What is a Bernoulli random variable?

What is a Bernoulli random variable?

• A. A random variable that can take any nonnegative integer value.
• B. A Bernoulli random variable takes two values, 0 and 1, with probability p of 1.
• C. A variable that is normally distributed with mean p.
• D. A continuous variable taking values from 0 to 1.

Answer: (B)

A Bernoulli random variable models a single yes/no trial. It can take exactly two values, typically 0 and 1. The probability of the value 1 is p (with 0 ≤ p ≤ 1), and the probability of 0 is 1 − p. This is captured by P(X = 1) = p and P(X = 0) = 1 − p. It’s the simplest discrete distribution and is used for binary outcomes like a biased coin flip, where 1 means success and 0 means failure. The expected value is p and the variance is p(1 − p). Other descriptions don’t fit: a variable that can take any nonnegative integer value describes a different discrete distribution (like Poisson or geometric); a normally distributed variable with mean p is continuous and follows a bell-shaped curve, not two-point outcomes; a continuous variable from 0 to 1 describes distributions like Uniform(0,1) or Beta, not a Bernoulli.

A Bernoulli random variable models a single yes/no trial. It can take exactly two values, typically 0 and 1. The probability of the value 1 is p (with 0 ≤ p ≤ 1), and the probability of 0 is 1 − p. This is captured by P(X = 1) = p and P(X = 0) = 1 − p. It’s the simplest discrete distribution and is used for binary outcomes like a biased coin flip, where 1 means success and 0 means failure. The expected value is p and the variance is p(1 − p).

Other descriptions don’t fit: a variable that can take any nonnegative integer value describes a different discrete distribution (like Poisson or geometric); a normally distributed variable with mean p is continuous and follows a bell-shaped curve, not two-point outcomes; a continuous variable from 0 to 1 describes distributions like Uniform(0,1) or Beta, not a Bernoulli.