In this topic you compute covariance and correlation for two random variables. Covariance is E[XY] minus E[X]E[Y]. A positive value means large values of one tend to come with large values of the other. Correlation rescales covariance to a number between -1 and 1. Recognition cue: the problem involves two random variables at once, for example the variance of a sum or a two-asset portfolio. You will practice: covariance definition, bilinearity and shift rules, variance of sums, correlation and independence.
A student's daily study hours X and coffee cups Y satisfy E[X] = 2, E[Y] = 3, and E[XY] = 10. Find Cov(X, Y).
Answer:
Apply the computing formula for covariance directly.
The positive covariance means more study hours tend to come with more coffee.
A commute has two legs with durations X and Y (minutes), where Var(X) = 4, Var(Y) = 9, and Cov(X, Y) = 2. Find Var(X + Y), the variance of the total commute time.
Answer:
Use the variance formula for a sum.
The positive covariance, from traffic delaying both legs, adds variance beyond the independent case.
Two exam scores satisfy Cov(X, Y) = 6, Var(X) = 9, Var(Y) = 16. Find the correlation ρ(X, Y).
Answer:
Take square roots of the variances to get the standard deviations.
This lies safely inside the interval from -1 to 1.
Two independent noise sources have Var(X) = 5 and Var(Y) = 3. Find Var(X − Y).
Answer:
Independence makes the covariance zero.
Variances add even for a difference. They never subtract.
Two food trucks on opposite sides of a city have daily sales X and Y. X and Y are independent. Find Cov(X, Y).
Answer:
Independent random variables have zero covariance.
The noon temperature X at a weather station has Var(X) = 6. Find Cov(X, X).
Answer:
The covariance of a variable with itself is its variance.
A kitchen scale reads exactly 100 grams too high, so a dough's recorded weight is W = X + 100, where X is the true weight. The bake time is Y, and Cov(X, Y) = 7. Find Cov(W, Y).
Answer:
Adding a constant does not change covariance.
Roll a fair four-sided die once and let X be the number shown. Let Y = 2X. Find Cov(X, Y).
Answer:
Pull the constant out: Cov(X, 2X) = 2 Var(X). Then compute Var(X) for the four-sided die.
Two swimmers race in the same pool. Their finishing times X and Y satisfy Var(X) = 10, Var(Y) = 6, and Cov(X, Y) = 3. Find Var(X − Y).
Answer:
Use the variance formula for a difference.
For a breed of dogs, shoulder height X has standard deviation 2 cm and weight Y has standard deviation 5 kg. The correlation is ρ(X, Y) = 0.6. Find Cov(X, Y).
Answer:
Covariance is the correlation times both standard deviations.
Random variables X and Y satisfy ρ(X, Y) = −0.4. Let U = −2X + 7 and V = 3Y − 1. Find ρ(U, V).
Answer:
Correlation is invariant under affine maps up to the sign of the product of the slopes. The slopes are −2 and 3, so the product is negative and the sign flips.
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