This distribution models the simplest random experiment: exactly one trial with two possible outcomes (success/failure, yes/no, pass/fail).
The Bernoulli Distribution is named after the mathematician Jacob Bernoulli (Bernoulli 1713) and describes a binary random variable \(X\) that can only be “success” or “failure.”
\[ \text{P}(X=1)=p, \quad \text{P}(X=0)=q=1-p \]
where \(p\) = probability of success and \(q\) = probability of failure.
Equivalent PMF form:
\[ \text{P}(X=x) = p^x(1-p)^{1-x}, \quad x \in \{0,1\} \]
\[ F(x)= \begin{cases} 0, & x < 0\\ q, & 0 \le x < 1\\ 1, & x \ge 1 \end{cases} \]
\[ \text{E}(X) = p \]
The mean value (or “expected” value) of the outcome \(X\) of the Bernoulli experiment is equal to the probability of obtaining a success. This result can be intuitively interpreted as the expected or average outcome of \(X\) when the Bernoulli experiment is repeated \(N\) times: the average of \(X\) becomes \(\frac{pN}{N} = p\).
\[ \begin{align*} \begin{cases}\text{Mo}(X) = 0 &\text{ if } q > p\\\text{Mo}(X) = 0,1 &\text{ if } q = p\\\text{Mo}(X) = 1 &\text{ if } q < p\end{cases} \end{align*} \]
When \(p=q=0.5\), both 0 and 1 are modes (bimodal tie).
\[ \begin{align*}\begin{cases}\text{Med}(X) = 0 &\text{ if } q > p\\\text{Med}(X) \in [0,1] &\text{ if } q = p\\\text{Med}(X) = 1 &\text{ if } q < p\end{cases}\end{align*} \]
At \(p=q=0.5\), any value in \([0,1]\) is a valid median because both median inequalities are satisfied: \(\text{P}(X \le m)\ge 0.5\) and \(\text{P}(X \ge m)\ge 0.5\).
\[ \text{V}(X) = p q \]
Intuition: variability is highest when outcomes are most uncertain (\(p=q=0.5\)), and it goes to zero as \(p \to 0\) or \(p \to 1\).
\[ M_X(t)=\text{E}(e^{tX})=q+pe^t \]
\[ g_1 = \frac{q-p}{\sqrt{pq}} \]
Interpretation: \(g_1>0\) when \(p<0.5\) (right-skewed toward 1), \(g_1<0\) when \(p>0.5\) (left-skewed toward 0), and \(g_1=0\) when \(p=0.5\).
\[ g_2 = \frac{1-3pq}{pq} \]
The corresponding excess kurtosis is \(\frac{1-6pq}{pq}\).
Interpretation: kurtosis is smallest at \(p=0.5\) (\(g_2=1\)), increases as outcomes become more imbalanced, and diverges as \(p \to 0\) or \(p \to 1\).
For a sample \(x_1,\dots,x_n\) with \(x_i \in \{0,1\}\), the maximum-likelihood estimator is
\[ \hat p = \bar x = \frac{1}{n}\sum_{i=1}^n x_i. \]
The Bernoulli model is the building block for many discrete models:
You can compute Bernoulli probabilities in R with dbinom using size = 1:
p_demo <- 0.35
cat("PMF values at x = 0 and x = 1:\n")
print(dbinom(c(0, 1), size = 1, prob = p_demo))
cat("\nCDF value P(X <= 0):", pbinom(0, size = 1, prob = p_demo), "\n")
cat("Random Bernoulli draws (n = 10):\n")
set.seed(123)
print(rbinom(10, size = 1, prob = p_demo))PMF values at x = 0 and x = 1:
[1] 0.65 0.35
CDF value P(X <= 0): 0.65
Random Bernoulli draws (n = 10):
[1] 0 1 0 1 1 0 0 1 0 0Suppose a quality-control test marks one product as either “pass” (=1) or “fail” (=0). If the pass probability is \(p = 0.7\), then:
p <- 0.7
probs <- c(`0` = 1 - p, `1` = p)
print(probs)
barplot(probs, col = "steelblue", ylab = "Probability",
main = "Bernoulli probabilities (p = 0.7)")
0 1
0.3 0.7