MATH 230/STAT 230: Homework 9
Due on 11/24, in class
Read Section 4.5-4.6, and Section 5.1
Problem 1. Suppose an electric circuit has two parts (as shown in the gure). Part I
consists of two branches, whereas Part II has only one branch. The electric circ
MATH 230/STAT 230: Practice Midterm I
Problem 1. A biometric security device using ngerprints erroneously refuses to admit 1 in 100
authorized persons from a facility containing classied information. The device will erroneously
admit 1 in 1,000 unauthoriz
Homework 3 Solution
2.1 4. Let A = cfw_3 sixes in the 8 rolls and B = cfw_2 sixes in the rst 5 rolls. Then
AB = cfw_2 sixes in the rst 5 rolls and 1 six in the last 3 rolls. The probability is
P (AB)
=
P (B|A) =
P (A)
5
2
3 1 1 5 2
( ) (6)
1 6
8 1 3 5 5
(
Problem Set for Lecture 18: Solutions
- Bivariate Normal 6.5 2. Let X and Y be the heights of daughters and mothers in the population, which
follows bivariate normal with X = Y = 5 4 , X = Y = 4 , = 0.5. The desired
probability can be written as P (X < Y
Homework 2 Solution
4
1.5 2. (a) The probability of drawing a white or a black ball in the 1st step is 10 and
6
respectively. Given the 1st draw is white, the probability of drawing a white ball in
10
7
the 2nd step is 13 . Given the 1st draw is black, th
Homework 8 Solution
1
4.4 4. The density of X is fX (x) = 2 I(x (1, 1). The function Y = g(X) = X 2 is
not monotone on [1, 1], but is monotone on [1, 0] and on (0, 1]. On [1, 0], g 1 (y) =
y; on [0, 1], g 1 (y) = y. The range of Y is [0, 1). Using the ch
Homework 6 Solution
3.2 14. Let Ii be the indicator that the elevator will stop on the ith oor, which is
equivalent to say that there are at least one person who will get out on the ith oor.
Therefore, the expectation of Ii is
9
10
E(Ii ) = P (Ii = 1) = 1
Homework 10 Solution
5.3 13. (a) We change the variables from (X, Y ) to the polar coordinate (R, ). We
need to multiply the the Jacobian of r. The joint density of (R, ) is
fR, (r, ) = fX,Y (x, y) r =
r r2
r x2 +y2
e 2 =
e 2.
2
2
for r (0, +) and (0, 2).
Homework 4 Solution
2.4 2. (a) n = 500, p = 0.02, = np = 10. By Poisson approximation
1
P (1 success) = e = 10 e10 = 0.0004540.
1!
(b)
P (2 or fewer successes)
= P (0 success) + P (1 success) + P (2 successes)
0 1 2
e + e + e
0!
1!
2!
2
e
= 1+
2
=
= 0.
5.2 9. (a) We start with the probability of the event cfw_X > x, Y y (because X is
the minimum and Y is the maximum).
P (X > x, Y y)
= P (min(S, T ) > x, max(S, T ) y)
= P (S > x, T > x, S y, T y)
= P (x < S y, x < T y)
= P (x < S y)P (x < T y) ( S, T ind
Homework 1 Solution
1.1 3. (a) If we sample with replacement, the outcome space has n2 elements:
= cfw_(1, 1), (1, 2), ., (1, n), (2, 1), ., (n, n).
The probability of (1, 2) is 1/n2 .
(b) The event of consecutive integers is
E = cfw_(1, 2), (2, 3), ., (
DECISION 518Q - Midterm 2017
Applied Probability and Statistics
Version A
Name
Date
Instructions:
You have two hours to complete the exam.
Write in your name and date on the exam.
Write in your name and date on the scantron (Akindi) form.
Bubble in yo
DECISION 518Q - Practice Midterm
Applied Probability and Statistics
Name
Date
Instructions:
This is an ungraded version of the upcoming midterm.
Solutions to the multiple choice questions are available on the last page of the document.
Video solutions t
Poisson Distribution
Binomial Approximation
Binomial Approximations
Last week we looked at the normal approximation for the binomial
distribution:
Lecture 5: Poisson, Hypergeometric, and Geometric
Distributions
Works well when n is large
Sta230/Mth230
Con
Example
Imagine you have a bag with 6 slips of paper numbered 1 to 6. How many
dierent pairs can you draw if you sample without replacement?
Lecture 3: Binomial Distribution
Sta230/Mth230
Colin Rundel
January 22, 2014
Sta230/Mth230 (Colin Rundel)
Permutat
More Conditional Probability
Lecture 2: More Condition Probability, Distributions
1
More Conditional Probability
2
Probability Distributions
Statistics 104
Colin Rundel
January 17, 2012
Statistics 104
Lecture 2: More Condition Probability, Distributions
M
Introduction to Probability
What does it mean to say that:
The probability of rolling snake eyes is P (S) = 1/36?
The probability of ipping a coin and getting heads is P (H) = 1/2?
