# Introduction to Bernoulli Distribution

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## What is a Bernoulli Distribution?

It is a statistical technique named after the Swiss Scientist, Jacob Bernoulli. Bernoulli Distribution is a probability distribution of a random variable. There are only two possible outcomes to this distribution: namely success, and failure. Bernoulli distribution is used for determining the solutions in cases where the two terms are yes or no, true or false, success or failure. Assuming the probability of success is denoted by ‘p,’ therefore the probability of failure is ‘1-p’. In other words, if the random variable takes the value of 1 for the probability of success, then the probability of failure automatically takes the value of 0. This random variable is termed as Bernoulli random variable and the probability distribution of this variable is known as ‘Bernoulli Distribution.’ For further explanations and examples visit the Bernoulli distribution assignment help section.

Unlike other binomial distribution, in the case of Bernoulli distribution, only a single experiment is conducted. In other words, n=1 for a Bernoulli Distribution. Visit the Bernoulli distribution homework help for self assessment.

### Important points: Bernoulli Distribution

Additional explanation are included in the Bernoulli Distribution assignment help section along with practical exercises in the homework help section.

In a nutshell, the important points of Bernoulli Distribution can be listed as below. Let us assume, for this discussion, that the Bernoulli variable is ‘x.’

- Probability of X to be zero is the same as 1 – Probability X=1

i.e. P(x=0) = 1 – P(x=1) or {p=(1-p) = q}

- Let us assume that ‘f’ represents the probability mass function where the number of outcomes is denoted by ‘k’

f(k:p) = p if k=1 and

f(k:p) = (1 – p) if k =0

Mathematically it can also be expressed as

f(k:p) = p ^{k} (1-p) ^{(1-k) } for k ∈ {0,1}

- The Bernoulli distributions for 0 ≤ p ≤ 1 from an exponential family distributions.

**Characteristics of a Bernoulli Distribution** **: Mean, Variance and Skewness ** ** **

This section aims to introduce the measures of central tendency and measures of distribution for a Bernoulli distribution. This is a crucial guide and must be kept in mind when approaching the problems on Bernoulli Distribution.

*Mean*

Based on the above notations we can say that P(x=1) = p

Likewise P(x=0) = q

E(x) = P(x=1)*1 + P(x=0)*0 = p*1 + q*0 = p

The expected value of a Bernoulli random variable ‘x’ can be said to be ‘p.’

E(x) = p.

*Variance*

Variance of the Bernoulli random variable can be calculated as below

E(x^{2}) = P(x=1) * 1^{2}+ P(x=0) * 0^{2} = p*1^{2} + q * 0^{2} = p

Based on this information we can calculated variance as

Var(X) = E(x^{2}) – E(x)^{2 } = p – p^{2} = p(1-p) = pq

- Skewness

Skewness is the measure of the asymmetry of a probability distribution. A distribution may be positively skewed or negatively skewed.

The skewness of a Bernoulli distribution can be represented as :

### Bernoulli Distribution and Binomial Distribution

We can say that a Binomial distribution is the sum total of identically distributed independent Bernoulli variables. For instance, consider the case when you flip a coin. The outcome can be only one of the two terms: heads or tails. These two terms are mutually exclusive, i.e.; the result is either ‘heads’ or ‘tails.’ If the coin is flipped 5 times (k as per the above notations). The result of each toss is an independent variable and is an example of Bernoulli Distribution. When the outcomes of all five coin tosses are considered, it can be termed as Binomial Distribution.

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