Can anyone do this proof?

**MathJack** · February 17th 2013, 01:23 PM

Let X denote a Geometric random variable with probability of success p. Show that the

sum of probabilities of all possible outcomes of X is 1.

**abender** · February 17th 2013, 02:03 PM

Originally Posted by MathJack

Let X denote a Geometric random variable with probability of success p. Show that the

sum of probabilities of all possible outcomes of X is 1.

Note that $0\leq p \leq1$ .

$\sum^{\infty}_{k=1}P(X=k) = \sum^{\infty}_{k=1}p(1-p)^{k-1} = p \sum^{\infty}_{k=1}(1-p)^{k-1} =$ $p\left(\dfrac{1}{1-(1-p)}\right)=\dfrac{p}{p}=1$

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