Covariance and Correlation (SOA Exam P – Probability – Multivariate Random Variables)
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After completing this video you should be able to:
Calculate joint moments, such as the covariance and the correlation coefficient for discrete random variables only.
Properties of Covariance:
𝐶𝑜𝑣(𝑋,𝑋)=𝑉𝑎𝑟(𝑋)
𝐶𝑜𝑣(𝑋,𝑌) can be negative
𝐶𝑜𝑣(𝑋,𝑌)=𝐶𝑜𝑣(𝑌,𝑋)
𝐶𝑜𝑣(𝑋,𝑘)=0 where 𝑘 is a constant
𝑋 and 𝑌 are independent ⇒𝐶𝑜𝑣(𝑋,𝑌)=0
𝐶𝑜𝑣(𝑎𝑋+𝑏𝑌,𝑐𝑊+𝑑𝑍)=𝑎𝑐𝐶𝑜𝑣(𝑋,𝑊)+𝑎𝑑𝐶𝑜𝑣(𝑋,𝑍)+𝑏𝑐𝐶𝑜𝑣(𝑌,𝑊)+𝑏𝑑𝐶𝑜𝑣(𝑌,𝑍)
