Bernoulli Distribution
- Consider a Bernoulli process with parameter p=1/3. Let T1 be the time of the first success and let T1+T2 be the time of the second success. We are told that the results of the two slots that follow the first success are failures, Statistics. An American roulette wheel has 38 slots, of which 18 are black, 18 are red, and 2 are green.
- Details The Bernoulli distribution with prob = p has density p (x) = p x (1 − p) 1 − x for x = 0 o r 1. If an element of x is not 0 or 1, the result of dbern is zero, without a warning. P (x) is computed using Loader's algorithm, see the reference below.
The probability that the Bernoulli trials yield four successes followed by a failure is: P(SSSSF)=ppppq =(0.70)4(0.30)=0.0720. Our next tool is extremely powerful and very useful in science. It is the binomial probability distribution. Suppose that we plan to perform/observe n Bernoulli trials. Let X denote the total number of successes in the.
Overview
The Bernoulli distribution is a discrete probability distribution with only two possible values for the random variable. Each instance of an event with a Bernoulli distribution is called a Bernoulli trial.
Parameters
Artinya poker face. The Bernoulli distribution uses the following parameter.
Parameter | Description | Support |
---|---|---|
p | Probability of success | 0≤p≤1 |
Probability Density Function
The probability density function (pdf) of the Bernoulli distribution is
For discrete distributions, the pdf is also known as the probability mass function (pmf).
For an example, see Compute Bernoulli Distribution pdf.
Cumulative Distribution Function
The cumulative distribution function (cdf) of the Bernoulli distribution is
For an example, see Compute Bernoulli Distribution cdf.
Descriptive Statistics
The mean of the Bernoulli distribution is p.
The variance of the Bernoulli distribution is p(1 – p).
Roulette Bernoulli Equation
Examples
Compute Bernoulli Distribution pdf
The Bernoulli distribution is a special case of the binomial distribution, where N = 1
. Use binopdf
to compute the pdf of the Bernoulli distribution with the probability of success 0.75
.
Plot the pdf with bars of width 1
.
The probability density function (pdf) of the Bernoulli distribution is
For discrete distributions, the pdf is also known as the probability mass function (pmf).
For an example, see Compute Bernoulli Distribution pdf.
Cumulative Distribution Function
The cumulative distribution function (cdf) of the Bernoulli distribution is
For an example, see Compute Bernoulli Distribution cdf.
Descriptive Statistics
The mean of the Bernoulli distribution is p.
The variance of the Bernoulli distribution is p(1 – p).
Roulette Bernoulli Equation
Examples
Compute Bernoulli Distribution pdf
The Bernoulli distribution is a special case of the binomial distribution, where N = 1
. Use binopdf
to compute the pdf of the Bernoulli distribution with the probability of success 0.75
.
Plot the pdf with bars of width 1
.
Compute Bernoulli Distribution cdf
The Bernoulli distribution is a special case of the binomial distribution, where N = 1
. Use binocdf
to compute the cdf of the Bernoulli distribution with the probability of success 0.75
.
Plot the cdf.
Related Distributions
Binomial Distribution — The binomial distribution is a two-parameter discrete distribution that models the total number of successes in repeated Bernoulli trials. The Bernoulli distribution occurs as a binomial distribution with N = 1.
Geometric Distribution — The geometric distribution is a one-parameter discrete distribution that models the total number of failures before the first success in repeated Bernoulli trials.
References
[1] Abramowitz, Milton, and Irene A. Stegun, eds. Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. 9. Dover print.; [Nachdr. der Ausg. von 1972]. Dover Books on Mathematics. New York, NY: Dover Publ, 2013.
[2] Evans, Merran, Nicholas Hastings, and Brian Peacock. Statistical Distributions. 2nd ed. New York: J. Wiley, 1993.