Statistics : Probability Distribution Tutorials

Normal Approximation to Binomial Distribution Let $X$ be a binomially distributed random variable with number of trials $n$ and probability of success $p$. The mean of $X$ is $\mu=E(X) = np$ and variance of $X$ is $\sigma^2=V(X)=np(1-p)$. The general rule of thumb to use normal approximation to binomial distribution is that the sample size $n$ is sufficiently large if $np \geq 5$ and $n(1-p)\geq 5$. For sufficiently large $n$, $X\sim N(\mu, \sigma^2)$.

Beta Type I Distribution A continuous random variable $X$ is said to have a beta type I distribution with parameters $\alpha$ and $\beta$ if its p.d.f. is given by $$ \begin{equation*} f(x)=\left\{ \begin{array}{ll} \frac{1}{B(\alpha,\beta)}x^{\alpha-1}(1-x)^{\beta-1}, & \hbox{$0\leq x\leq 1$;} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{equation*} $$ where, $B(\alpha,\beta) =\frac{\Gamma \alpha \Gamma \beta}{\Gamma (\alpha+\beta)}=\int_0^1 x^{\alpha-1}(1-x)^{\beta-1}\; dx$ is a beta function and $\Gamma \alpha$ is a gamma function. In notation, it can be written as $X\sim \beta_1(\alpha,\beta)$.

Introduction In the theory of probability and statistics, a Bernoulli trial or Bernoulli Experiment is a random experiment with exactly two mutually exclusive outcomes, “Success” and “Failure” with the probability of success remains same every time the experiment is conducted. The name Bernoulli trial or Bernoulli distribution named after a Swiss scientist Jacob Bernoulli. Bernoulli Trials Trials of random experiment are called Bernoulli trials, if they satisfy the following conditions

Definition of Beta Type I Distribution A continuous random variable $X$ is said to have a beta type I distribution with parameters $\alpha$ and $\beta$ if its p.d.f. is given by $$ \begin{equation*} f(x)=\left\{ \begin{array}{ll} \frac{1}{B(\alpha,\beta)}x^{\alpha-1}(1-x)^{\beta-1}, & \hbox{$0\leq x\leq 1$;} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{equation*} $$ where, $B(\alpha,\beta) =\frac{\Gamma \alpha \Gamma \beta}{\Gamma (\alpha+\beta)}=\int_0^1 x^{\alpha-1}(1-x)^{\beta-1}\; dx$ is a beta function and $\Gamma (\alpha)$ is a gamma function. Mean of Beta Type I Distribution The mean of beta type I distribution is $E(X) = \dfrac{\alpha}{\alpha+\beta}$.

Beta Type II Distribution A continuous random variable $X$ is said to have a beta type II distribution with parameters $\alpha$ and $\beta$ if its p.d.f. is given by $$ \begin{equation*} f(x)=\left\{ \begin{array}{ll} \frac{1}{B(\alpha,\beta)}\cdot\frac{x^{\alpha-1}}{(1+x)^{\alpha+\beta}}, & \hbox{$0\leq x\leq\infty$;} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{equation*} $$ where, $B(\alpha,\beta) =\frac{\Gamma \alpha \Gamma \beta}{\Gamma (\alpha+\beta)}=\int_0^1 x^{\alpha-1}(1-x)^{\beta-1}\; dx$ is a beta function and $\Gamma \alpha$ is a gamma function. In notation, it can be written as $X\sim \beta_2(\alpha,\beta)$.

Beta Type II Distribution A continuous random variable $X$ is said to have a beta type II distribution with parameters $\alpha$ and $\beta$ if its p.d.f. is given by $$ \begin{equation*} f(x)=\left\{ \begin{array}{ll} \frac{1}{B(\alpha,\beta)}\cdot\frac{x^{\alpha-1}}{(1+x)^{\alpha+\beta}}, & \hbox{$0\leq x\leq\infty$;} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{equation*} $$ where, $B(\alpha,\beta) =\frac{\Gamma \alpha \Gamma \beta}{\Gamma (\alpha+\beta)}=\int_0^1 x^{\alpha-1}(1-x)^{\beta-1}\; dx$ is a beta function and $\Gamma \alpha$ is a gamma function. Mean of Beta Type II Distribution The mean of beta type II distribution is $E(X) = \dfrac{\alpha}{\beta-1}$.

