Choosing Priors
Probability Intervals
18.05 Spring 2013
Concept question
Say we have a bent coin with unknown probability of heads .
We are convinced that .7.
Our prior is uniform on [0,.7] and 0 from .7 to 1.
We ip the coin 65 times and get 60 heads.
Whi

18.05 Problem Set 4, Spring 2013
(due in 2-285 Monday, Mar. 4 at 4:30 PM)
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

Exam 1 Practice Questions I, 18.05, Spring 2013
Note: This is a set of practice problems for exam 1. The actual exam will be much
shorter.
1. There are 3 arrangements of the word DAD, namely DAD, ADD, and DDA. How
many arrangements are there of the word P

18.05 Problem Set 1, Spring 2013 Solutions
Problem 0. (10 pts.) Thank you for taking the Statistics Concept Inventory.
Problem 1. (10 pts.)
answer: (reasons below)
P (two-pair) = .047539, P (three-of-a-kind) = 0.021128,
two pairs is more likely
We create

18.05 Problem Set 8, Spring 2013
(due in 2-285 Wed, May 1 at 4:30 PM)
Problem 1. (10 pts.)
Condent coin: III (Quote taken from Information
Theory, Inference, and Learning Algorithms by David J. C. Mackay.)
A statistical statement appeared in The Guardian

18.05 Problem Set 4, Spring 2013 Solutions
Problem 1. (10 pts.) a)
x
ex dx = 1 (1 ex ) = ex .
P (X x) = 1 P (X < x) = 1
0
b) For t 0, we know that T t if and only if both X1 t and X2 t. So
P (T t) = P (X1 t, X2 t). Since X1 and X2 are independent,
P (X1

Exam 1 Practice Questions II, 18.05, Spring 2013
Note: This is a set of practice problems for exam 1. The actual exam will be much
shorter.
1. A full house in poker is a hand where three cards share one rank and two cards
share another rank. How many ways

18.05 Problem Set 6, Spring 2013 Solutions
Problem 1. (0 pts.) No problem: no solution
Problem 2. (10 pts.) a) Throughout this problem we will let x be the data of 140
heads out of 250 tosses. We have 140/250 = .56. Computing the likelihoods:
p(x|H0 ) =
2

18.05 Problem Set 9, Spring 2013
(due in 2-285 Thursday, May 9 in class)
Problem 1. (10 pts.) An experiment can produce outcomes in the range cfw_1, 2, 3,
4, 5, 6 The gure shows a likelihood table for 5 hypotheses = cfw_.1, .3, .5, .7, .9.
Somewhat arbitr

18.05 Problem Set 8, Spring 2013 Solutions
Code for all Matlab computations and gures is posted alongside these solutions.
Problem 1. (10 pts.) a) H0 : = .5
HA : one-sided > .5, two-sided = .5.
Test statistic: x = number of heads in 250 spins.
Data: x = 1

18.05 Problem Set 9, Spring 2013 Solutions
Problem 1. (10 pts.) a) By denition the condence for a hypothesis is the
probability that if the hypothesis is true then it is in the condence interval generated
by the random data. For example if x = 4 then .3 i

18.05 Problem Set 7, Spring 2013 Solutions
Problem 1. (10 pts.) a) We have x binomial(n, ), so E(X) = n and Var(X) =
n(1 ). The rule-of-thumb variance is just n . So the distributions being plotted are
4
binomial(250, ), N(250, 250(1 ), N(250, 250/4).
Not

18.05 Problem Set 3, Spring 2013 Solutions
Problem 1. (10 pts.)
a) We have P (A) = P (B) = P (C) = 1/2. Writing the outcome of die 1 rst, we can
easily list all outcomes in the following intersections.
A B = cfw_(1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3,

Exam 1 Practice Questions II, 18.05, Spring 2013
Note: This is a set of practice problems for exam 1. The actual exam will be much
shorter.
1.
4
2
Number of ways to get a full-house:
Probability:
4
2
13
1
4
3
13
1
4
3
12
1
12
1
52
5
2. a) There are 20 way

Exam 1 Practice Questions I, 18.05, Spring 2013
Note: This is a set of practice problems for exam 1. The actual exam will be much
shorter.
1.
11!
2!2!
2.
6
4
7
4!
4
3.
We are given P (E F ) = 3/4.
E c F c = (E F )c P (E c F c ) = 1 P (E F ) = 1/4.
4. a) A

Exam 1 All Questions, 18.05, Spring 2013
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 a ful