A random variable with two possible values, 0 and 1, is called a Bernoulli variable, its discrete probability distribution (i.e. probability mass function) is called a Bernoulli distribution, and any experiment with a binary outcome is called a Bernoulli trial

Probability Mass Function

For 𝑥 = 0 or 1

• $P(X=x)=p^x\ast (1-p{)}^{(}1-x)$
• $P(X=x)=px+(1-p)(1-x)$

Expectation

the expected value of a Bernoulli Variable 𝑋 with a probability of success 𝑃(𝑋=1) = 𝑝:

𝐄(𝑋) = 𝑝

proof

• 𝐄(𝑋) = 𝛴_𝑥∈𝑋[𝑥𝑃(𝑋=𝑥)]
• 𝐄(𝑋) = (0)𝑃(𝑋=0) + (1)𝑃(𝑋=1)
• 𝐄(𝑋) = (0)(1−𝑝) + (1)(𝑝)
• 𝐄(𝑋) = 𝑝

Variance

the variance of a Bernoulli Variable 𝑋 with a probability of success 𝑃(𝑋=1) = 𝑝:

𝑉𝑎𝑟(𝑋) = 𝑝(1−𝑝)

proof

• 𝑉𝑎𝑟(𝑋) = 𝐄[(𝑋 − 𝜇)²]
• 𝑉𝑎𝑟(𝑋) = 𝐄[(𝑋 − 𝐄(𝑋))²]
• 𝑉𝑎𝑟(𝑋) = 𝛴_𝑥∈𝑋 [(𝑥−𝐄(𝑋))²𝑃(𝑋=𝑥)]
• 𝑉𝑎𝑟(𝑋) = 𝛴_x∈𝑋[(𝑥−𝑝)²𝑃(𝑋=𝑥)]
• 𝑉𝑎𝑟(𝑋) = (0−𝑝)²𝑃(𝑋=0) + (1−𝑝)²𝑃(𝑋=1)
• 𝑉𝑎𝑟(𝑋) = (0−𝑝)²(1−𝑝) + (1−𝑝)²𝑝
• 𝑉𝑎𝑟(𝑋) = 𝑝²(1−𝑝) + (1−𝑝)²𝑝
• 𝑉𝑎𝑟(𝑋) = (1−𝑝)(𝑝² + (1−𝑝)𝑝)
• 𝑉𝑎𝑟(𝑋) = (1−𝑝)(𝑝² + 𝑝 − 𝑝²)
• 𝑉𝑎𝑟(𝑋) = (1−𝑝)(𝑝)
• 𝑉𝑎𝑟(𝑋) = 𝑝(1−𝑝)

Moment-Generating Function

• 𝑀_𝑋(𝑡) = 1 - 𝑝 + 𝑒^𝑡𝑝

See: Moment-Generating Function - Bernoulli Distribution

Other Distributions Using Bernoulli Distribution

The Bernoulli Distribution is the simplest discrete distribution, and it is the building block for other more complicated discrete distributions