I need to prove this
σ(hat)^2(x,y) = (1/n) Σ (Xi - X (bar) ) Yi
using
σ(hat)^2(x,y)= (1/n) Σ (Xi - X(bar)) (Yi-Y(bar))
and
X(bar)= (1/n) Σ Xi
σ(hat)^2 (x,y) is the estimator of covariance
To prove the equation σ(hat)^2(x,y) = (1/n) Σ (Xi - X(bar)) Yi, using the given equations σ(hat)^2(x,y) = (1/n) Σ (Xi - X(bar))(Yi - Y(bar)) and X(bar) = (1/n) Σ Xi, we can follow these steps:
Step 1: Start with the equation σ(hat)^2(x,y) = (1/n) Σ (Xi - X(bar))(Yi - Y(bar))
Step 2: Expand the equation by multiplying (Xi - X(bar)) and (Yi - Y(bar))
σ(hat)^2(x,y) = (1/n) Σ (XiYi - XiY(bar) - X(bar)Yi + X(bar)Y(bar))
Step 3: Rearrange the terms
σ(hat)^2(x,y) = (1/n) [Σ (XiYi - XiY(bar)) - Σ (X(bar)Yi - X(bar)Y(bar))]
Step 4: Separate the two summations and rewrite the equation as
σ(hat)^2(x,y) = (1/n) [Σ (XiYi - XiY(bar))] - (1/n) [Σ (X(bar)Yi - X(bar)Y(bar))]
Step 5: Simplify the first summation
σ(hat)^2(x,y) = (1/n) Σ (XiYi - XiY(bar))
Step 6: Simplify the second summation
σ(hat)^2(x,y) = -(1/n) [Σ (X(bar)Yi - X(bar)Y(bar))]
Step 7: Substitute X(bar) = (1/n) Σ Xi into the equation
σ(hat)^2(x,y) = -(1/n) [Σ (X(bar)Yi - X(bar)Y(bar))]
= -(1/n) [Σ ((1/n) Σ XiYi - (1/n) Σ XiY(bar))]
Step 8: Distribute the terms
σ(hat)^2(x,y) = -(1/n) [ Σ (1/n) Σ XiYi - Σ (1/n) Σ XiY(bar)]
Step 9: Rearrange the terms
σ(hat)^2(x,y) = -(1/n^2) [ Σ Σ XiYi - Σ Σ XiY(bar)]
Step 10: Now, expand the first summation inside the brackets
σ(hat)^2(x,y) = -(1/n^2) [ Σ Σ XiYi - Σ Σ XiY(bar)]
= -(1/n^2) [ Σ Σ XiYi - Σ Xi (1/n) Σ Yi]
Step 11: Notice that Σ Xi (1/n) Σ Yi is just X(bar) times Y(bar), so we can rewrite the equation as
σ(hat)^2(x,y) = -(1/n^2) [ Σ Σ XiYi - X(bar)Y(bar)]
Step 12: Finally, we can simplify the equation by multiplying by -1/n^2 and rearranging the terms
σ(hat)^2(x,y) = (1/n) [ Σ Σ XiYi - X(bar)Y(bar)]
= (1/n) Σ (Xi - X(bar)) Y(bar)
= (1/n) Σ (Xi - X(bar)) Yi
Therefore, we have proved that σ(hat)^2(x,y) = (1/n) Σ (Xi - X(bar)) Yi using the given equations.