IE300 - Homework 1 (AL1)
Due: Thursday, February 6, 2014 in class
2-30. A sample of two items is selected without replacement from a batch. Describe the (ordered) sample space for
each of the following batches:
(a) (5 pts) The batch contains the items cfw
IE300 - Homework 4 (AL1)
Prof. Farzad Yousean
Due: Thursday, October 3, 2013 in class
3-108. A computer system uses passwords constructed from the 26 letters (a-z) or 10 integers (0-9). Suppose
there are 10,000 users of the system with unique passwords. A
IE300 - Homework 11 (AL1)
Prof. Farzad Yousean
Due: Thursday, December 5, 2013 in class
8-46. (10 pts) The sugar content of the syrup in canned peaches is normally distributed. A random sample
of n = 10 cans yields a sample standard deviation of s = 4.8 m
IE300 - Homework 3 (AL1)
Prof. Farzad Yousean
Due: Thursday, September 26, 2013 in class
3-18. Verify that the following function is a probability mass function, and determine the requested probabilities. f (x) = (3/4)(1/4)x , x = 0, 1, 2, 3, 4
(a) (5 pts
IE300, AL1, Midterm Exam 1, October 1
Name:
Discussion Section:
1. 20 points Consider the design of a communication system with 10-digit phone numbers.
All numbers have the same 3-digit prefix, which is 217.
(a) 5 points How many different phone numbers c
IE 300
Homework 3
Due: Tuesday, February 7, 2017 at 8:00 AM
6 questions
1. 10 points Let c be a constant X be a random variable. Prove:
(a) Var[cX] = c2 Var[X]
(b) Var[c + X] = Var[X]
2. 20 points Let X be a nonnegative integer valued random variable (i.e
IE 300
Homework 1
Due: Tuesday, January 24, 2017 at 8:00 AM
13 questions
Compute the following:
1.
df
,
dx
where f (x) = log(x).
2.
df
,
dx
where f (x) = ecx
3.
df
,
dx
where f (x) = xn
4.
where f (x) =
df
,
dx
x.
5.
n
X
1
5
n=0
,
(infinite sum of a geom
IE300, AL1, Midterm Exam 1, October 1
Name:
Discussion Section:
1. 20 points Consider the design of a communication system with 10-digit phone numbers.
All numbers have the same 3-digit prex, which is 217.
(a) 5 points How many dierent phone numbers can b
IE300, AL1, Midterm Exam 2, November 12
Name:
Discussion Section:
1. 40 points Determine the value of c that makes the function f (x, y) = c(x + y) a joint
probability density function over the range 0 < x < 3 and x < y < x+2. Then determine
the following
Fall 2015
IE300
Practice Midterm 1
Problem
1: Let
cxX 22 Gamma(5, 3). Recall that (k) = k! when k is an integer. Hint:
R 2 cx
3
x e dx = 1/c
e (c x 2cx + 2).
a) What is P (X 1)?
b) What is P (X 2)?
c) What is P (X 2|X 1)?
d) What is E[X 2 ]? Hint: dont c
_author_ = 'jmd'
import numpy as np
from scipy.stats import norm
from scipy.stats import binom
from matplotlib import pyplot as plt
import math
#to see what each object is USE PRINT STATEMENTS
#print standard normal cdf
print(norm.cdf(1)
#clears plot
plt.
IE 300
Homework 5
Due: Tuesday, February 21, 2017 at 8:00 AM
6 questions
1. 20 points Let X and Y be independent exponential random variables with probability
density functions 1 e1 x and 2 e2 y , respectively. Let Z = min(X, Y ). Prove that the
probabili
IE 300
Homework 2
Due: Tuesday, January 31, 2017 at 8:00 AM
5 questions
1. 20 points Let X be a random variable with probability mass function:
1
,
2
1,
P[X = x] = 41
,
81
,
8
if
if
if
if
x = 1.
x = 2.
x = 3.
x = 4.
Calculate E[X], Var[X], and P[X > 1].
IE 300
Summer 2017
Midterm Exam
07/07/2017
Time Limit: 120 Minutes
Name:
Show work to get partial credit.
No cell phones, notes, etc.