Introduction Binomial distribution is one of the most important discrete distribution in statistics. In this tutorial we will discuss about theory of Binomial distribution along with proof of some important results related to binomial distribution. Binomial Experiment Binomial experiment is a random experiment that has following properties: The random experiment consists of n repeated Bernoulli trial. Each trial has only two possible outcomes like success ($S$) and failure ($F$). All the trials are independent, i.

Cauchy Distribution A continuous random variable $X$ is said to follow Cauchy distribution with parameters $\mu$ and $\lambda$ if its probability density function is given by $$ \begin{equation*} f(x) =\left\{ \begin{array}{ll} \frac{\lambda}{\pi}\cdot \frac{1}{\lambda^2+(x-\mu)^2}, & \hbox{$-\infty < x< \infty$;} \\ & \hbox{$-\infty < \mu< \infty$, $\lambda>0$;} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{equation*} $$ In notation it can be written as $X\sim C(\mu, \lambda)$. The parameter $\mu$ and $\lambda$ are location and scale parameters respectively.

Cauchy Distribution A continuous random variable $X$ is said to follow Cauchy distribution with parameters $\mu$ and $\lambda$ if its probability density function is given by $$ \begin{equation*} f(x; \mu, \lambda) =\left\{ \begin{array}{ll} \frac{\lambda}{\pi}\cdot \frac{1}{\lambda^2+(x-\mu)^2}, & \hbox{$-\infty < x< \infty$;} \\ & \hbox{$-\infty < \mu< \infty$, $\lambda>0$;} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{equation*} $$ Distribution Function of Cauchy Distribution The distribution function of Cauchy distribution is $$ \begin{equation*} F(x) =\frac{1}{\pi}\tan^{-1}\bigg(\frac{x-\mu}{\lambda}\bigg) + \frac{1}{2}.

Chebyshev’s Inequality Chebyshev’s Theorem If $g(x)$ is a non-negative function and $f(x)$ be p.m.f. or p.d.f. of a random variable $X$, having finite expectation and if $k$ is any positive real constant, then $$ \begin{equation*} P[g(x)\geq k] \leq \frac{E[g(x)]}{k}\quad \text{ and }\quad P[g(x) < k] \geq 1-\frac{E[g(x)]}{k} \end{equation*} $$ Proof Discrete Case Let $X$ be a discrete random variable with p.m.f. $f(x)$. Let $g(x)$ be a non-negative function of $X$. Then $E[g(x)] = \sum g(x)f(x)$.

Chebyshev’s Inequality Let $X$ be a random variable with mean $\mu$ and finite variance $\sigma^2$. Then for any real constant $k>0$, $$ \begin{equation*} P[|X-\mu| \geq k\sigma] \leq \frac{1}{k^2}\quad \text{ and }\quad P[|X-\mu|< k\sigma] \geq 1-\frac{1}{k^2} \end{equation*} $$ OR If $\mu$ and $\sigma$ are the mean and the standard deviation of a random variable $X,$ then for any positive constant $k$, the probability is at least $1-\dfrac{1}{k^2}$ that $X$ will take on a value within $k$ standard deviations of the mean.

Continuous Uniform Distribution A random variable has a uniform distribution if each value of the random variable is equally likely and the values of the random variable are uniformly distributed throughout some specified interval. A uniform distribution is a distribution with constant probability. This tutorial will help you understand the theory and proof of theoretical results of uniform distribution. Uniform distribution is also known as rectangular distribution (see the graph below).