Covariance from a triangular density

Let (X1,X2)(X_1, X_2) have joint density f(x1,x2)=c(1x2)f(x_1, x_2) = c\,(1 - x_2) on the region 0x1x210 \le x_1 \le x_2 \le 1 (and 00 otherwise), where cc normalizes the density. Compute Cov(X1,X2)\operatorname{Cov}(X_1, X_2).

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  1. Find cc from f=1\int f=1 over 0x1x210\le x_1\le x_2\le 1 (inner limit is x2x_2, not 11): c=6c=6.
  2. Compute E[X1]=14E[X_1]=\tfrac14, E[X2]=12E[X_2]=\tfrac12, E[X1X2]=320E[X_1X_2]=\tfrac{3}{20}; then Cov=32018\operatorname{Cov}=\tfrac{3}{20}-\tfrac18.

Answer

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0.025 (± 0.001)

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Asked at: Citadel, Two Sigma

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