Problem set 6: Covariance, Correlation and the Weak Law of Large Numbers

During the seminars, we will work on problems of three different difficulty levels. Category 1 (green) contains introductory problems, Category 2 (blue) contains standard problems, and Category 3 (orange) contains the most challenging problems.

1 Category 1

Suppose that \(X\) and \(Y\) have the following joint probability mass function:

\[ \begin{array}{c|ccc} X \backslash Y & 1 & 2 & 3\\ \hline 1 & 0.25 & 0.25 & 0\\ 2 & 0 & 0.25 & 0.25 \end{array} \]

What is the correlation coefficient?

The probability distribution of a random variable \(X\) is:

\[ \begin{array}{c|ccc} X & -1 & 0 & 1\\ \hline P_X(x) & a & b & a \end{array} \]

What is the correlation coefficient between \(X\) and \(X^2\)?

2 Category 2

A box contains 100 balls, where 25 of them are white. Two balls are successively taken out of the box. Let \(\xi_i\) be the number of white balls appearing in the \(i\)-th removal, \(i=1,2\).

Find the correlation coefficient between \(\xi_1\) and \(\xi_2\).

Two dice are rolled. Let \(\xi_1\) correspond to the value obtained on the first die, and let \(\xi_2\) correspond to the value obtained on the second die. Define

\[ \eta_1=\xi_1+\xi_2, \qquad \eta_2=\xi_1-\xi_2. \]

Find \(\operatorname{cov}(\eta_1,\eta_2)\). Are \(\eta_1\) and \(\eta_2\) independent?

Consider two random variables \(X\) and \(Y\). They both take the values \(0\), \(1\), and \(2\). Joint probabilities for each pair are given by the following table:

\[ \begin{array}{c|ccc} Y \backslash X & 0 & 1 & 2\\ \hline 0 & 0 & 0.2 & 0.2\\ 1 & 0.2 & 0 & 0.1\\ 2 & 0.2 & 0.1 & 0 \end{array} \]

  1. Calculate marginal distributions, expected values, and covariance of \(X\) and \(Y\).
  2. Calculate covariance of the random variables \(X\) and \(V\), where \(V=X-Y\).
  3. Calculate \(\mathbb E(X\mid Y=0)\) and \(\mathbb E(X\mid V=1)\).
  4. The random variable \(W\) has the same marginal distribution as \(X\) and the random variable \(Z\) has the same distribution as \(Y\). It is also known that \(W\) and \(Z\) are independent. Write down the table for the joint probabilities of \(W\) and \(Z\).

3 Category 3

Let \(\xi\) and \(\eta\) be two random variables such that

\[ \mathsf P(\xi\eta=0)=1, \]

and

\[ \mathsf P(\xi=1)=\mathsf P(\xi=-1)=\mathsf P(\eta=1)=\mathsf P(\eta=-1)=\frac14. \]

Find the joint probability distribution of these two random variables and compute

\[ \mathbb E\xi, \qquad \mathbb E\eta, \qquad \operatorname{Var}\xi, \qquad \operatorname{Var}\eta, \qquad \operatorname{cov}(\xi,\eta). \]

A die is rolled \(n\) times. Let \(\xi\) and \(\eta\) correspond to the number of times we obtained \(1\) and \(6\), respectively.

Find the correlation coefficient of these two random variables.