scope=“col” width=“225px” scope = “row” height=“75px” | Distribution |
Parameter(s) |
Mean |
Variance |
Distribution Function f(x) |
When to use |
|---|---|---|---|---|---|
scope = “row” height=“75px” | Bernoulli |
p probability of success. |
p |
p(1−p) |
px(1−p)1−x |
One trial with probability of success p and probability of failure 1−p. |
scope = “row” height=“75px” | Binomial |
nnumber of Bernoulli Trials,pprobability of success. |
np |
np(1−p) |
\displaystyle\binom nx p^x(1-p)^{n-x} |
n trials with probability of success p and probability of failure 1-p. |
scope = “row” height=“75px” | Poisson |
\begin{align} &\lambda \text{ the average rate} \qquad \textit{or} \\ &\lambda = np \text{ when approximating the} \\ &\text{ the binomial distribution.} \end{align} |
\lambda |
\lambda |
\dfrac{\lambda^x e^{-\lambda}}{x} |
When given an average rate or to approximating the binomial distribution when n is large and p is small. |
scope = “row” height=“75px” | Normal |
\begin{align} &\mu \text{ the mean,} \\ &\sigma^2 \text{ the variance.} \end{align} |
\mu |
\sigma ^2 |
\dfrac{1}{ \sigma \sqrt{2 \pi}} \mathrm{exp}\left(-\frac{1}{2} \left( \frac{(x-\mu)}{ \sigma} \right)^2 \right) |
N/A |
scope = “row” height=“75px” | Standard Normal |
N/A |
0 |
1 |
\dfrac{1}{\sqrt{2 \pi} }\mathrm{exp}\left(-\frac{1}{2}x^2\right) |
When population mean and standard deviation are given. |
scope = “row” height=“75px” | Student - t |
v the degrees of freedom. |
0 |
\dfrac{v}{v-2} |
Very complicated and beyond the scope of this wiki |
N/A |