Bernoulli Distribution

Definition and Properties

An experiment of chance whose result has only two possibilities is called a Bernoulli trial (or Bernoulli experiment). Let p denote the probability of success in a Bernoulli trial (0 < p < 1). Then, a random variable X that assigns value 1 for a success with probability p and value 0 for a failure with probability 1 – p is called a Bernoulli random variable, and it follows the Bernoulli distribution with probability p, which is denoted by X ∼ Ber(p). The probability mass function of Ber(p) is given by

The mean of X is p, and the variance is p(1 – p). Figure 1 shows the probability mass function of Ber(.7). The horizontal axis represents values of X, and the vertical axis represents the corresponding probabilities. Thus, the height is .7 at X = 1, and .3 for X = 0. The mean of Ber(0.7) is 0.7, and the variance is .21.

Suppose that a Bernoulli trial with probability p is independently repeated for n times, and we obtain a random sample X1, X2, …, Xn. Then, the number of successes Y = X1 + X2 + … + Xn follows the binomial distribution with probability [Page 79]p and the number of trials n, which is denoted by Y Bin(n,p). Stated in the opposite way, the Bernoulli distribution is a special case of the binomial distribution in which the number of trials n is 1. The probability mass function of Bin(n, p) is given by

where

Figure 1 Probability Mass Function of the Bernoulli Distribution With p = .7

Relationship to Other Probability Distributions

The Bernoulli distribution is a basis for many probability distributions, as well as for the binomial distribution. The number of failures before observing a success t times in independent Bernoulli trials follows the negative binomial distribution with probability p and the number of successes t. The geometric distribution is a special case of the negative binomial distribution in which the number of failures is counted before observing the first success (i.e., t = 1).

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