// Home / Documentation / Gallery of Distributions / Bernoulli Distribution

# Bernoulli Distribution

## Where do you meet this distribution?

• Coin toss
• 1-dim. random walk

## Shape of Distribution

### Basic Properties

• A parameter $p$ is required.
$0
• Discrete distribution defined at $x=\{0,1\}$

### Probability

• Cumulative distribution function
$F(x)=\begin{cases}1-p\;&(x=0)\\1\;&(x=1)\end{cases}$
• Probability mass function
$f(x)=\begin{cases}1-p\;&(x=0)\\p\;&(x=1)\end{cases}$
• How to compute these on Excel.

1
2
3
4
5
6
A B
Data Description
1 Value for which you want the distribution
0.6 Value of parameter p
Formula Description (Result)
=IF(A2=0,1-A3,1) Cumulative distribution function for the terms above
=IF(A2=0,1-A3,A3) Probability mass function for the terms above

## Characteristics

### Mean – Where is the “center” of the distribution? (Definition)

• Mean of the distribution is given as $p$

### Standard Deviation – How wide does the distribution spread? (Definition)

• Variance of the distribution
$p(1-p)$

Standard Deviation is a positive square root of Variance

• How to compute this on Excel

1
2
3
4
A B
Data Description
0.6 Value of parameter p
Formula Description (Result)
=SQRT(A2*(1-A2)) Variance of the distribution for the terms above

### Skewness – Which side is the distribution distorted into? (Definition)

• Skewness
$\frac{1-2p}{\sqrt{p(1-p)}}$
• How to compute this on Excel.

1
2
3
4
A B
Data Description
0.6 Value of parameter p
Formula Description (Result)
=(1-2*A2)/(SQRT(A2*(1-A2)) Skewness of the distribution for the terms above

### Kurtosis – Sharp or Dull, consequently Fat Tail or Thin Tail (Definition)

• Kurtosis
$\frac{6p^2-6p+1}{p(1-p)}$
• How to compute this on Excel

1
2
3
4
A B
Data Description
0.6 Value of parameter p
Formula Description (Result)
=(6*A2^2-6*A2+1)/(A2*(1-A2)) Kurtosis of the distribution for the terms above

## Random Numbers

• How to generate random numbers on Excel.

1
2
3
4

A B
Data Description
0.6 Value of parameter p
Formula Description (Result)
=IF(NTRAND(100)<1-A2,0,1) 100 Bernoulli deviates based on Mersenne-Twister algorithm for which the parameters above

Note The formula in the example must be entered as an array formula. After copying the example to a blank worksheet, select the range A4:A103 starting with the formula cell. Press F2, and then press CTRL+SHIFT+ENTER.

## NtRand Functions

Not supported yet