Real random variable

Variance and standard deviation

The variance of a random variable is defined as

𝜎2𝑋=Var⁡(𝑋)=⟨(𝑋−⟨𝑋⟩)2⟩=⟨𝑋−2𝜇𝑋+𝜇2⟩=⟨𝑋⟩−2𝜇⟨𝑋⟩−𝜇2=⟨𝑋2⟩−⟨𝑋⟩2

or in terms of moments

𝜎2𝑋=Var⁡(𝑋)=𝜇2−𝜇2

and standard deviation as

𝜎𝑋=√Var⁡(𝑋)

Both of these are a measure of the spread of a random variable.

Properties

  1. 0 ≥Var⁡(𝑋)
  2. Var⁡(𝑎𝑋 +𝑏) =𝑎2Var⁡(𝑋)
  3. Var⁡(𝑋 +𝑌) =Var⁡(𝑋) +Var⁡(𝑌) given 𝑋 and 𝑌 are independent

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