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Statistics : Probability Distribution Tutorials

Poisson Distribution

  • STATISTICS
Poisson Distribution Poisson distribution helps to describe the probability of occurrence of a number of events in some given time interval or in a specified region. The time interval may be of any length, such as a minutes, a day, a week etc. Definition of Poisson Distribution A discrete random variable $X$ is said to have Poisson distribution with parameter $\lambda$ if its probability mass function is $$ \begin{equation*} P(X=x)= \left\{ \begin{array}{ll} \frac{e^{-\lambda}\lambda^x}{x!
6 min Read 4

Weibull Distribution

  • STATISTICS
Weibull Distribution A continuous random variable $X$ is said to have a Weibull distribution with three parameters $\mu$, $\alpha$ and $\beta$ if the random variable $Y =\big(\frac{X-\mu}{\beta}\big)^\alpha$ has the exponential distribution with p.d.f. $$ \begin{equation*} f(y) = e^{-y}, y>0. \end{equation*} $$ The probability density function of Weibull random variable $X$ is $$ \begin{equation*} f(x;\alpha, \beta)=\left\{ \begin{array}{ll} \frac{\alpha}{\beta} \big(\frac{x-\mu}{\beta}\big)^{\alpha-1}e^{-\big(\frac{x-\mu}{\beta}\big)^\alpha}, & \hbox{$x>\mu$, $\alpha, \beta>0$;} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{equation*} $$
5 min Read 3

Bernoulli Distribution Example

  • STATISTICS
Bernoulli Distribution In this tutorial, we will discuss example on Bernoulli’s distribution. The discrete random variable $X$ is said to have Bernoulli distribution if its probability mass function is given by $$ \begin{equation*} P(X=x) = p^x q^{1-x}, \; x=0,1; 0<p,q<1; q=1-p. \end{equation*} $$ Example The battery from a lot of batteries has 85% chances of non-defective. We select a batter at random from a lot. a. What is the probability distribution of non-defective battery?
2 min Read 0

Binomial Distribution Examples

  • STATISTICS
Binomial Distribution In this tutorial, we will provide you step by step solution to some numerical examples on Binomial distribution to make sure you understand the Binomial distribution clearly and correctly. Upon successful completion of this tutorial, you will be able to understand how to calculate binomial probabilities. Definition of Binomial Distribution A discrete random variable $X$ is said to have Binomial distribution with parameters $n$ and $p$ if the probability mass function of $X$ is $$ \begin{eqnarray*} P(X=x) & =& \binom{n}{x} p^x q^{n-x},\\ & & \qquad \; x = 0,1,2, \cdots, n; \;\\ & & 0 \leq p \leq 1, q = 1-p \end{eqnarray*} $$ where,
4 min Read 2

Continuous Uniform Distribution Examples

  • STATISTICS
Continuous Uniform Distribution The continuous uniform distribution is the simplest probability distribution where all the values belonging to its support have the same probability density. It is also known as rectangular distribution. This tutorial will help you understand how to solve the numerical examples based on continuous uniform distribution. Definition of Uniform Distribution A continuous random variable $X$ is said to have a Uniform distribution (or rectangular distribution) with parameters $\alpha$ and $\beta$ if its p.
3 min Read 2

Discrete Uniform Distribution Examples

  • STATISTICS
Discrete uniform distribution In this tutorial we will discuss some examples on discrete uniform distribution and learn how to compute mean of uniform distribution, variance of uniform distribution and probabilities related to uniform distribution. Definition of Discrete Uniform Distribution A discrete random variable $X$ is said to have a uniform distribution if its probability mass function (pmf) is given by $$ \begin{aligned} P(X=x)&=\frac{1}{N},\;\; x=1,2, \cdots, N. \end{aligned} $$ The expected value of discrete uniform random variable is $E(X) =\dfrac{N+1}{2}$.
3 min Read 2

Exponential Distribution Examples

  • STATISTICS
Exponential Distribution In this tutorial, we will provide you step by step solution to some numerical examples on exponential distribution to make sure you understand the exponential distribution clearly and correctly. Definition of Exponential Distribution 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.
4 min Read 2

Geometric Distribution Examples

  • STATISTICS
Geometric Distribution In this tutorial, we will provide you step by step solution to some numerical examples on geometric distribution to make sure you understand the geometric distribution clearly and correctly. Definition of Geometric Distribution A discrete random variable $X$ is said to have geometric distribution with parameter $p$ if its probability mass function is given by $$ \begin{equation*} P(X=x) =\left\{ \begin{array}{ll} q^x p, & \hbox{$x=0,1,2,\ldots$} \\ & \hbox{$0<p<1$, $q=1-p$} \\ 0, & \hbox{Otherwise.
3 min Read 2

Laplace Distribution Examples

  • STATISTICS
Laplace Distribution A continuous random variable $X$ is said to have a Laplace distribution, if its p.d.f. is given by $$ \begin{equation*} f(x;\mu, \lambda)=\left\{ \begin{array}{ll} \frac{1}{2\lambda}e^{-\frac{|x-\mu|}{\lambda}}, & \hbox{$-\infty < x< \infty$;} \\ & \hbox{$-\infty < \mu < \infty$, $\lambda >0$;} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{equation*} $$ Distribution Function Distribution function of $L(\mu,\lambda)$ distribution is $$ \begin{equation*} F(x) = \left\{ \begin{array}{ll} \frac{1}{2}e^{\frac{(x-\mu)}{\lambda}}, & \hbox{$x< \mu$;} \\ &\\ 1-\frac{1}{2}e^{-\frac{(x-\mu)}{\lambda} }, & \hbox{$x\geq \mu$;} \end{array} \right.
2 min Read 1

Log-normal Distribution

  • STATISTICS
Definition of Log-normal distribution The continuous random variable $X$ has a log-normal distribution if the random variable $Y=\ln (X)$ has a normal distribution with mean $\mu$ and standard deviation $\sigma$. The probability density function of $X$ is $$ \begin{aligned} f(x) & = \frac{1}{\sqrt{2\pi}\sigma x}e^{-\frac{1}{2\sigma^2}(\ln x -\mu)^2},x\geq 0 \end{aligned} $$ In Log-normal distribution $\mu$ is called location parameter, since it locates the curve of the distribution, and $\sigma$ is called scale parameter, since the shape of the curve depends on the value of $\sigma$.
3 min Read 2

Log-normal Distribution Examples

  • STATISTICS
Log-normal distribution Normal distribution is not suitable when the data are highly skewed or data contains outliers. In such a situation, log-normal distribution is often a good choice. If $Y\sim N(\mu,\sigma^2)$ then $X= e^Y$ follows a log-normal distribution with parameter $\mu$ and $\sigma^2$. Definition of Log-normal Distribution The continuous random variable $X$ has a Log-normal Distribution if the random variable $Y=\ln (X)$ has a normal distribution with mean $\mu$ and standard deviation $\sigma$.
3 min Read 1

Negative Binomial Distribution Examples

  • STATISTICS
Negative Binomial Distribution In this tutorial, we will provide you step by step solution to some numerical examples on negative binomial distribution to make sure you understand the negative binomial distribution clearly and correctly. Definition of Negative Binomial Distribution A discrete random variable $X$ is said to have negative binomial distribution if its p.m.f. is given by $$ \begin{aligned} P(X=x)&= \binom{x+r-1}{r-1} p^{r} q^{x},\\ & \quad \quad x=0,1,2,\ldots; r=1,2,\ldots\\ & \quad\quad \qquad 0<p, q<1, p+q=1.
4 min Read 3