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MTH4120: Midterm 2 Practice 1

Problem 1. Construct an example of a probability space \( \left(\Omega, \mathcal F, \mathbb P\right) \) and two independent random variables \( Q \) and \( R \) on \( \left(\Omega, \mathcal F, \mathbb P\right) \) such that \( \mathbb E\left[Q\right]=\mathbb E\left[R\right]=0 \), \( \mbox{Var}(Q)=41 \), and \( \mbox{Var}(R)=73 \).

Problem 2. The joint probability mass function of the random variables \( X \) and \( Y \) is given by the following table: \begin{eqnarray*} \begin{array}{cc}&Y\\ \begin{array}{c} \\ X\\ \\\end{array}& \begin{array}{c||c|c|} & 20&41\\ \hline\hline 2& 0.13&0.15\\ \hline 4&0.18&0.17\\ \hline 5&0.23&0.14\\\hline \end{array} \end{array} \end{eqnarray*} Calculate \( \mathbb P\left(2X^2+Y< 71\right) \).

Problem 3. \( Z \) is a standard normal random variable and \( X \) a random variable defined as \begin{eqnarray*} X&=&\left\{\begin{array}{ll}e^{Z},\mbox{ if }e^Z > \frac{2}{10},\\ 0,\mbox{ if }e^Z\leq \frac{2}{10}.\end{array}\right. \end{eqnarray*} Evaluate \( \mathbb E\left[X\right] \).

Problem 4. The covariance matrix for random variables \( X \) and \( Y \) is given by \[ \left[\begin{array}{cc}\mbox{cov}(X,X)& \mbox{cov}(X,Y)\\ \mbox{cov}(Y,X)&\mbox{cov}(Y,Y)\end{array}\right]=\left[\begin{array}{cc}4& -1\\ -1&3\end{array}\right].\] Evaluate \( \mbox{cov}\left( 7X+5Y,3Y+4\right) \).

Problem 5. The moment generating function of the random variable \( X \) is given by \[ M_X(t)=\frac{ 3e^{-5t}+7+9e^{9t}+4e^{12t}}{23}.\] Calculate \( \mathbb P\left(X\leq 0\right)\).

Problem 6.
  • (a) Let \(k\) be a positive integer. There are two boxes each containing the numbers \(\{1,2,\dots, 2k\}\). Two numbers are selected at random - one number from each of the boxes (each number has equal probability of being selected). Let \(A\) be the event that the number from the first box is even. Let \(B\) be the event that the sum of the two numbers is even. Are the events \(A\) and \(B\) independent? Explain your answer!
  • (b) Prove that in the probability space from the previous problem there are three events \(A_1\), \(A_2\), \(A_3\) that satisfy: Each two of \(A_1\), \(A_2\), \(A_3\) are independent and \[\mathbb P(A_1\cap A_2\cap A_3)\neq \mathbb P(A_1)\cdot \mathbb P(A_2)\cdot \mathbb P(A_3).\]

Problem 7. Assume that \( X \) and \( Y \) are independent normal random variables such that \( \mathbb E[X]=\mathbb E[Y] \) and \( \mbox{var}(X)=\mbox{var}(Y) \). If \( \mathbb P\left(X > 54\right)=0.5 \) and \( \mathbb P\left(X > 49\right)=0.55 \) calculate \( \mathbb P\left(\mbox{max}\{X,Y\} > 59\right) \).

Problem 8. The random variables \( X_1 \), \( X_2 \), \( \dots \), \( X_{50} \) are identically distributed. It is known that \( \mathbb E\left[X_1\right]=0 \), \( \mathbb E\left[X_1^2\right]=3.4 \), and that for every different \( i \), \( j\in\{1 \), \( 2 \), \( \dots \), \( 50\} \) the covariance between \( X_i \) and \( X_j \) is \( 1 \). Calculate the variance of the random variable \( X_1+X_2+\cdots+X_{50} \).

Problem 9. What is the expected number of rolls of a standard die until each of the numbers appears at least once?

Problem 10. Construct an example of two binomial random variables \(X\) and \(Y\) that have covariance \(7\).