More Discrete Distributions
Stub: Chapter coming soon!
Geometric Random Variable
| Notation: | $X \sim \Geo(p)$ |
|---|---|
| Description: | Number of experiments until a success. Assumes independent experiments each with probability of success $p$. |
| Parameters: | $p \in [0, 1]$, the probability that a single experiment gives a "success". |
| Support: | $x \in \{1, \dots, \infty\}$ |
| PMF equation: | $$\p(X=x) = (1-p)^{x-1} p$$ |
| Expectation: | $\E[X] = \frac{1}{p}$ |
| Variance: | $\var(X) = \frac{1-p}{p^2}$ |
| PMF graph: |
Parameter $p$:
Negative Binomial Random Variable
| Notation: | $X \sim \NegBin(r, p)$ |
|---|---|
| Description: | Number of experiments until $r$ successes. Assumes each experiment is independent with probability of success $p$. |
| Parameters: | $r > 0$, the number of success we are waiting for. $p \in [0, 1]$, the probability that a single experiment gives a "success". |
| Support: | $x \in \{r, \dots, \infty\}$ |
| PMF equation: | $$\p(X=x) = {x - 1 \choose r - 1}p^r(1-p)^{x-r}$$ |
| Expectation: | $\E[X] = \frac{r}{p}$ |
| Variance: | $\var(X) = \frac{r \cdot (1-p)}{p^2}$ |
| PMF graph: |
Parameter $r$:
Parameter $p$: