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  • STA256 Lecture 19
  • Special Distributions

Negative Binomial Distribution

• Distribution for modelling the trials to get the $r^{\text{th}}$ success.
  • $p$ is the probability of a success
  • $r$ is the last success
  • $x$ is the number of failures
• Properties:
  • Mean variance…
  • #tk
• $(\genfrac{}{}{0ex}{}{\stackrel{\text{\# of Trials}}{\overbrace{x+r-1}}}{\underset{\text{\# of successes}}{\underbrace{r-1}}})p^{\stackrel{\begin{array}{c}\text{\# of}\\ \text{successes}\end{array}}{\overbrace{r-1}}}(1-p{)}^{\stackrel{\text{\# of failures}}{\overbrace{x}}}\cdot p^1$
• $(\genfrac{}{}{0ex}{}{x+r-1}{r-1})p^{r-1+1}(1-p{)}^x$
• $(\genfrac{}{}{0ex}{}{x+r-1}{r-1})p^r(1-p{)}^x\text{ for }x=0,1,\dots$
• Example:
  • $p=0.20$
  • Find the probability that the third oil strike, comes on the fifth well drilled.
  • Striking oil is a success
  • We want the third success on the 5th trial
  • So there are $2$ failures in this combination
  • Failures: $x=2$
  • Success: $r=3$
  • Fifth Trial: $x+r$
  • Fourth Trial: $x+r-1$
  • $x+r=5$
  • $x+r-1=4$
  • $x=2$ and $r=3$
  • $P(X=x)=(\genfrac{}{}{0ex}{}{x+r-1}{r-1})p^r(1-p{)}^x$
  • $P(X=x)=(\genfrac{}{}{0ex}{}{4}{2})(0.2{)}^3(0.8{)}^2\approx 0.0307$
  • STA256 Lecture 20#^fe467b