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Statistics : Probability Distribution Tutorials

Discrete Uniform Distribution Guide

  • STATISTICS
Discrete Uniform Distribution A discrete random variable has a discrete 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. In this article, I will walk you through discrete uniform distribution and proof related to discrete uniform. You can use discrete uniform distribution Calculator. Discrete Uniform Distribution Definition A discrete random variable $X$ is said to have a uniform distribution if its probability mass function (pmf) is given by
4 min Read 2

Expected value and variance of probability distribution

  • STATISTICS
Example Find the expected value and variance of $X$ for the following probability distribution. $x$ 1 2 3 4 $P(X=x)$ 0.0625 0.1875 0.3125 0.4375 Solution The Expected value of $X$ is $$ \begin{aligned} E(X) &= \sum_x x*P(X=x) \end{aligned} $$ The expected value of $X^2$ is $$ \begin{aligned} E(X^2) &= \sum_x x^2*P(X=x) \end{aligned} $$ The variance of $X$ is $$ \begin{aligned} V(X) &= E[X-E(X)]^2 \\ &= E(X^2) - [E(X)]^2 \end{aligned} $$
1 min Read 0

Exponential Distribution

  • STATISTICS
Exponential distribution Exponential Distribution Definition A continuous random variable $X$ is said to have an exponential distribution with parameter $\theta$ if its p.d.f. is given by $$ \begin{equation*} f(x)=\left\{ \begin{array}{ll} \theta e^{-\theta x}, & \hbox{$x\geq 0;\theta>0$;} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{equation*} $$ In notation, it can be written as $X\sim \exp(\theta)$. Graph of pdf of exponential distribution Following is the graph of probability density function of exponential distribution with parameter $\theta=0.
5 min Read 3

Gamma Distribution

  • STATISTICS
Gamma Distribution Gamma distribution is used to model a continuous random variable which takes positive values. Gamma distribution is widely used in science and engineering to model a skewed distribution. In this tutorial, we are going to discuss various important statistical properties of gamma distribution like graph of gamma distribution for various parameter combination, derivation of mean, variance, harmonic mean, mode, moment generating function and cumulant generating function. Gamma Distribution Definition A continuous random variable $X$ is said to have an gamma distribution with parameters $\alpha$ and $\beta$ if its p.
5 min Read 4

Gamma Distribution Examples

  • STATISTICS
Definition of Gamma Distribution A continuous random variable $X$ is said to have an gamma distribution with parameters $\alpha$ and $\beta$ if its p.d.f. is given by $$ \begin{equation*} f(x)=\left\{ \begin{array}{ll} \frac{\alpha^\beta}{\Gamma(\beta)}x^{\beta -1}e^{-\alpha x}, & \hbox{$x>0;\alpha, \beta >0$;} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{equation*} $$ Mean of Gamma Distribution The mean or expected value of gamma random variable is $E(X)= \dfrac{\beta}{\alpha}$ Variance of Gamma distribution The variance of gamma random variable is $V(X) = \dfrac{\beta}{\alpha^2}$.
1 min Read 0

Geometric Distribution

  • STATISTICS
Introduction Geometric distribution is used to model the situation where we are interested in finding the probability of number failures before first success or number of trials (attempts) to get first success in a repeated mutually independent Beronulli’s trials, each with probability of success $p$. There are two different definitions of geometric distributions one based on number of failures before first success and other based on number of trials (attempts) to get first success.
9 min Read 7

Hypergeometric Distribution

  • STATISTICS
Hypergeometric Experiment A hypergeometric experiment is an experiment which satisfies each of the following conditions: The population or set to be sampled consists of $N$ individuals, objects, or elements (a finite population). Each object can be characterized as a “defective” or “non-defective”, and there are $M$ defectives in the population. A sample of $n$ individuals is drawn in such a way that each subset of size $n$ is equally likely to be chosen.
3 min Read 2

Laplace Distribution

  • STATISTICS
Laplace Distribution A continuous random variable $X$ is said to have a Laplace distribution (Double exponential distribution or bilateral exponential distribution), if its p.d.f. is given by $$ \begin{equation*} f(x;\mu, \lambda)=\left\{ \begin{array}{ll} \frac{1}{2\lambda}e^{-\frac{|x-\mu|}{\lambda}}, & \hbox{$-\infty < x< \infty$;} \\ & \hbox{$-\infty < \mu < \infty$, $\lambda >0$;} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{equation*} $$ In Laplace distribution $\mu$ is called location parameter, since it locates the curve of the distribution, and $\lambda$ is called scale parameter, since the shape of the curve depends on the value of $\lambda$.
8 min Read 6

Negative Binomial Distribution

  • STATISTICS
Introduction A negative binomial distribution is based on an experiment which satisfies the following three conditions: An experiment consists of q sequence of independent Bernoulli’s trials (i.e., each trial can result in a success ($S$) or a failure ($F$)), The probability of success is constant from trial to trial, i.e., $P(S\text{ on trial } i) = p$, for $i=1,2,3,\cdots$, The experiment continues until a total of $r$ successes have been observed, where $r$ is a specified positive integer.
8 min Read 6

Normal Approximation to Poisson Distribution

  • STATISTICS
Normal approximation to Poisson distribution In this tutorial we will discuss some numerical examples on Poisson distribution where normal approximation is applicable. For large value of the $\lambda$ (mean of Poisson variate), the Poisson distribution can be well approximated by a normal distribution with the same mean and variance. Let $X$ be a Poisson distributed random variable with mean $\lambda$. The mean of $X$ is $\mu=E(X) = \lambda$ and variance of $X$ is $\sigma^2=V(X)=\lambda$.
3 min Read 2

Normal Distribution

  • STATISTICS
Normal Distribution Normal distribution is one of the most fundamental distribution in Statistics. It is also known as Gaussian distribution. Definition A continuous random variable $X$ is said to have a normal distribution with parameters $\mu$ and $\sigma^2$ if its probability density function is given by $$ \begin{equation*} f(x;\mu, \sigma^2) = \left\{ \begin{array}{ll} \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2\sigma^2}(x-\mu)^2}, & \hbox{$-\infty< x<\infty$,} \\ & \hbox{$-\infty<\mu<\infty$, $\sigma^2>0$;} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{equation*} $$ where $e= 2.
10 min Read 8

Poisson approximation to binomial distribution

  • STATISTICS
Poisson 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 Poisson approximation to binomial distribution is that the sample size $n$ is sufficiently large and $p$ is sufficiently small such that $\lambda=np$ (finite). For sufficiently large $n$ and small $p$, $X\sim P(\lambda)$.
3 min Read 2