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MTH4120: Final Practice 1

Problem 1. Assume that \(X\) and \(Y\) are two random variables. What type of object is \(X\cdot Y\)?
(A) probability measure \(\quad\) (B) event \(\quad\) (C) random variable \(\quad\) (D) sample space \(\quad\) (E) improperly defined object

Problem 2. It is known that \(S\) and \(T\) are independent events that satisfy \(\mathbb P\left(S\right)=\frac{12}{23}\) and \(\mathbb P\left(S\cap T^C\right)=\frac{5}{23}\). What is \(\mathbb P(T)\)?
(A) \(\frac{5}{12}\) \(\quad\quad\) (B) \(\frac{7}{18}\) \(\quad\quad\) (C) \(\frac{5}{16}\) \(\quad\quad\) (D) \(\frac{11}{18}\) \(\quad\quad\) (E) \(\frac{7}{12}\)

Problem 3. \(Z\) is a normal random variable with expectation \(81\) and variance \(16\). What is \(\mathbb P\left(Z\geq 5\right)\)?
(A) \(\Phi(21)\) \(\quad\quad\) (B) \(e^{-\frac12\cdot 21^2}\) \(\quad\quad\) (C) \(e^{-\frac12\cdot 19^2}\) \(\quad\quad\) (D) \(\Phi(19)\) \(\quad\quad\) (E) \(\frac1{\sqrt{2\pi}}e^{-\frac12t^2}\)

Problem 4. The moment generating function of the random variable \(X\) is given by \[M_X(t)=\frac{3e^{-3t}+2+2e^{6t}+9e^{9t}}{16}.\] If \(m\) and \(n\) are relatively prime positive integers such that \(\mathbb E\left[X\right]=\frac{m}{n}\), what is \(m+n\)?

Problem 5. Assume that \(\mathbb E\left[X\right]=3\) and \(\mbox{var}(X)=37\). Evaluate \(\mathbb E\left[(X+9)(X+2)\right]\).

Problem 6. The covariance matrix for random variables \(X\) and \(Y\) is \(\left[\begin{array}{cc}10& -1\\ -1 & 9\end{array}\right]\). Evaluate \(\mbox{cov}(9X+7Y,5Y+6)\).
(A) \(25\) \(\quad\quad\) (B) \(32\) \(\quad\quad\) (C) \(98\) \(\quad\quad\) (D) \(168\) \(\quad\quad\) (E) \(270\)

Problem 7. Assume that the joint probability density function between random variables \(X\) and \(Y\) satisfies \[f(x,y)=\left\{\begin{array}{ll}\frac1{20},&\mbox{ if } 0\leq x \leq 1\mbox{ and }0\leq y \leq 8\\ \frac1{80},&\mbox{ if }1 < x \leq 13 \mbox{ and }0 \leq y \leq 4\\ 0,&\mbox{ otherwise.}\end{array}\right.\] Calculate \(\mathbb P\left(X+Y\leq 9\right)\).
(A) \(\frac{17}{26}\) \(\quad\quad\) (B) \(\frac{7}{10}\) \(\quad\quad\) (C) \(\frac{3}{4}\) \(\quad\quad\) (D) \(\frac{13}{16}\) \(\quad\quad\) (E) \(\frac{17}{20}\)

Problem 8. A total of \(N\) balls is placed in \(N\) boxes in such a way that each ball is equally likely to be placed in each of the boxes and the placements are independent from each other. Find the expected number and the variance of the number of empty boxes.