The probability Apples stock price goes up today is P (+) = 3/4?
Lecture 2
Homework 7 Solution
4.2 4.(a) Let X be the lifetime of the component. Then X Exp() with = 1/10.
(a)
P (X 20) = e20 = 0.1353.
(b)
Median(X) =
log 2
= 6.931.
(c)
1
= 10.
(e) Suppose the two lifetimes are X1 and X2 . Then the probability we want is
SD(X) =
P
MATH 230/STAT 230: Homework 8
Due on 11/10, in class
Read Section 4.2, and Section 4.4
Problem 1. Assume that calls arrive at a call centre according to a Poisson arrival process
with a rate of 15 calls per hour. For 0 s < t, let N (s, t] denote the numbe
MATH 230/STAT 230: Homework 7
Due on 10/29, in class
Read Section 3.4, Section 3.5, and Section 4.1
Problem 1. Suppose X takes values in (0, 1) and has a density
f (x) = cx2 (1 x)2
x (0, 1)
for some c > 0.
(a) Find c.
(b) Find E(X).
(c) Find Var(X).
Probl
MATH 230/STAT 230: Homework 6
Due on 10/22, in class
Read Section 3.3-Section 3.4
1
Problem 1. Let A1 , A2 , and A3 be events with probabilities 1 , 4 , and 1 respectively. Let
5
3
N be the number of these events that occur. For example, N = 2 if two of A
MATH 230/STAT 230: Homework 5
Due on 10/15, in class
Read Section 3.1-Section 3.2
Problem 1. Let us assume that we have n + m independent Bernoulli (p) trials. For any
r > 0, let Sr be the number of successes in the rst r trials, Tr the number of successe
MATH 230/STAT 230: Homework 4
Due on 09/29, in class
Read Chapter 2 (excluding Section 2.3), Appendix 1, and Section 3.1 (up to page 148)
1
Problem 1. Let S be the number of successes in 25 independent trials with probability 10
of success on each trial.
MATH 230/STAT 230: Homework 3
Due on 09/17, in class
Read Section 1.5-Section 1.6, and Section 2.1
Problem 1. A biased coin lands head with probability 2/3. It is tossed three times.
(a) Given that there is at least one head in three tosses, what is the c
MATH 230/STAT 230: Homework 2
Due on 09/10, in class
Read Section 1.1-Section 1.5
Problem 1. Show that for any collection of events cfw_Ai n+1 ,
i=1
n+1
n
i=1
n
Ai + P(An+1 ) P
Ai = P
P
i=1
(Ai An+1 ) .
(1)
i=1
Problem 2. (i) For two events A and B show t
MATH 230/STA 230: Homework 1
Problem 1. Show that (i)
n
1
j = n(n + 1),
2
j=1
and
(ii)
n
j2 =
j=1
n(n + 1)(2n + 1)
.
6
Hint: (i) Notice that 1 + n = n + 1, 2 + (n 1) = n + 1, 3 + (n 2) = n + 1 and so on.
(ii) Note (j + 1)3 j 3 = 1 + 3j + 3j 2 .
Problem 2.
Name
Math 135 Midterm 1 Solutions
Math 135.01 Probability
Feb. 13, 2007
There are ve questions on this test. Partial credit will be given on questions 2 through 5. DO use
calculators if you need them. You are allowed one sheet both sides with whatever you
Key to Final 2013
Note: Stared questions (*) are not in the range of our nal.
1. (a)
P (T > t) = P (T > t|Roof)P (Roof) + P (T > t|Tree)P (Tree)
10
1
1
= et/10 +
2 10 + t 2
1 t/10
10
=
e
+
.
2
10 + t
Therefore for t > 0,
dP (T > t)
dt
1
10
1
= et/10
2
10
2+3
A <- 2
2*a
Runif (pdf for uniform distribution in R) or type ?uniform in R console to get help
Rnorm
R = no. of times we sample
N = sample size
(out of total students, I take 10 people and take their average so n=10). If I sample 10 people a 100
times
DECISION 518Q
Applied Probability and Statistics
Writing Assignment 1: Identifying Probabilistic Mistakes
Assignment Overview: The purpose of this assignment is to help you identify
probabilistic mistakes made in the reporting of current event
Class 2
Predictive Modeling: Regression
Decision 618: Data Analytics for Business
Predictive Modeling
Regression Paradigm
Linear and Quantile Regressions
Dealing with Linearity
Healthcare Analytics Case
Visualization
Testing independence across dummy vari
Toxic Release Inventory
Basic Data File Format
Documentation v15
Prepared by:
The Environmental Protection Agency
Office of Environmental Information
Office of Information Analysis and Access
Environmental Analysis Division
Toxics Release Information Bran
Class 3 Conditional Probabilities
1
I had a feeling once about mathematics - that I saw it
all. Depth beyond depth was revealed to me - the
Byss and Abyss. I saw - as one might see the transit
of Venus or even the Lord Mayor's Show - a quantity
passing th