2.5.1 Bernoulli

Question 1

What is the Bernoulli distribution?

Answer 1

A discrete probability distribution with only two possible outcomes: 0 (failure) and 1 (success).

Question 2

What is the probability of success in a Bernoulli distribution?

Answer 2

It is denoted by ( p ), where ( P(X = 1) = p ).

Question 3

What is the probability of failure in a Bernoulli distribution?

Answer 3

It is ( 1 - p ), where ( P(X = 0) = 1 - p ).

Question 4

How do we denote a Bernoulli-distributed random variable mathematically?

Answer 4

( X sim ext{Bernoulli}(p) ), meaning ( X ) follows a Bernoulli distribution with probability ( p ).

Question 5

What is the probability mass function (PMF) of a Bernoulli random variable?

Answer 5

• ( P(X = 1) = p )
• ( P(X = 0) = 1 - p )

Question 6

What is the mean (expected value) of a Bernoulli random variable?

Answer 6

( E[X] = p ), which is the probability of success.

Question 7

What is the variance of a Bernoulli random variable?

Answer 7

( ext{Var}(X) = p(1 - p) ).

Question 8

What is the Bernoulli shortcut used for?

Answer 8

It helps quickly compute the variance of a random variable that takes only two possible values.

Question 9

How do we compute the variance of a random variable that takes values ( a ) and ( b ) using the Bernoulli shortcut?

Answer 9

[
\text{Var}(X) = (b - a)^2 \cdot p(1 - p)
]