# Can anyone do this proof?

• February 17th 2013, 01:23 PM
MathJack
Can anyone do this proof?
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.

• February 17th 2013, 02:03 PM
abender
Re: Can anyone do this proof?
Quote:

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 $0\leq p \leq1$.

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