Negative Binomial Distribution - lec. 12

ucla | MATH 170E | 2023-02-08T17:35

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Table of Contents

• Definitions
• Big Ideas
  • Negative Binomial R.V.
• Resources

Definitions

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Big Ideas

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Negative Binomial R.V.

Probability of $n^{th}$ success on $r^{th}$ trial

• independent, identical Bernoulli trials with probability $p\in 0,1$ of success
• let $r\ge 1$ and let $X$ be the d.r.v. the first trial on which we first achieve the $r^{th}$ success takes values from $S=r,r+1,r+2,\dots$

$X\sim \text{Negative Binomial($r,p$)}$

• PMF

$p_X(x)=(\genfrac{}{}{0ex}{}{x-1}{r-1})p^r(1-p{)}^{x-r}x\in S$

• CDF

$\text{Missing end{cases}}$

• Lemma for finding sum

${\left(\frac{1}{1-s}\right)}^r=\sum _{x=r}^{\text{infin}}(\genfrac{}{}{0ex}{}{x-1}{r-1})s^{x-r}$

• MGF

$M_X(t)={\left(\frac{p{e}^t}{1-(1-p)e^t}\right)}^{r}t<-\log (1-p)$

• mean and variance

$\mathbb E[X]=\frac rp\space$

Resources

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**SUMMARY
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