18.05 Problem Set 9, Spring 2014
Problem 1. (10 pts.) Condent coin.
Recall the condent coin from pset 7. It was spun on its edge 250 times and came up
heads 140 times. Our null hypothesis was that the coin was fair and the two-sided
p-value came out as 0.
18.05 Problem Set 8, Spring 2014
Problem 1. (10 pts.) Jerry steals a JP Licks token and asks Jon to perform a test
at signicance level = 0.05 to investigate whether the coin is fair or biased toward
tails (the side that says Token). Jon records the follow
18.440 Midterm 2, Spring 2014: 50 minutes, 100 points
1. (20 points) Consider a sequence of independent tosses of a coin that is
biased so that it comes up heads with probability 3/4 and tails with
probability 1/4. Let Xi be 1 if the ith toss comes up hea
18.440 Midterm 2, Fall 2012: 50 minutes, 100 points
1. (10 points) Suppose that a fair die is rolled 18000 times. Each roll turns
up a uniformly random member of the set cfw_1, 2, 3, 4, 5, 6 and the rolls are
independent of each other. Let X be the total
18.440 Midterm 2 Solutions Fall 2009
1.
(a) P cfw_X < a = a2 for a 2 (0, 1) and thus FX (a) = a2 . Dierentiating
gives
2a a 2 [0, 1]
fX (a) =
0
otherwise
(b)
[X] =
R1
0
fX (x)xdx =
R1
0
2x2 dx = 2/3.
(c) The pair (X, Y ) is uniformly distributed on the tr
18.05 Problem Set 7, Spring 2014 Solutions
Problem 1. (10 pts.) (a) H0 : = 0.5
HA : one-sided > 0.5, two-sided = 0.5.
6
Test statistic: x = number of heads in 250 spins.
Data: x = 140.
One-sided data at least as extreme: x
p-value is
p = P (x
140. Using R
18.05 Problem Set 8, Spring 2014 Solutions
Problem 1. (10 pts.) (a) Let x = number of heads
Model: x binomial(12, ).
Null distribution binomial(12, 0.5).
Data: 3 heads in 12 tosses.
Since HA is one-sided the rejection region is one-sided. Since HA says th
18.05 Problem Set 7, Spring 2014
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 on Friday January 4, 2002:
When spun
18.440 Midterm 1, Spring 2014: 50 minutes, 100 points
1. (10 points) How many quintuples (a1 , a2 , a3 , a4 , a5 ) of non-negative
integers satisfy a1 + a2 + a3 + a4 + a5 = 100? ANSWER: This the number
104!
of ways to make a list of 100 stars and 4 bars,
18.05 Problem Set 9, Spring 2014 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
4
plotted are
binomial(250, ), N(250, 250(1 ), N(250, 250/4).
No
18.05 Problem Set 6, Spring 2014
Problem 1. (10 pts.) Beta try again. (Adapted from Information Theory, Inference, and Learning Algorithms by David J. C. Mackay.)
A statistical statement appeared in The Guardian on Friday January 4, 2002:
When spun on edg
18.05 Final Exam Solutions
Part I: Concept questions (58 points)
These questions are all multiple choice or short answer. You dont have to show any work.
Work through them quickly. Each answer is worth 2 points.
Concept 1.
answer: C. (i) and (ii)
Concept
18.05 Problem Set 6, Spring 2014 Solutions
Problem 1. (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:
250
250
250
p(x|H0 ) =
(.5)
p(x|H1 ) =
(.56)140 (.4
18.05 Final Exam
2
Part I: Concept questions (58 points)
These questions are all multiple choice or short answer. You dont have to show any work.
Work through them quickly. Each answer is worth 2 points.
Concept 1. Which of the following represents a vali
Practice Final
This exam is closed book, no books, papers or recording devices permitted. You may use theorems
from class, or the book, provided you can recall them correctly.
Problem 1
Suppose
and
Show thatt
for all simple measurable functions
on
alm