Given X=0, Y equals 0 or 10 with equal probability. Given X=1, Y equals 10 or 20 with equal probability. X=0 and X=1 each have probability 0.5. What is Var(Y)?

Explanation

Conditional means: 5 and 15, so E[Y]=10. Conditional variances: each is 25 (outcomes 5 away from the mean), so E[Var(Y|X)]=25. Var(E[Y|X])=0.5(25)+0.5(25)=25. Total variance via the law of total variance is 25+25=50. Check directly: Y takes 0,10,10,20 with probabilities 0.25,0.25,0.25,0.25; variance=(100+0+0+100)/4=50.

More Multivariate Random Variables questions

• A risk analyst models returns on a portfolio of two assets, each with standard deviation 5%, as bivariate normal with correlation 0.30. The …
• Two assets have return standard deviations of 10% and 20%, and their correlation is -0.25. A portfolio holds 60% in the first asset and 40% …
• Random variables X and Y have E[X] = 4, E[Y] = 10, Var(X) = 9, Var(Y) = 16 and Cov(X,Y) = 6. What is the variance of Z = 2X + 3Y?
• An analyst observes that the correlation between daily returns of two stocks is 0.80 in calm markets but the relationship becomes much stron…
• Annual returns on X and Y are bivariate normal with means of 12% and 2%, both standard deviations equal to 10%, and a correlation of 0.5. Wh…
• X takes values 1 or 2 and Y takes values 10 or 20 with joint probabilities: P(1,10)=0.30, P(1,20)=0.20, P(2,10)=0.10, P(2,20)=0.40. What is …