ENGINEERING PROBABILITY Spring 2012 Homework #6 _ Solutions Topic More on Bayes Rule and Independence
2155. 2157. Let A = excellent surface finish; B = excellent length a) P(A) = 82/100 = 0.82 b) P(B) = 90/100 = 0.90 c) P(A') = 1 0.82 = 0.18 d) P(AB) =
ENGINEERING PROBABILITY Spring 2012 Homework #8 _ Solutions Topic Binomial Distribution
375. A binomial distribution is based on independent trials with two outcomes and a constant probability of success on each trial. a) reasonable b) independence assu
ENGINEERING PROBABILITY Spring 2012 Homework #5 _ Solutions Topic Independence and Bayes Rule
2136. Let A denote the upper devices function. Let B denote the lower devices function. P(A) = (0.9)(0.8)(0.7) = 0.504 P(B) = (0.95)(0.95)(0.95) = 0.8574 P(AB)
Welcome to Engineering Probability 540:210 Professor Susan
What is probability?
Branch of math that deals with uncertainty Repeat operation again & again Each time, different outcome IE: How to operate efficiently under uncertainty
IEs Make Decisions Unde
Rutgers University
College of Engineering
Department of IE
540:210 Engr. Probability
Instructor: Dr. S. L. Albin
Midterm II - Spring 2015
Define all random variables.
1. The weight of a liquid filled into a container is normally distributed with mean 12.1
Homework Solution for Week 2
Due on February 3rd, 1:20 pm
2-63
Solution:
(a)
S = cfw_1,2,3,4,5,6,7,8
(b)
2/8
(c)
6/8
2-86
Solution:
(a)
P(High temperature and high conductivity)= 74/100 =0.74
(b)
P(Low temperature or low conductivity)
= P(Low temperature)
Homework Solution for Week 1
Due on January 30th, 1:20 pm
2-2
Each of four transmitted bits is classified as either in error or not in error.
Solution:
Let e and o denote a bit in error and not in error (o denotes okay), respectively.
eeee , eoee , oeee ,
Homework Solution for Week 4
Due on February 17th, 1:20 pm
2-168
Solution:
Let F denote a fraudulent user and let T denote a user that originates calls from
two or moremetropolitan areas in a day. Then,
(|)()
0.3(0.0001)
P(F|T) =
=
= 0.003
(|)() + (|)() 0
Homework Solution for Week 6
Due on March 3rd, 1:20 pm
3-100
Solution:
Let X denote the number of defective circuits.
Then, X has a binomial distribution with = 40 and = 0.01
40
( = 0) = ( ) (0.01)0 (0.99)40 = 0.6690
0
3-108
Solution:
() = 20(0.01) = 0.2
Rutgers University
Dept. of Industrial & Systems Engineering
School of Engineering
Professor Susan Albin
CORE Bldg. 206
x5-2238; [email protected]
Engineering Probability 540:210 - Exam I Spring 2015
Define all random variables and events. Specify probability
di
1
Discrete Random Variables
Discrete Random Variable X
2
ex: flip a coin twice r.v. X = number of heads (now each possible outcome is attached to a number)
S = cfw_ HH , TH , HT , TT
2 1 1 0
Sample Space set of possible outcomes Assign a numerical value
1
Bayes Rule
2
Revisiting the Suppliers WITH A NEW QUESTION
A supplies 60% of parts with fraction defective 0.02 and B supplies 40% of parts with fraction defective 0.05. If I find a defective part, what is the probability it came from supplier A? Given:
1
Mean and Standard Deviation of a Random Variable
2
Mean: Measure of Central Tendency
Guess the Mean
x 0 1 2 f(x) .15 .70 .15
Is this mean higher or lower
x 0 1 2 100 f(x) .14 .70 .14 .02
Expected Value of discrete r.v. X
3
Called:
Expected Value of X
1
Binomial Distribution
2
Binomial Distribution: the Situation
Experiment
There are N independent trials. The outcome for each trial is success with probability p and failure with probability 1-p.
Define r.v. X = number of successes in N independent trial
1
Geometric Distribution
Geometric: Experiment and r.v.
2
Experiment
Conduct independent trials. The outcome on each trial is success with probability p and failure with probability 1-p. Stop at the first success.
R.v. X=number of trials to the first succ
1
Poisson Distribution
Poisson Distribution: Examples
2
r.v. X = number of events in an interval Possible values X=0, 1, 2, .
r.v. X = number of defects on a square inch of painted surface r.v. X = number of earthquakes in a year r.v. X = number of typo
1
Mean & Std Deviation of a Function of a Discrete random variable
2
Example: What is a function of a random variable?
r.v. X= number of defects on a part. Here is f(x):
x f(x) 0 1 .3 .5 2 .2
The rework cost is $2 per defect r.v. C = rework cost r.v. C(
1
Total Probability rule and Random Variables
2
Use the Total Probability Rule and random variables
An airport limo has 4 seats. Depending on demand, the number of reservations made has the following distribution: 5 6 r 3 4
f(r) .1 .2 .3 .4
From previous
Homework #3 Solutions Topic Conditional probability and Multiplication rule
287. a) P(A) = 86/100 c) P( A B ) = b) P(B) = 79/100
Assignment 2-87, 89, 90,93, 95, 97, 104, 105, 106
P( A B ) 70 / 100 70 P( B) 79 / 100 79 P( A B) 70 / 100 70 P( A) 86 / 100
Homework #4 Solutions Topic Total probability rule and independence
2109. Let R denote the event that a product exhibits surface roughness. Let N,A, and W denote the events that the blades are new, average, and worn, respectively. Then, 2111. a) (0.88)(
ENGINEERING PROBABILITY Spring 2012 Homework #7 _ Solutions Topic Discrete random variables & Probability distribution functions & Expected values
32. 34. 37. 315. 316. All probabilities are greater than or equal to zero and sum to one. a) P(X 1)=P(X=1)