Exercise: Histograms - Estimate of a Probability Distribution Function 1. Make a histogram of the 30 call center service times below. Rule of Thumb: For n observations, make n/5 categories Are service call times discrete or continuous? Label the horizontal axis and the vertical axis. Comment on the shape – does it look like the bell shaped curve? Use histogram to estimate the probability the service time is greater than 15? Use histogram to estimate the probability the service time is less than 8 Find the average and sample standard deviation of the service times: 6 1 12 4 2 37 1 1 17 19 3 26 29 6 23 5 11 5 9 0 8 3 2 35 3 4
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Exercise: Histograms - Estimate of a Probability Distribution Function
1. Make a histogram of the 30 call center service times below. Rule of Thumb: For n observations, make n/5 categories
Are service call times discrete or continuous?
Label the horizontal axis and the vertical axis.
Comment on the shape – does it look like the bell shaped curve?
Use histogram to estimate the probability the service time is greater than 15?
Use histogram to estimate the probability the service time is less than 8
Find the average and sample standard deviation of the service times:
2. Are the following discrete or continuous random variables?a) Number of texts you send per dayb) Letters per text that you sendc) Time to text messaged) Time per phone calle) Customers who enter Apple Store per hourf) Fraction of customers who enter store that buy something per hour
Exercise: Sample Spaces, Events, and Set Operations
Restaurants are classified by 2 characteristics: parking and food. The results of 110 restaurants are shown below:
foodgood bad
parking good 85 7bad 10 8
a) Define an event that the food is good and an event that the parking is good. Use set notation to name the set of restaurants that are good in both characteristics. How many restaurants are in the set?
b) Use set notation to identify parts that are bad in at least one characteristic. How many restaurants are in the set?
A system consists of 2 components in series. Let E 1 denote the event that component 1 works. Let E 2 denote the event that component 2 works. Write the events below in terms of E 1 and E2. Draw a Venn Diagram also.
a) the system works
b) the system doesn’t work
c) at least one component works
d) component 1 works and component 2 doesn’t
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e) both components work
f) Are events E1 and E2 mutually exclusive?
g) Is E1 a subset of E2 ?
A system consists of 50 components in series. Write the events below in terms of E 1… E50.
a) the system works
b) the system doesn’t work
c) at least one component works
d) all components work
e) None of the components work
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Exercise: Definition of Probability & The Addition Rule
Call center customer representatives are classified by 2 characteristics: Speed and accuracy. The ratings for 200 representatives are shown below:
accuracygood bad
speed good 150 10bad 15 25
1) Define events.2) Find the probability that a representative is good in both speed and accuracy.3) Find the probability that a representative bad in both speed and accuracy.
A system is subject to two types of shock: A and B. The probability of shock A is 0.2 and the probability of shock B is 0.3. The probability of both occurring is 0.05.
1. Define notation and write the given information.
2. Find the probability that no shocks occur.
3. Find the probability that at least one type of shock occurs.
4. Find the probability that shock A occurs or shock B occurs.
5. Find the probability that shock A occurs and shock B occurs.
6. Find the probability that shock A occurs and shock B does not occur.
The backup system for a hospital electric system consists of two generators in parallel. The probability that the first generator fails is 0.001. The probability that the second one fails is 0.03. The probability that both fail is 0.00005
7. Define notation and write the given information.
8. Find the probability the system operates.
The process of placing a phone call consists of two activities. The caller’s phone has to work and the receiver’s phone has to work. The probability that the caller’s phone fails is 0.01. The probability that the receiver’s phone fails is 0.01. The probability that both fail is 0.002.
9. Define notation and write the given information.
10. Find the probability a call is completed.
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Exercise: Conditional Probability and the Multiplication Rule
A system is subject to two types of shock: A and B. The probability of shock A is 0.2 and the probability of shock B is 0.3. The probability of both occurring is 0.05. Define events and use notation to solve the problems.
1. Find the probability that no shocks occur.
2. Find the probability that shock A occurs. Find the probability A occurs given shock B occurred. Compare the numbers. What do they tell you?
3. Find the probability that shock B occurs. Find the probability B occurs given shock A occurred.
4. Find the probability that shock B does not occur. Find the probability that shock B does not occur given shock A occurred.
Consider students at Rutgers. Ten percent major in engineering. 20% of engineering students are women. Fifty percent of students are women. Define events.
