# MTH4120: Midterm 2 Practice 2

Problem 1.

Assume that $$F$$ is the cumulative distribution function of a binomial random variable with parameters $$n$$ and $$p$$ equal to $$6$$ and $$\frac{2}{3}$$. Calculate $$F(3)$$.

(A) $$\frac{26} { 27}$$ $$\quad\quad$$ (B) $$\frac{ 233} {729}$$ $$\quad\quad$$ (C) $$\frac{19}{27}$$ $$\quad\quad$$ (D) $$\frac{19}{729}$$ $$\quad\quad$$ (E) $$\frac{ 8 }{ 27}$$ $$\quad\quad$$ (F) $$\frac{ 64 } {729 }$$ $$\quad\quad$$ (G) $$\sqrt{2\pi}$$

Problem 2. If $$X$$ and $$Y$$ are independent random variables such that $$\mathbb E[X]=5$$, $$\mathbb[Y]=4$$, $$\mbox{var}(X)=100$$, cacluate $$\mathbb E\left[X\cdot(2X+3Y\right)]$$.

Problem 3. If $$X$$ is a standard normal random variable, calculate $$\mathbb E\left[\left(2X+5\right)^2\right]$$.

Problem 4. It is known that $$g:\mathbb R^2\to \mathbb R$$ is a positive function of two variables. Let $$T$$ be the triangle with vertices $$(3,5)$$, $$(4,10)$$, and $$(5,17)$$. If there exists a constant $$M$$ such that the joint density function of random variables $$X$$ and $$Y$$ satisfies \begin{eqnarray*} f_{X,Y}(x,y)&=&\left\{\begin{array}{ll} \frac{g(x,y)}{M},& \mbox{ if } (x,y)\in T\\ 0,&\mbox{ otherwise,} \end{array}\right. \end{eqnarray*} which of the following equalities is correct?
• (A) $$M=\int_{3}^{4}\int_{5+5(x-3)}^{5+6(x-3)}g(x,y)\,dydx+\int_{4}^{5}\int_{10+7(x-4)}^{5+6(x-3)}g(x,y)\,dydx$$
• (B) $$M=\int_{3}^{4}\int_{5+\frac{5}{2}(x-3)}^{5+12(x-3)}g(x,y)\,dydx + \int_{4}^{5}\int_{5+\frac{5}{2}(x-3)}^{17-7(x-4)}g(x,y)\,dydx$$
• (C) $$M=\int_{3}^{4}\int_{17-12(x-3)}^{17-\frac{7}2(x-3)}g(x,y)\,dydx+\int_{4}^{5}\int_{5+5(x-4)}^{17-\frac{7}2(x-3)}g(x,y)\,dydx$$
• (D) $$M=\int_{3}^{4}\int_{17-12(x-3)}^{17-7(x-3)}g(x,y)\,dxdy+\int_{4}^{5}\int_{10+7(x-4)}^{5+6(x-3)}g(x,y)\,dxdy$$
• (E) $$M=\int_{3}^{4}\int_{5+\frac{5}{2}(x-3)}^{5+12(x-3)}g(x,y)\,dxdy+\int_{4}^{5}\int_{10+7(x-4)}^{5+6(x-3)}g(x,y)\,dxdy$$
• (F) $$M=\int_{3}^{5}\int_{5 }^{17}g(x,y)\,dxdy$$
• (G) $$M=\int_{3}^{5}\int_{5 }^{17}g(x,y)\,dydx$$

Problem 5. It is known that $$X$$ and $$Y$$ are two random variables whose joint density function satisfies \begin{eqnarray*} f_{X,Y}(x,y)&=&\left\{\begin{array}{ll} Cx^2y,& \mbox{ if } 6 < x < 1000, \; 72 < y < 100, \; \mbox{ and } y \leq 2x^2\\ 0,&\mbox{ otherwise,} \end{array}\right. \end{eqnarray*} for some real number $$C$$. Which of the following is the correct formula for $$\mathbb P\left(Y\leq X^2\right)$$?
• (A) $$\int_{6}^{5\sqrt 2}\int_{72}^{2x^2}Cx^2y\,dydx + \int_{5\sqrt 2}^{1000}\int_{72}^{100}Cx^2y\,dydx$$
• (B) $$\int_{6}^{5\sqrt 2}\int_{72}^{2x^2}Cx^2y\,dydx + \int_{5\sqrt 2}^{1000}\int_{72}^{100}Cx^4y^2\,dydx$$
• (C) $$\int_{6\sqrt 2}^{10}\int_{72}^{x^2}Cx^2y\,dydx + \int_{10}^{1000}\int_{72}^{100}Cx^2y\,dydx$$
• (D) $$\int_{6}^{5\sqrt 2}\int_{72}^{2x^2}Cx^4y^2\,dydx + \int_{5\sqrt 2}^{1000}\int_{72}^{100}Cx^4y^2\,dydx$$
• (E) $$\int_{6\sqrt 2}^{10}\int_{72}^{x^2}Cx^4y^2\,dydx + \int_{10}^{1000}\int_{72}^{100}Cx^4y^2\,dydx$$

Problem 6. Provide an example of a probability space $$\left(\Omega,\mathcal F,\mathbb P\right)$$ and two random variables $$X$$ and $$Y$$ on this probability space whose joint cumulative distribution function $$F_{X,Y}$$ satisfies $F_{X,Y}\left( 28, 18\right)= 0.35 \quad\quad \mbox{and}\quad\quad F_{X,Y}\left( 38, 58\right)= 0.78.$

Problem 7. The derivative of the moment generating function $$M_X(t)$$ of the discrete random variable $$X$$ satisfies $M_X'(t)=-15e^{-42t}+9e^{42t}+8e^{56t}.$ Evaluate $$14\cdot \mathbb P\left( X \leq 0\right)$$.

Problem 8. Prove that there exists a probability space $$(\Omega, \mathcal F, \mathbb P)$$ and two random variables $$X$$ and $$Y$$ on this probability space that are not independent and that satisfy $$\mbox{var}(X)\neq 0$$, $$\mbox{var}(Y)\neq 0$$, $$\mathbb E\left[X\right]=4$$, $$\mathbb E\left[Y\right]=7$$, and $$\mathbb E\left[XY\right]=28$$.