Bernoulli Distribution Calculator

Bernoulli Distribution Calculator

Bernoulli's Process Calculator can help you to calculate the mean, variance and probability for Bernoulli's distribution with parameter probability of success $p$.

Bernoulli Process Calculator
Probability of success (p):
Number of success (x):
Probability : P(X = x)
Mean : E(X)
Variance : V(X)
Standard Deviation :

How to use Bernoulli Process Calculator?

Step 1 - Enter the Probability of success

Step 2 - Enter the number of success

Step 3 - Click Bernoulli Process Calculator button

Step 4 - Calculate mean of Bernoulli distribution

Step 5 - Calculate variance of Bernoulli distribution

Step 6 - Calculate standard deviation of Bernoulli distribution

Bernoulli's Distribution Theory

A random experiment having two outcomes, viz., success or failure with respective probabilities $p$ and $q$ is called Bernoulli trial or Bernoulli experiment.

The probability distribution of the random variable $X$ representing the number of success obtained in a Bernoulli experiment is called Bernoulli distribution. Thus the random variable $X$ takes the value 0 and 1 with respective probabilities $q$ and $p$, i.e.,

$$ \begin{equation*} P(X=0) = q, \text{ and } P(X=1) = p. \end{equation*} $$

Bernoulli's Distribution Definition

The discrete random variable $X$ is said to have Bernoulli distribution if its probability mass function (p.m.f.) is given by

$$ \begin{equation*} P(X=x) = p^x q^{1-x},\; x=0,1; 0 < p < 1; q=1-p. \end{equation*} $$


  • $P(X=x)\geq 0$ for all $x$
  • $\sum_{x} P(X=x) = P(X=0) + P(X=1) = q+p =1$.

Hence $P(X=x)$ is a probability mass function (p.m.f.).

Key features of Bernoulli's Distribution

  • There are only two outcomes for a random experiment like success ($S$) and failure ($F$).
  • The outcomes are mutually exclusive.
  • The probability of success is $p$.
  • The random variable $X$ is the total number of success.

Mean of Bernoulli's Distribution

The mean of Bernoulli distribution is

$$ \begin{eqnarray*} \text{mean }= E(X) &=& \sum_{x=0}^1 x P(X=x) \\ &=& 0\times P(X=0) + 1\times P(X=1)\\ &=& 0\times q + 1\times p = p. \end{eqnarray*} $$

Variance of Bernoulli's Distribution

To find variance of Bernoulli distribution $X$, we need to find $E(X^2)$.

$$ \begin{eqnarray*} E(X^2) &=& \sum_{x=0}^1 x^2 P(X=x) \\ &=& 0\times P(X=0) + 1\times P(X=1)\\ &=& 0\times q + 1\times p\\ &=& p. \end{eqnarray*} $$

Hence, the variance of Bernoulli distribution is

$$ \begin{eqnarray*} \text{ Variance }= V(X) &=& E(X^2)-[E(x)]^2\\ &=& p-p^2 \\ &=& p(1-p)\\ &=& pq. \end{eqnarray*} $$

Thus if $X$ is Bernoulli random variable with parameter $p$, then mean is $E(X)=p$ and variance is $V(X)= pq$.

Mean $>$ Variance.

M.G.F. of Bernoulli Distribution

The m.g.f. of Bernoulli Distribution is given by

$$ \begin{eqnarray*} M_X(t) &=& E(e^{tX}) \\ &=& \sum_{x=0}^1 e^{tx}P(X=x)\\ &=& e^0 P(X=0) + e^tP(X=1)\\ &=& q+pe^t. \end{eqnarray*} $$

Thus, the moment generating function of Bernoulli's distribution is $M_X(t) = q+pe^t$.

P.G.F. of Bernoulli Distribution

The p.g.f. of Bernoulli Distribution is given by

$$ \begin{eqnarray*} P_X(t) &=& E(t^{X}) \\ &=& \sum_{x=0}^1 t^xP(X=x)\\ &=& t^0 P(X=0) + t^1P(X=1)\\ &=& q+pt. \end{eqnarray*} $$

Thus, the probability generating function of Bernoulli's distribution is $P_X(t) = q+pt$.


I hope you find above article on Bernoulli Distribution Calculator helpful and educational.

Let me know in the comments if you have any questions on Bernoulli Process Calculator and your thought on this article.

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