5. Find the fraction of students at Rutgers that are both engineering majors and women.
6. Find the fraction of women that are engineers.
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A part can have good or bad surface finish and good or bad length. The probabilities are given in the table below. Define events and use notation to solve the problem.
lengthBad Good
surface bad .05 .02finish good .03 .90
7. Find the probability that the surface finish is bad given the length is bad.
8. Find the probability that the surface finish and the length is bad
9. If the surface finish is bad, find the probability that length is bad.
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Exercise : Law of Total Probability
DEFINE ALL NOTATION-WRITE THE GIVEN
There are three assembly lines. Line A makes 40% of the units, line B makes 50%, and line C makes 10%. The defective rate is 0.03 for A and 0.07 for B and 0.12 for C.
1. What is the fraction defective of the total units made?
2. What is the fraction of good units?
3. Find the fraction of units that are defective and from line B.
4. Find the fraction of units that are made on line A or B.
5. Of the units produced by line B, what fraction are defective?
6. Of the units that are defective, what fraction came from line B?
7. 60% of computers have anti-virus software. Among those with the anti-virus software 1 percent have virus-related crashes. Among those without the anti-virus software, 8 percent have virus-related crashes. Find the percent of virus-related crashes.
Consider a sample space S and two events, E and F.
8. Write the addition rule.9. Write the multiplication rule.10. Write the definition of independence.11. Write the law of total probability.
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Quiz– Name_____________________________________
Consider sample space S and two events, G and H, that are subsets of S. Define:
A company that tracks the use of its website determines that the more pages a visitor views, the more likely the visitor is to provide contact information. Here is the data:
Pages viewed 1 2 3 4+% visitors 40 30 20 10% visitors that give contact info
10 10 20 40
3. If a visitor provides contact information, find the probability that the visitor viewed 4 or more pages.
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Review Quiz
At Rutgers, 5% of students are in Engineering. In Engineering, 20% are women. . Among students not in engineering, 55% are women.
Define notation and write the given information. Is gender (man or woman) independent of major (Engineering or not)? Find the fraction of men at Rutgers across all majors Suppose I randomly choose a woman student. Find the probability she is in
engineering. Find the percent of students at Rutgers that are women engineers.
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Random Variables: PDF, CDF
x -1 0 1 2 3f(x)
0.1 0.2 0.4 0.2 0.1
Verify that f(x) is a probability distribution function.
For each item below, write the question in terms of f(x) and F(x) and find the answer.
1. ex: P(X=0)=f(0)=0.2
2. P(X-33)
3. P(X=0.4)
4. P(X203)
5. P(X2)
6. P(0<X2)
7. P(X>0)
8. A random variable can have the values 1, 2 or 3 with equal probabilities. Give the pdf and the cdf as a
Table
Formula
Graph
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9. Write the pdf f(x).
F ( x )={0 x<0.2
0.2 0.2≤x<1.20.5 1.2≤x<2.20.7 2.2≤ x<3.2
13.2≤ x
10. What are the possible values of the random variable X?
11. Find f(2)
12. f(0.7)
13. F(0.5)
14. F(0.6)
15. F(2)
16. F(-1.6)
17. F(3.99)
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Exercise: Expected Value and Variance of a r.v.
Here is the probability distribution function of the random variable X.
x -1 2 4f(x) .3 .5 .2
1. Make a graph of the probability mass function.
2. Make a graph of the cumulative distribution function.
3. Write the cdf as a formula - F(x) = {
4. Find the expected value of X.
5. Find the standard deviation & variance of X using the definition V(X)=E(X-µ)2.
6. Find the expected value of X2 and confirm that V(X)=E(X2)-E(X)2 .
7. Find the expected value of the r.v. Z with the following cdf:
F ( z )={ 0−∞<z<400.5 40≤ z<50
150≤ z<∞ }
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Binomial Distribution
USE the following format for Binomial define random variable; i.e., r.v. X = number of successes in N independent trials give distribution and parameters; i.e., X~Binomial (N= , p= ) write question in notation.
Among first year engineers, 20% are women. At orientation, groups of 4 new students are randomly chosen to tour the campus together.
1. Find the probability that a group one woman and three men.
2. Find the probability that a group has all women.
3. Find the probability that a group is all men.
4. Find the probabililty that a group has at least one women.
5. Find the probability that a group is 2+2.
6. Find the probability a group has at least one man.
7. What is the expected number of women in each group?
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Users attempt to logon to a network. Ninety percent of attempts to logon succeed. (Is this Binomial – are attempts independent – are you willing to assume independence?)
8. Find the probability 2 out of 8 attempts fail.
9. Find the probability at least 2 out of 8 attempts fail.
10. Find the expected number and standard deviation of attempts out of 8 that work.
11. A sample of 6 packages is taken from a production line. Historically, 5% of packages are defective. If more than one package in the sample is defective, the production line is stopped to investigate possible problems. Assume the process is working at its usual 5% defective rate. Find the probability of a false alarm – stopping the line when it is operating as usual.
