# 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 $$7$$.