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Negative Binomial Distribution

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Find the probability mass function, lower cumulative distribution function, and upper cumulative distribution function of the negative binomial distribution.
Number of failures before k successes: x
x=0,1,2,..
Number of successes: k
k=1,2,..
Probability of success in a single trial p
0≦p≦1
\(\normalsize Negative\ Binomial\ distribution\\ (1)\ probability\ mass\\ \hspace{30px}f(x,k,p)={}_{x+k-1}C_xp^k(1-p)^x\\ (2)\ lower\ cumulative\ distribution\\ \hspace{30px}P(x,k,p)={\large\displaystyle \sum_{t=0}^{x}}f(t,k,p)\\ (3)\ upper\ cumulative\ distribution\\ \hspace{30px}Q(x,k,p)={\large\displaystyle \sum_{t=x}^{\infty}}f(t,k,p)\\ (4)\ expectation(mean):\hspace{10px}k{\large\frac{1-p}{p}}\\\)

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