175 Points Total.
1. (30 points) Suppose that we have a die with
P (1) =
1
1
1
3
1
1
, P (2) = , P (3) = , P (4) = , P (5
IE 300
Summer 2017
Quiz 2
06/16/2017
Time Limit: 10 Minutes
Name:
1. (10 points) A medical test is to detect genetic mutation. If the patient has mutation, the
test will detect it with probability of 0.99; If the patient doesnt have genetic mutation,
the
IE 300
Summer 2017
Quiz 7
06/30/2017
Time Limit: 10 Minutes
Name:
1. Suppose X is an Exponential random variable with mean 5. What is density of X, fX (x)?
2. Suppose X has a uniform distribution over [2.5, 4.5]:
(a) Compute P (X 3.7)
(b) Compute P (X 4.8
IE 300
Summer 2017
Quiz 6
06/29/2017
Time Limit: 15 Minutes
Name:
1. (30 points) X is a continuous random variable with the following density:
fX (x) =
2.5
0
48 x 52, 0.05x y 2.6
ow
(a) (5 points) Whats the marginal distribution of X?
(b) (5 points) Whats
IE 300
Summer 2017
Quiz 5
06/28/2017
Time Limit: 15 Minutes
Name:
1. (5 points) Suppose that X is an Exponential random variable with mean 5. What is
the probability density function, fX (x), of X?
2. (20 points) Suppose that X is uniformly distributed ov
IE 300
Summer 2017
Quiz 1
06/14/2017
Time Limit: 10 Minutes
Name:
1. (15 points) Suppose that we have die with
1
1
1
3
1
1
, P (2) = , P (3) = , P (4) = , P (5) = , P (6) =
24
6
12
24
3
4
Lets define random variable X which is 2 if the outcome of the die
IE 300
Summer 2017
Quiz 4
06/26/2017
Time Limit: 10 Minutes
Name:
1. (20 points) X is a Cont. R.V such that fX (x) = Cx for 0 < x < 1.
What value(s) of C makes fX (x) valid?
Find CDF of X.
IE 300
Quiz 4 - Page 2 of 2
Answer:
Z 1
Cxdx = 1
0
C/2 = 1
C=2
Z
x
IE 300
Summer 2017
Quiz 3
06/21/2017
Time Limit: 10 Minutes
Name:
1. (10 points) Suppose that X is Poisson with parameter = 2. What is P (X = 5)?
IE 300
Answer:
P (X = 5) =
Quiz 3 - Page 2 of 2
e2 25
5!
06/21/2017
IE 300
Homework 4
Due: Tuesday, February 14, 2017 at 8:00 AM
5 questions
1. 20 points Let X be a continuous random variable with support on [0, 2] and with
probability density function f (x) = cx2 . What is the value of c? What is the cumulative
distribut
IE 300
Homework 6
Due: Tuesday, February 28, 2017 at 8:00 AM
8 questions
1. 20 points Let X1 , X2 , X3 , . . . be random variables. Each Xi is uniformly distributed
on [0, 1]. The random variables X1 , X2 , X3 , . . . are independent. The random variable
Analysis of Data
IE300, Fall 2013
Douglas M. King
Department of Industrial and
Enterprise Systems Engineering
University of Illinois
(C) Copyright 2013
IE300
1
WHAT IS PROBABILITY?
We encounter it regularly
Probability of passing this course
Probabilit
IE 300
Homework 7
Due: Tuesday, March 7, 2017 at 8:00 AM
4 questions
1. 25 points Let x1 , x2 , . . . , xn be observations of a random variable X with a Gaussian
distribution with mean and variance 2 . Derive the maximum likelihood estimator for
the varia
Analysis of Data
IE300, Spring 2017
Douglas M. King
Department of Industrial and
Enterprise Systems Engineering
University of Illinois
(C) Copyright 2013
IE300 (D. M. King)
1
HYPOTHESIS TESTING
Frequently, we ask questions about the population parameters
Analysis of Data
IE300, Spring 2017
Douglas M. King
Department of Industrial and
Enterprise Systems Engineering
University of Illinois
(C) Copyright 2013
IE300 (D. M. King)
1
INTERVAL ESTIMATION
Point estimation estimates a single value for an unknown pop
IE300 BL1 Analysis of Data
Assignment #7 Solution
QUESTION 1:
(a)
-+ f(x) dx
(b)
= 0 3x2 / 3 dx
= x3 / 3 |0
= (3 / 3) (0 / 3)
=1
Since we have one unknown parameter, we will estimate it by equating the first sample and population moments.
The first sample
IE300 BL1 Analysis of Data
Assignment #7 Solution
QUESTION 1:
(a)
-+ f(x) dx
(b)
= + e x + dx
= -e x + | +
= 0 (-e 0)
=1
Since we have one unknown parameter, we will estimate it by equating the first sample and population moments.
The first sample moment
Analysis of Data
IE300, Spring 2017
Douglas M. King
Department of Industrial and
Enterprise Systems Engineering
University of Illinois
(C) Copyright 2013
IE300 (D. M. King)
1
JOINT PROBABILITY DISTRIBUTIONS
We have analyzed
Single random variables
Groups