Properties of Expected Value / Mean

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• 𝐄[𝑋 + 𝑌] = 𝐄[𝑋] + 𝐄[𝑌]
• 𝐄[𝑎𝑋] = 𝑎𝐄[𝑋]
• 𝐄[𝑐] = 𝑐
• 𝐄[𝑎𝑋 + 𝑏𝑌 + 𝑐] = 𝑎𝐄[𝑋] + 𝑏𝐄[𝑌] + 𝑐
• 𝐄[𝑋𝑌] = 𝐄[𝑋]𝐄[𝑌] # for independent 𝑋 and 𝑌
• 𝐄[𝑋𝑌] = 𝐶𝑜𝑣(𝑋,𝑌) + 𝐄[𝑋]𝐄[𝑌] # for non-independent 𝑋 and 𝑌, 𝐶𝑜𝑣(𝑋,𝑌) is the covariance
• 𝐄[𝑋²] = 𝑉𝑎𝑟(𝑋) + 𝐄[𝑋]²# derived from the second central moment

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Properties of Variance and Standard Deviation

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• 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, …, 𝑋_𝑛}:

• $Var[X_1{X}_2...X_n]={\prod }_{i=1}^n\left(Var[X_i]+\mathbf{E}[X_i{]}^2\right)-{\prod }_{i=1}^n\mathbf{E}[X_i{]}^2$ # see here

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Properties of Covariance

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• 𝐶𝑜𝑣(𝑋,𝑌) = 𝐄[(𝑋-𝐄[𝑋])(𝑌-𝐄[𝑌])] = 𝐄[𝑋𝑌] - 𝐄[𝑋]𝐄[𝑌]
• 𝐶𝑜𝑣(𝑋,𝑌) = 𝐄[(𝑋-𝜇_𝑋)(𝑌-𝜇_𝑌)] = 𝐄[𝑋𝑌] - 𝜇_𝑋𝜇_𝑌
• 𝐶𝑜𝑣(𝑋,𝑌) = 𝐶𝑜𝑣(𝑌,𝑋)
• 𝐶𝑜𝑣(𝑎𝑋 + 𝑏, 𝑐𝑌 + 𝑑) = 𝑎𝑐𝐶𝑜𝑣(𝑋,𝑌)
• 𝐶𝑜𝑣(𝑎𝑋 + 𝑏𝑌, 𝑐𝑍 + 𝑑𝑊) = 𝑎𝑐𝐶𝑜𝑣(𝑋,𝑍) + 𝑎𝑑𝐶𝑜𝑣(𝑋,𝑊) + 𝑏𝑐𝐶𝑜𝑣(𝑌,𝑍) + 𝑏𝑑𝐶𝑜𝑣(𝑌,𝑊)
• 𝐶𝑜𝑣(𝑋,𝑌) = 0 # for independent 𝑋 and 𝑌

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Properties of Correlation

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• 𝜌(𝑋,𝑌) = 𝜌(𝑌,𝑋)
• 𝜌(𝑎𝑋 + 𝑏, 𝑐𝑌 + 𝑑) = 𝜌(𝑋,𝑌)

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Resources

• Continuous Probability Distribution - Calculating Statistics