}{1-(1-p)}=\frac{1}{p}$.

The easy way: We have $X\gt n$ precisely if we get $n$ failures in a row, probability $(1-p)^n$.

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