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Properties of Variance

Properties of Variance

Jan 31, 2024

properties of variance and standard deviation

  • 𝑉𝑎𝑟(𝑋 + 𝑌) = 𝐶𝑜𝑣(𝑋,𝑋) + 𝐶𝑜𝑣(𝑌,𝑌) + 𝐶𝑜𝑣(𝑋,𝑌) + 𝐶𝑜𝑣(𝑌,𝑋)

  • 𝑉𝑎𝑟(𝑋 + 𝑌) = 𝑉𝑎𝑟(𝑋) + 𝑉𝑎𝑟(𝑌) + 2𝐶𝑜𝑣(𝑋,𝑌)
  • 𝑉𝑎𝑟(𝑋 + 𝑌) = 𝑉𝑎𝑟(𝑋) + 𝑉𝑎𝑟(𝑌) + 2𝐶𝑜𝑟(𝑋,𝑌)𝑠𝑑(𝑋)𝑠𝑑(𝑌)
  • 𝑉𝑎𝑟(𝑋 + 𝑌) = 𝑉𝑎𝑟(𝑋) + 𝑉𝑎𝑟(𝑌) # when 𝑋 and 𝑌 are uncorrelated

  • 𝑉𝑎𝑟(𝑋 - 𝑌) = 𝐶𝑜𝑣(𝑋,𝑋) + 𝐶𝑜𝑣(𝑌,𝑌) - 𝐶𝑜𝑣(𝑋,𝑌) - 𝐶𝑜𝑣(𝑌,𝑋)

  • 𝑉𝑎𝑟(𝑋 - 𝑌) = 𝑉𝑎𝑟(𝑋) + 𝑉𝑎𝑟(𝑌) - 2𝐶𝑜𝑣(𝑋,𝑌)
  • 𝑉𝑎𝑟(𝑋 - 𝑌) = 𝑉𝑎𝑟(𝑋) + 𝑉𝑎𝑟(𝑌) - 2𝐶𝑜𝑟(𝑋,𝑌)𝑠𝑑(𝑋)𝑠𝑑(𝑌)
  • 𝑉𝑎𝑟(𝑋 - 𝑌) = 𝑉𝑎𝑟(𝑋) + 𝑉𝑎𝑟(𝑌) # when 𝑋 and 𝑌 are uncorrelated

  • 𝑉𝑎𝑟(𝑎𝑋) = 𝑎2𝑉𝑎𝑟(𝑋)

  • 𝑉𝑎𝑟(𝑐) = 0

together:

  • 𝑉𝑎𝑟(𝑎𝑋 + 𝑏𝑌 + 𝑐) = 𝑎2𝑉𝑎𝑟(𝑋) + 𝑏2𝑉𝑎𝑟(𝑌) + 2𝑎𝑏𝐶𝑜𝑣(𝑋,𝑌)

for independent 𝑋 and 𝑌:

  • 𝑉𝑎𝑟(𝑋 + 𝑌) = 𝑉𝑎𝑟(𝑋) + 𝑉𝑎𝑟(𝑌) # because 𝐶𝑜𝑣(𝑋,𝑌) = 0
  • 𝑉𝑎𝑟(𝑋𝑌) = 𝑉𝑎𝑟(𝑋)𝑉𝑎𝑟(𝑌) + 𝑉𝑎𝑟[𝑋]𝐄[𝑌]2 + 𝑉𝑎𝑟[𝑌]𝐄[𝑋]2 # see here

for independent {𝑋1, 𝑋2, ..., 𝑋𝑛}:

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     # see here