Binomial Distribution

The Bernoulli distribution is the probability distribution of a random variable which takes the value 1 with success probability of $p$ and the value 0 with failure probability of $1-p$ . It is frequently used to represent binary experiments, such as a coin toss.

The binomial distribution describes the result of $n$ independent trials with probability $p$ . It is frequently used to model the number of successes in a specific number of identical binary experiments, such as the number of heads in five coin tosses.

The negative binomial distribution is the probability distribution of the number of successes in a sequence of independent Bernoulli trials with probability $p$ before $r$ failures occur. For example this distribution could be used to model the number of heads before three tails in a sequence of coin tosses.

The geometric distribution is the probability distribution that the first success is after exactly $k$ failures in a sequence of independent Bernoulli trials with probability $p$ . For example this distribution can be used to model the number of rolls of a dice before the first six is rolled.

The Poisson distribution describes the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate $\lambda$ , and independently of the time since the last event. This distribution has been used to model events such as meteor showers and goals in a soccer match.

The chi-squared distribution arises from the sum of $k$ independent and identically distributed squared standard normal distributions. It is used frequently in hypothesis testing and in the construction of confidence intervals.

PMF

Mean

$\left\{\begin{array}{ll} (1-p) \text{ for } x = 0\\ p \qquad \text{ for } x = 1 \end{array}\right.$

$\binom{n}{x}p^{x}(1-p)^{n-x}$

$\binom{x + r -1}{x}p^{x}(1-p)^{r}$

$(1-p)^{k}p$

$\dfrac{\lambda^{x}e^{-\lambda}}{x!}$

$\left\{\begin{array}{ll} \dfrac{1}{b-a} \text{ for } x \in [a,b]\\ 0 \qquad \text{ otherwise } \end{array}\right.$

$\dfrac{1}{\sqrt{2\sigma^{2}\pi}} e^{-\dfrac{(x-\mu)^{2}}{2\sigma^{2}}}$

$\dfrac{Z}{\sqrt{U/k}} \qquad \begin{array}{ll} Z \sim N(0,1)\\ U \sim \chi_{k} \end{array}$

$\sum_{i=1}^{k}Z_{i}^{2} \qquad Z_{i} \overset{i.i.d.}{\sim} N(0,1)$

$\lambda e^{-\lambda x}$

$\dfrac{U_{1}/d_{1}}{U_{2}/d_{2}} \qquad \begin{array}{ll} U_{1} \sim \chi_{d_{1}}\\ U_{2} \sim \chi_{d_{2}} \end{array}$

$\dfrac{1}{\Gamma(k)\theta^{k}}x^{k-1}e^{-\dfrac{x}{\theta}}$

$\dfrac{\Gamma(\alpha + \beta)x^{\alpha - 1}(1-x)^{\beta - 1}}{\Gamma(\alpha)\Gamma(\beta)}$

$p$

$np$

$\dfrac{pr}{1-p}$

$\dfrac{1-p}{p}$

$\lambda$

$\dfrac{a+b}{2}$

$\mu$

$0$

$k$

$\lambda^{-1}$

$\dfrac{d_{2}}{d_{2}-2}$

$k\theta$

$\dfrac{\alpha}{\alpha + \beta}$

$\large p$ = 0.5

$\large n$ = 5

$\large p$ = 0.5

$\large r$ = 5

$\large p$ = 0.5

$\large\lambda$ = 5

$\large a$ = -5

$\large b$ = 5

$\large \mu$ = 0

$\large \sigma$ = 1

$\large k$ = 5

$\large \lambda$ = 5

$\large d_{1}$ = 5

$\large d_{2}$ = 5

$\large k$ = 5

$\large \theta$ = 5

$\large \alpha$ = 5

$\large \beta$ = 5

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