Practice Problems

Problem 1:

H properties of covariance

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Easy

Bakers Ellen and Mina each begin with 1 pound of flour. Ellen chooses a quantity A at random from a uniform distribution on $[0,1]$. Ellen gives $A/2$ to Mina and uses $A/2$ to bake a cake. Mina chooses a quantity B at random from a uniform distribution on $[0,1]$ (A and B are independent). Mina gives $B/2$ to Ellen and uses $B/2$ to bake a cake. Let E be the amount of flour Ellen is left with at the end and let M be the amount of flour Mina is left with at the end. Compute $cov[E,M]$

Solution:

Following the steps, Ellen is left with $E = 1 - A + B/2$. Mina is left with $M = 1 + A/2 - B$.

Using linearity of covariance,

$$\begin{align}cov[E,M] = cov[1 - A + B/2, 1 + A/2 - B] &= cov[1, 1] + cov[1,A/2] + cov[1,-B] \\ &+cov[-A,1] + cov[-A, A/2] + cov[-A, -B] \\ &+ cov[B/2,1] + cov[B/2, A/2] + cov[B/2, -B]\end{align}$$

we know the covariance of a constant with anything is 0, so each covariance that includes a 1 is zero. We also know A and B are independent, which means that any function of A is independent of any function of B, so all the covariances that include an A and an B are zero. Thus,

$$cov[E,M] = cov[-A, A/2] + cov[B/2, -B]$$

Again, applying linearity,

$$cov[E,M] = \frac{-1}{2} cov[A, A] + \frac{-1}{2} cov[B, B] = \frac{-1}{2} var[A] + \frac{-1}{2} var[B]$$

We know the variance of a uniform random variable on the interval $[a,b]$ is $(1/12)(b-a)^2$, so both variances above are $1/12$. Thus,

$$cov[E,M] = \frac{-1}{2} (1/12) + \frac{-1}{2} (1/12) = -1/12$$