Discrete random variable

Bernoulli trial

A Bernoulli trial is an experiment with two possible outcomes, namely success and failure. prob For any event 𝐴 ∈F in a probability model has an associated Bernoulli random variable 1𝐴 called its indicator random variable¹, where

1𝐴:𝐴↦{1}𝐴𝑐↦{0}

We say 1𝐴 ∼Bern⁡(𝑝) where 𝑝 =ℙ(𝐴). The sum of repeated independent but identical Bernoulli trials follows a Binomial distribution.

Properties

Let 𝑋 ∼Bern⁡(𝑝) and 𝑞 =1 −𝑝

1. Expectation: 𝜇 =𝔼⁡[𝑋] =𝑝
2. Variance: 𝜎2 =Var⁡[𝑋] =𝑝𝑞
3. Moment-generating function: 𝑀𝑋 :ℝ →ℝ :𝑡 ↦𝑝𝑒𝑡 +𝑞
4. Probability-generating function: 𝑔𝑋(𝑡) =𝑝𝑡 +𝑞

We have the further properties

1. 𝑋𝑘 =𝑋 for any 𝑘 ∈ℕ

Indicator random variables

Let 𝐴,𝐵 ∈F in a probability model

1. 1𝐴𝑐 =1 −1𝐴
2. 1𝐴∩𝐵 =1𝐴1𝐵
3. 1𝐴∪𝐵 =1𝐴 +1𝐵 −1𝐴1𝐵
4. 𝔼⁡[1𝐴] =ℙ(𝐴)

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Footnotes

1. This is a special case of an indicator function. ↩