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.
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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 $\displaystyle 0\leq p \leq1$. $\displaystyle \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} =$$\displaystyle p\left(\dfrac{1}{1-(1-p)}\right)=\dfrac{p}{p}=1$
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