Post Exam 2 Practice Questions, 18.05,
Spring 2014
Jeremy Orlo and Jonathan Bloom
Note: This is a set of practice problems for the material that came after exam 2. In
preparing for the nal you should use the previous review materials, the psets, the
notes
Practice Exam 1: Long List
18.05, Spring 2014
1
Counting and Probability
1. A full house in poker is a hand where three cards share one rank and two cards
share another rank. How many ways are there to get a full-house? What is the
probability of getting
Left from Monday: the Central Limit Theorem
18.05 Spring 2017
0.5
0.4
0.3
0.2
0.1
0
-4
-3
-2
-1
0
1
2
3
4
Exam next Wednesday
Exam 1 on Wednesday March 8.
Designed for 1 hour. You will have the full 80 minutes.
Class on Monday will be review.
Practice mat
Discrete Random Variables
Class 4, 18.05, Spring 2014
Jeremy Orlo and Jonathan Bloom
1
Learning Goals
1. Know the denition of a discrete random variable.
2. Know the Bernoulli, binomial, Poisson and geometric distributions and examples of what
they model.
Discrete Random Variables: Expected Value
Class 4, 18.05, Spring 2014
Jeremy Orlo and Jonathan Bloom
1
Expected Value
In the R reading questions for this lecture, you simulated the average value of rolling a die
many times. You should have gotten a value
Variance of Discrete Random Variables
Class 5, 18.05, Spring 2014
Jeremy Orlo and Jonathan Bloom
1
Learning Goals
1. Be able to compute the variance and standard deviation of a random variable.
2. Understand that standard deviation is a measure of scale o
Gallery of Continuous Random Variables
Class 5, 18.05, Spring 2014
Jeremy Orlo and Jonathan Bloom
1
Learning Goals
1. Be able to give examples of what uniform, exponential and normal distributions are used
to model.
2. Be able to give the range and pdfs o
Counting and Sets
Class 1, 18.05, Spring 2014
Jeremy Orlo and Jonathan Bloom
1
Learning Goals
1. Know the denitions and notation for sets, intersection, union, complement.
2. Be able to visualize set operations using Venn diagrams.
3. Understand how count
Continuous Random Variables
Class 5, 18.05, Spring 2014
Jeremy Orlo and Jonathan Bloom
1
Learning Goals
1. Know the denition of a continuous random variable.
2. Know the denition of the probability density function (pdf) and cumulative distribution
functi
Introduction
Class 1, 18.05, Spring 2014
Jeremy Orlo and Jonathan Bloom
1
Probability vs. Statistics
In this introduction we will preview what we will be studying in 18.05. Dont worry
if many of the terms are unfamiliar, they will be explained as the cour
Manipulating Continuous Random Variables
Class 5, 18.05, Spring 2014
Jeremy Orlo and Jonathan Bloom
1
Learning Goals
1. Be able to nd the pdf and cdf of a random variable dened in terms of a random variable
with known pdf and cdf.
2
Transformations of Ran
Appendix
Class 6, 18.05, Spring 2014
Jeremy Orlo and Jonathan Bloom
1
Introduction
In this appendix we give more formal mathematical material that is not strictly a part of
18.05. This will not be on homework or tests. We give this material to emphasize t
Bayesian Updating with Discrete Priors
Class 11, 18.05, Spring 2014
Jeremy Orlo and Jonathan Bloom
1
Learning Goals
1. Be able to apply Bayes theorem to compute probabilities.
2. Be able to identify the denition and roles of prior probability, likelihood
Joint Distributions, Independence
Class 7, 18.05, Spring 2014
Jeremy Orlo and Jonathan Bloom
1
Learning Goals
1. Understand what is meant by a joint pmf, pdf and cdf of two random variables.
2. Be able to compute probabilities and marginals from a joint p
Null Hypothesis Signicance Testing II
Class 18, MIT 18.05, Spring 2014
Jeremy Orlo and Jonathan Bloom
1
Learning Goals
1. Be able to list the steps common to all null hypothesis signicance tests.
2. Be able to dene and compute the probability of Type I an
Probability intervals
Class 16, 18.05, Spring 2014
Jeremy Orlo and Jonathan Bloom
1
Learning Goals
1. Be able to nd probability intervals given a pmf or pdf.
2. Understand how probability intervals summarize belief in Bayesian updating.
3. Be able to use
Central Limit Theorem and the Law of Large Numbers
Class 6, 18.05, Spring 2014
Jeremy Orlo and Jonathan Bloom
1
Learning Goals
1. Understand the statement of the law of large numbers.
2. Understand the statement of the central limit theorem.
3. Be able to
Comparison of frequentist and Bayesian inference.
Class 20, 18.05, Spring 2014
Jeremy Orlo and Jonathan Bloom
1
Learning Goals
1. Be able to explain the dierence between the p-value and a posterior probability to a
doctor.
2
Introduction
We have now learn
Covariance and Correlation
Class 7, 18.05, Spring 2014
Jeremy Orlo and Jonathan Bloom
1
Learning Goals
1. Understand the meaning of covariance and correlation.
2. Be able to compute the covariance and correlation of two random variables.
2
Covariance
Cova
18.05 Exam 2 Solutions
Problem 1. (10 pts: 4,2,2,2) Concept questions
(a) Yes and yes. Frequentist statistics dont give the probability an hypothesis is true.
(b) True. Bayesian updating involves multiplying the likelihood and the prior. If the prior
is 0
Conditional Probability, Independence and Bayes Theorem
Class 3, 18.05, Spring 2014
Jeremy Orlo and Jonathan Bloom
1
Learning Goals
1. Know the denitions of conditional probability and independence of events.
2. Be able to compute conditional probability
18.05 Problem Set 6, Spring 2017 Solutions
Problem 1. (20 pts.) Hypotheses and data.
(a) (i) The possible hypotheses are that the coin is fair or has probability 0.4 of
landing heads.
(ii) The data x is the result of the experiment: toss the chosen coin 3
Covariance and Correlation
Class 7, 18.05
Jeremy Orloff and Jonathan Bloom
1
Learning Goals
1. Understand the meaning of covariance and correlation.
2. Be able to compute the covariance and correlation of two random variables.
2
Covariance
Covariance is a
Notational conventions;
Bayesian updating with continuous priors
Class 13, 18.05
Jeremy Orloff and Jonathan Bloom
1
Learning Goals
1. Be able to work with the various notations and terms we use to describe probabilities
and likelihood.
2. Understand a par
Choosing Priors
Probability Intervals
18.05 Spring 2017
Two-parameter tables: Malaria
In the 1950s scientists injected 30 African volunteers with malaria.
S = carrier of sickle-cell gene
N = non-carrier of sickle-cell gene
D+ = developed malaria
D = did n
18.05 Problem Set 2, Spring 2014
Problem 1. (10 pts.) Boy or girl paradox.
The following pair of questions appeared in a column by Martin Gardner in Scientic
American in 1959.
(a) Mr. Jones has two children. The older child is a girl. What is the probabil
18.05 Problem Set 4, Spring 2014
Problem 1. (10 pts.) Time to failure. Recall that an exponential random variable
1
X exp() has mean and pdf given by f (x) = ex on x 0.
(a) Compute P (X x).
(b) Suppose that X1 and X2 are independent exponential random var
18.05 Problem Set 1, Spring 2014
Problem 1. (10 pts.) Poker hands.
After one-pair, the next most common hands are two-pair and three-of-a-kind:
Two-pair: Two cards have one rank, two cards have another rank, and
the remaining card has a third rank. e.g. c