Homework 7

1. Show that if \(X = (X_1, X_2)\) is a random vector, then \(\mathbb{E}\left[aX_1 + bX_2 + c\right] = a\mathbb{E}X_1 + b\mathbb{E}X_2 + c\). Establish the result in both the discrete and continuous cases.

2. (Covariance formula) Show that \(\mathbb{E}\left[(X - \mathbb{E}X)(Y - \mathbb{E}Y)\right] = \mathbb{E}(XY) - \mathbb{E}X\mathbb{E}Y\).

3. Suppose \((X_1, X_2)\) are uniformly distributed on the unit square, i.e., \[f(x_1, x_2) = 1\;,\qquad 0 < x_1 < 1, 0 < x_2 < 1\] Find the distribution of \(Y = X_1 + X_2\) by finding the CDF of \(Y\).

4. Consider again the house hunting example from class where \(X_1\) denotes the number of bedrooms and \(X_2\) denotes the number of bathrooms, and for a randomly selected listing the vector \((X_1, X_2)\) has joint distribution:

	\(x_1 = 0\)	\(x_1 = 1\)	\(x_1 = 2\)	\(x_1 = 3\)
\(x_2 = 1\)	0.1	0.1	0.2	0
\(x_2 = 1.5\)	0	0.1	0.2	0
\(x_2 = 2\)	0	0	0	0.3
\(x_2 = 2.5\)	0	0	0	0

Find the covariance and correlation of \(X_1\) and \(X_2\).

5. Let \((X_1, X_2)\) be independent exponential random variables with parameter \(\beta = 2\), so that they have joint distribution \[f(x_1, x_2) = \frac{1}{4} \exp\left\{-\frac{1}{2}(x_1 + x_2)\right\}\;,\qquad x_1 > 0, x_2 > 0\] Let \(Y_1, Y_2\) denote the sum and difference, respectively, of \(X_1, X_2\). Find the correlation \(\text{corr}(Y_1, Y_2)\).

6. Suppose that you arrive at work within a 2-minute window of your expected arrival time uniformly at random, and your expected arrival time may shift slightly depending on whether there are traffic delays. That is, if \(X\) denotes your arrival time where \(X = 0\) indicates you are exactly on time, and \(Y\) denotes the traffic delay, then assume: \[ X|Y \sim \text{uniform}(Y - 1, Y + 1) \] If \(Y \sim \text{exponential}(1)\), find the mean arrival time \(\mathbb{E}X\).