Midterm 2 Study Guide

Midterm Exam 2 will take place on Thursday November 13 during your lab section. There will be 6 problems, and they will break down as follows:

  1. (an undisclosed check on your visual intuition)
  2. Discrete random variables
  3. Continuous random variables
  4. The moment-generating function
  5. Transformations
  6. Joint distributions

Below are practice problems for each of these areas, as well as guidance about what course materials to refer to if you want more review on a particular topic.

Give your cheat sheet some TLC!

You are allowed two resources on the exam:

  • a “dumb” calculator (no wi-fi). You will not need it, but if you want it as a security blanket, be my guest;
  • both sides of one 8.5” x 11” sheet of notes, prepared by you and you alone. You can create it however you wish: handwritten, on a computer, etc.

Do not skimp on the cheat sheet! Students in past semesters have neglected their cheat sheets to their detriment. Get an early start, and give it some elbow grease.

What special distribution families should I know?

Binomial, geometric, Poisson, continuous uniform, normal, gamma (and its special cases like exponential). Anything else will be introduced from scratch within the exam as if you had never seen it before.

Discrete random variables

Problem 1

Let \(A_1\), \(A_2\), \(A_3\), \(A_4\), \(A_5\), and \(A_6\) be events in some sample space, each with the same probability \(0<p<1/6\). Define the indicator random variables

\[ I_k = \begin{cases} 0 & \text{if }A_k^c\text{ happens}\\ 1 & \text{if }A_k\text{ happens}, \end{cases} \]

and let \(X\) be their sum

\[ X = I_1 + I_2 + I_3 + I_4+I_5+I_6. \]

  1. Assume the events \(A_i\) are mutually disjoint and compute…

    1. \(E(X)\).
    2. \(P(X=1)\).
    3. \(P(X=3)\).

  2. Assume the \(A_i\) are independent and compute…

    1. \(E(X)\).
    2. \(P(X=1)\).
    3. \(P(X=3)\).

Problem 2

Let \(X\sim\text{Poisson}(\lambda)\) and define a new random variable \(Y\) that is a zero-truncated version of \(X\). That is, we start with \(\text{Range}(X)=\{0,\,1,\,2,\,3,\,...\}\), and then we rig it so that \(\text{Range}(Y)=\{1,\,2,\,3,\,...\}\). To do this, we define the PMF of the new random variable \(Y\) by

\[ P(Y=k) = P(X=k\mid X > 0),\quad k=1,\,2,\,3,\,4,\,... \]

  1. Show that the new PMF sums to one;
  2. Compute the mean of \(Y\);
  3. Compute the variance of \(Y\).

Want more review?

Continuous random variables

Problem 3

Consider a nonnegative, absolutely continuous random variable with cdf

\[ F_X(x) = \begin{cases} 1 - \exp\left(-\frac{x^2}{2}\right) & x\geq 0 \\ 0 & x<0. \end{cases} \]

  1. What is \(P(2\leq X\leq 3)\);
  2. What is the density of \(X\)?
  3. What is \(E(X^n)\) for any \(n\in\mathbb{N}\)?
  4. What is \(\text{var}(X)\)?
  5. What is the median of \(X\)?

Problem 4

Here is the pdf of some absolutely continuous random variable:

\[ f(x)=\begin{cases} ax^2e^{-bx^2} & x\geq 0\\ 0 & x< 0. \end{cases} \]

Solve for \(a\) in terms of \(b\). So your final answer will look like

\[ a=\text{``some formula including $b$ and a bunch of constants''} \]

Want more review?

Moment-generating function

Problem 5

Let \(X\) have the discrete uniform on \(\{1,\,2,\,...,\,n\}\). This means \(P(X=i)=1/n\) for all \(i = 1,\,2,\,...,\,n\).

  1. Compute the MGF of \(X\);
  2. Use it to compute the mean;
  3. Use it to compute the variance.

Problem 6

Consider a continuous random variable \(X\) with density

\[ f(x)=\frac{1}{\beta}\exp\left(-\frac{x}{\beta}-e^{-x/\beta}\right),\quad x\in\mathbb{R}. \]

The parameter \(\beta>0\) is just a positive constant. Compute the moment-generating function of \(X\).

Want more review?

  • worked examples from lecture;
  • new theorems and examples from lecture about updating MGFs under linear transformations;
  • Problem Set 5: problems 9 - 10;
  • Problem Set 6: problems 3 - 5.
  • Odd-numbered exercises in DeGroot & Schervish Chapter 4.4.

Transformations

Problem 7

This is an important technical skill, so let’s drill it. For each of these, plot the transformation and then derive the range and the density of the new distribution:

  1. Let \(X\sim\text{Unif}(0,\,1)\) and consider \(Y=1/X\);
  2. Let \(X\) have density \(f_X(x)=42x^5(1-x)\) for \(0<x<1\) and consider \(Y=X^3\);
  3. Let \(X\sim\text{Exponential}(\lambda)\) and consider \(Y=4X+3\);
  4. Let \(X\) have cdf \(F_X(x)=(x+1)^3/8\) for \(-1<x<1\) and consider \(Y=1-X^2\);
  5. Let \(X\sim\text{N}(0,\,1)\) and consider \(Y=|X|\).

Want more review?

  • Lab 7;
  • Problem Set 6: problem 6;
  • Read DeGroot & Schervish Chapter 3.8 and do the odd-numbered exercises.

Joint distributions

Problem 8

Here is the joint density for a random pair:

\[ f_{XY}(x,\,y) = \frac{24}{7} x(2-y),\quad 0<x<1\text{ and }0<y<1-x. \]

  1. Sketch the joint range of \((X,\,Y)\).
  2. Find the marginal density of \(Y\).
  3. Find the marginal density of \(X\).
  4. Find the conditional density of \(X\) given \(Y=y\).
  5. Use this to compute \(P(X>1/2\,|\,Y=1/3)\).
  6. What is \(E(X\,|\,Y=y)\)?

Problem 9

Consider this joint distribution for random variables \(X\) and \(Y\), written in hierarchical form:

\[ \begin{aligned} X&\sim\text{Gamma}\left(\alpha,\,\beta\right)\\ Y\mid X=x&\sim \text{Exponential}(x). \end{aligned} \]

  1. Sketch the joint range of \((X,\, Y)\).
  2. What is the joint density of the random pair \((X,\, Y)\)?
  3. What is the marginal density of \(Y\)?
  4. What is the density of \(X\) conditional on the event \(Y=y\)? Is it recognizable?

Want more review?

  • jointly discrete examples from lecture;
  • jointly contnuous examples from lecture;
  • mixed Poisson-gamma example from Lab 9;
  • Problem Set 6: problems 7 - 10;
  • Odd-numbered exercises in DeGroot & Schervish Chapters 3.4 - 3.6.