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Geometric Distribution

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Calculate the probability mass function, lower cumulative distribution function, and upper cumulative distribution function of the geometric distribution.
Number of failures until success: x
x=0,1,2,...
Probability of success in a single trial p
0≦p≦1
\(\normalsize Geometric\ distribution\\ (1)\ probability\ mass\\ \hspace{30px}f(x,p)=p(1-p)^x\\ (2)\ lower\ cumulative\ distribution\\ \hspace{30px}P(x,p)={\large\displaystyle \sum_{t=0}^{x}}f(t,p)\\ (3)\ upper\ cumulative\ distribution\\ \hspace{30px}Q(x,p)={\large\displaystyle \sum_{t=x}^{\infty}}f(t,p)\\ (4)\ expectation(mean):\hspace{10px}{\large\frac{1-p}{p}}\\\)

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