Counting Solutions
1. For this, you want to use the BCR. Order counts here since we are dealing with words. Note the
following:
There are 5 ways to pick a vowel.
There are 4 spaces to put that vowel in.
There are (21)3 ways to pick 3 distinct consonant
Counting Problems
How many 4-letter words can you form if no letter is used twice
and each word must contain exactly one vowel (A, E, I , O or U)?.
Suppose there are N people in the class and that I randomly pass
back the rst exam. a) What is the probabil
MA581 Practice problems Solutions
Problem 1 The sample space is all groups of 3 people from the oce. Every group of 3 people in the
oce has an equally likely chance of getting picked, so the size of our sample space is | = 16 . For both
3
a) and b), the s
The least squareslinej} = n + ,911 is the line that has the following two properties:
1. The sum of the errors equals 0, i.e.,mean error of prediction : 0.
L The sum of squared errors (SSE) is smaller than that for anyI other straight-
line model
Inter
CAS MA 581
Instructions:
Exam I
Spring 2017
D. Weiner
Work problems; problems are weighted uniformly, as are subdivisions within a problem.
Justify your solutions as indicated in lecture for fullest credit.
1.
Suppose the probability that it takes you exa
Intervals for Real Numbers
( a , b ) =cfw_ x R :a< x< b
[ a , b ] = cfw_ x R :a x b
():bounded open interval
[]:bounded closed interval
c
c
( A B) = A B
from the m
objects.
r
n
distinct objects
(m)r or mPr .
n
Independence
If P ( B| A )=P( B)
If
P ( A B
BETA
Chapter 5
Discrete Random Variables and
Their Distributions
5.1
From Variables to Random Variables
Basic Exercises
5.1 The answers will vary. One such example is the variable that measures time until arrival of
next customer to, say, a given bookstor
CAS MA 581
Probability
Spring 2017
SYLLABUS/ASSIGNMENTS
Prerequisite:
Instructor:
Contact:
MA225 or MA230 (or equivalent: multivariate calculus)
Dan Weiner and TA
Room 246 MCS Building, 111 Cummington Street
617-353-9546, [email protected]
Office Hours: Wed &
MA581 Fall 2015
PROBABILITY
Instructor: Prof. Murad Taqqu, MCS 252, Tel 617-353-3022. Email: [email protected]
T ime and place: M ,Wd, F 11-12, CAS 224.
U pdated inf ormation on blackboard
T extbook: Neil A. Weiss, A Course in Probability. Pearson: Addiso
Discrete distribution problems
1. Bill and Ted play a best-of-5 game series of rock-paper-scissors.
Since Bill trains in the US Rock-Paper-Scissors League, he has a 0.6
probability of winning each individual game. What is the probability
that Bill wins th
Probability Practice Problems
Problem 1
Using concepts from this class, compute the following integrals. (tricks like u-substitution, integration by
parts, etc. wont work)
2
e2x dx
a)
x1/2 ex dx
b)
0
Problem 2
Let X and Y be independent Poisson random var
MA581 Practice problems
Problem 1
Jim and Pam enter a rae where 3 dierent people from their oce are randomly selected to win a free trip.
There are 16 people total in their oce.
a) What is the probability that both Jim and Pam are selected?
b) Dwight, ano
Solutions to practice problems
Problem 1 I should have asked for the probability of getting swept :(
a) p = 95/162 = 0.586.
b) Say all 5 games are played regardless of how many the Red Sox win. Then they win the best-of-5 if
they win 3,4, or 5 of these ga
Probability Practice Problems Solutions
Problem 1
Probability densities integrate to 1. The trick here is to rewrite these in terms of known probability densities.
a) Write it as a normal density with mean = 0 and standard deviation = 1/2.
e
2x2
dx = 2
1
Discrete distribution solutions
1. There are 2 ways of approaching this problem.
Method 1:
Consider the games they play Bernoulli trials with probability of success p = 0.6. In a best-of-5 series,
Bill wins if he wins 3 games. So if you dene the random va