Definition 3.3.1 (Bernoulli distribution). An r.v. X is said to have the Bernoulli distribution with parameter p if P(X = 1) = p and P(X = 0) = 1 − p, where 0 < p < 1. We write this as X ~ Bern(p). The symbol ~ is read “is distributed as”. Any r.v. whose possible values are 0 and 1 has a Bern(p) distribution, with p the probability of the r.v. equaling 1. This number p in Bern(p) is called the parameter of the distribution; it determines which specific Bernoulli distribution we have. Thus there is not just one Bernoulli distribution, but rather a family of Bernoulli distributions, indexed by p. For example, if X ~ Bern(1/3), it would be correct but incomplete to say “X is Bernoulli”; to fully specify the distribution of X, we should both say its name (Bernoulli) and its parameter value (1/3), which is the point of the notation X ~ Bern(1/3). Any event has a Bernoulli r.v. that is naturally associated with it, equal to 1 if the event happens and 0 otherwise. This is called the indicator random variable of the event; we will see that such r.v.s are extremely useful.

Definition 3.3.2 (Indicator random variable). The indicator random variable of an event A is the r.v. which equals 1 if A occurs and 0 otherwise. We will denote the indicator r.v. of A by IA or I(A). Note that IA ~ Bern(p) with p = P(A).

These are the definitions in Introduction to Probability book, however, as been stated in the Definition of Bernoulli distribution that 0 < p < 1 so that means it exclude 1 and 0 but in the Indicator random variable definition it states that when some event A occur P(A) = 1 and P(A) = p ( the small p here is the parameter) which leads us that p = 1 which contradicts what I understand from the first definition that 0 < p < 1 , I think it suppose to be p is bigger or equal to 0 and less or equal to 1, correct me if I said something wrong and explain to me what exactly I am missing?


1 Answer 1


The first definition describes the Bernoulli random variable $X$ that follows the Bernoulli distribution. It has the following probability mass function

$$ f_X(x) = \begin{cases} p & \text{if }x=1, \\[6pt] 1-p & \text {if }x=0.\end{cases} $$

The second quote says that you can use Bernoulli distribution for random variables defined in terms of indicator random variable. In such a case, $IA$ is equal to $1$ if $A$ occurs and $0$ otherwise. The Bernoulli probability mass function is

$$ f_{IA}(Ia) = \begin{cases} p & \text{if }Ia=1, \\[6pt] 1-p & \text {if }Ia=0.\end{cases} $$

$P(IA=1) = p$ saying it differently.

The quotes do not say that the probabilities could be $1$ or $0$. In fact, probability equal to $1$ would mean that the random variable is constant, so it's degenerate, it wouldn't follow Bernoulli distribution because there would be only one (not two) event possible for it.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.