The Bernoulli distribution is one of the simplest probability models in statistics, but it is also one of the most important. It describes a single trial that has exactly two possible outcomes, usually called success and failure. The probability of success is written as p, and the probability of failure is 1 - p.
This model matters because many real situations can be simplified into a yes or no outcome, such as pass or fail, defective or not defective, and heads or tails.
Key Facts
- A Bernoulli random variable X has only two possible values: X = 1 for success and X = 0 for failure.
- P(X = 1) = p and P(X = 0) = 1 - p, where 0 <= p <= 1.
- The probability mass function is P(X = x) = p^x(1 - p)^(1 - x) for x = 0 or x = 1.
- The mean of a Bernoulli distribution is E(X) = p.
- The variance of a Bernoulli distribution is Var(X) = p(1 - p).
- A binomial random variable counts successes in n independent Bernoulli trials: Y = X1 + X2 + ... + Xn.
Vocabulary
- Bernoulli trial
- A Bernoulli trial is one experiment with exactly two possible outcomes, success or failure.
- Success probability
- The success probability p is the chance that the Bernoulli trial results in the outcome labeled success.
- Random variable
- A random variable is a rule that assigns a numerical value to the outcome of a random experiment.
- Expected value
- The expected value is the long-run average value of a random variable over many repetitions.
- Variance
- Variance measures how spread out the values of a random variable are around its expected value.
Common Mistakes to Avoid
- Treating p as the probability of failure is wrong because p is usually defined as the probability of success, while failure has probability 1 - p.
- Using Bernoulli for more than one trial is wrong because a Bernoulli distribution models exactly one trial, not a count across many trials.
- Forgetting that X can only be 0 or 1 is wrong because Bernoulli random variables do not take values like 2, 3, or negative numbers.
- Writing the variance as p^2 is wrong because the correct variance is p(1 - p), which depends on both success and failure probabilities.
Practice Questions
- 1 A free throw is made with probability p = 0.75. Let X = 1 if the shot is made and X = 0 if it is missed. Find P(X = 1), P(X = 0), E(X), and Var(X).
- 2 A light bulb passes inspection with probability 0.92. Model one inspected bulb as a Bernoulli random variable. What is the probability it fails inspection, and what is the variance of the random variable?
- 3 A student wants to model the number of heads in 10 coin flips using a Bernoulli distribution. Explain why this is not a Bernoulli distribution and identify the distribution that would be more appropriate.