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The Bernoulli distribution is one of the simplest probability distributions. It is based on the idea of a Bernoulli trial. This is an experiment with only two possible outcomes: "success" which occurs with probability p, and "failure", which occurs with probability 1 − p. The random variable X which equals 1 for a success and 0 for a failure has a Bernoulli(p) distribution.

Parameter Range Description
p 0 ≤ p ≤ 1 Probability of success

Probability Mass Function

P ( X = x ; p ) = p x ⁢ ( 1 − p ) 1 − x
F ( x ; p ) = { 0 if x < 0 1 − p if 0 ≤ x < 1 1 if x ≥ 1

Support

x = 0 , 1

Mean

Variance

Example p
A fair coin is tossed. Let X = 1 for heads and X = 0 for tails. 0.5000
A fair six-sided die is thrown. Let X = 1 if a 6 is thrown, and 0 otherwise. 0.1667
The probability that an LED light bulb will fail this year is 0.2. Let X = 1 if the bulb fails this year and X = 0 if the bulb continues working. 0.2000

X ∼ Bernoulli(p)

p =
Chart of the Bernoulli distribution Chart area for displaying the Bernoulli pmf, cdf, visualizations, and simulations

E(X) = , Var(X) =

The expected value can be changed directly by dragging left or right on the chart. This can also be done using the box below the chart which shows the mean ± one standard deviation. As the expected value gets closer to 0 or 1, the variance also approaches zero.

The expected value can be changed directly by dragging left or right on the chart. This can also be done using the box below the chart which shows the mean ± one standard deviation. As the expected value gets closer to 0 or 1, the variance also approaches zero.

The illustration above shows an experiment with only two possible outcomes:

❌
X = 0 indicates "failure" which occurs with probability 1 − p.
✅
X = 1 indicates "success" which occurs with probability p.

The simulation above shows a number U chose randomly from the interval [0, 1]. The light blue line shows the Bernoulli(p) random variable X which equals 1 if U falls in the green interval of length p, and 0 if U falls in the grey interval of length 1 − p. The histogram accumulates the results of each simulation.

Y = max Xᵢ ∼ Bernoulli(1 − (1 − p)ⁿ) min Xᵢ ∼ Bernoulli(pⁿ) ∑ Xᵢ ∼ Binomial(n, p)

Chart of the related distribution Chart area for displaying the related pdf, cdf, and simulation

E(Y) = , Var(Y) =

The Bernoulli(p) distribution is shown in red superimposed over the distribution of the sample maximum. Note that a maximum value of 1 means that the sample contains at least one success. As the sample size increases it can be seen that the probability of at least one success becomes increasingly likely.

The Bernoulli(p) distribution is shown in red superimposed over the distribution of the sample minimum. Note that a minimum value of 0 means that the sample contains at least one failure. As the sample size increases it can be seen that the probability of at least one failure becomes increasingly likely.

Note that the sum of n independent Bernoulli(p) random variables is equivalent to counting the total number of successes in n independent Bernoulli(p) trials. This is the definition of a binomial(n, p) random variable.