12. A product consists of 4 components in parallel – these are independent of one another. Each component has probability 0.8 of working. If I sample 10 of these products, find the probability that at least one doesn’t work.
13. Create a Binomial situation, give adequate information, ask a question and give the answer.
Hint: the outcome is this or that with prob p and 1-p. N independent trials. Find the probability of at least one “this.”
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Binomial & Geometric DistributionBinomial define the random variable r.v. X = number of successes in N independent trials give the distribution and the parameters; i.e., XBinomial (N= , p= )Geometric define the random variable r.v. X = number of trials to first success give the distribution and the parameters; i.e., XGeometric( p= )
Users attempt to logon to a network. Ninety percent of attempts to logon succeed.
1. Find the probability that it takes at least three attempts until you logon.
2. Find the expected number of attempts until successfully logging on.
A system consists of 5 components in series and parallel. The system works if at least 3 out of the five components work. (This is called a k out of n system.) The components are independent and each one fails with probability 0.1.
3. After assembly each system is tested. Find the probability that the number of systems tested until the first failed system is 2 or more.
4. Parts can have two types of defects: A and B. Five percent have type A; 3 percent have type B; and 1 percent have both. Find the expected number of parts inspected until I find one that is defect free.
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Exercise: Binomial, Geometric, Poisson
Shipments of raw materials are on time with probability 0.9 and late with probability 0.1.
1. Find the probability that exactly one shipment out of the next five is LATE.
r.v. X=
X~
P(X=
2. Find the probability that the fifth shipment is the first LATE one.
Name the distribution of the following random variables and you give the parameters:3. Number of texts per hour
4. Number of traffic lights to the first red one
5. Number of disabled cars per mile of turnpike
6. Number of dirty spots on a plate in a restaurant
7. Number of shooting stars per night
8. Number of keys to first one that works in a box of 1000 keys
9. Number of first years students at a table for four in student center
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10. A unit contains 3 components in parallel, each with probability of failure 0.4. Find the expected number of units inspected until the first failed unit.
11. The probability a subject lies is 0.10. A lie detector has two kinds of errors: The probability of detecting a lie when the subject is telling the truth is 0.05. The probability of not detecting a lie when the subject is lying is 0.15. What is the probability a subject is telling the truth if the lie detector indicates a subject is telling the truth.
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Poisson Distribution
Requests to a warehouse have a Poisson distribution with mean 0.7 per hour.
1. Find the probability of 2 requests in an hour.
2. Find the probability of at least 2 requests in an hour.
3. Find the probability of 2 requests in 1.5 hours.
4. Find the probability of 1 request in 45 minutes.
Emma, the newspaper seller, has demand that is Poisson with mean 4.0 per morning. F(x) for selected values of x are given below. If Emma buys too many papers from the supplier, she loses money when they are left unsold. If she buys too few, she will not get her profit and may even lose business in the future.
5. Find the probability of more than 7 newspapers of demand per day.
6. Find the probability of four or less newspapers day.
7. How many newspapers should Emma order such that the probability of running out of product is less than 2%?
Mean and Variance of a function of a discrete random variable
The pdf of r.v. X is f(x)=8 /7 (1/2)x x=1,2,3
1. Find F(2)
2. Find P(X<1.5)
3. Find E(X) and SD(X).
4. Find E(X3/2)
5. Find V( X3/2 )
6. Find mean and variance of the random variable Y=2X+6
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7. A product is seasonal and yields a net profit of $100 for each unit sold and a net loss of $50 for each unit left unsold at the end of the season. The number of customers is a r.v. with the following distribution. The mean number of customers is 2.8 and the variance is 1.96 and the sd is 1.4. If I stock 4 items, find mean of the profit.
r.v. C=number of customers
c 1 3 5f(c) 0.3 0.5 0.2
Note: Is the function linear (that would save a lot of trouble)?
Conditioning on Random Variables to Solve Problems
A bicycle built for two is used as an airport limousine. The company accepts up to three reservations per trip, though only two riders are possible, since previous records indicate 20% of passengers with reservations do not show up.
The distribution of r.v. R, the number of reservations made is:
r 1 2 3f(r) .1 .6 .3
1. Consider the random variable S, the number that show up. The possible values for S are: 0, 1, 2, 3. Find the distribution of S by conditioning on the number of reservations. You can verify that P(S=0)=.0464 and P(S=1)=.3008. Find P(S=2) and P(S=3).
2. Consider the random variable, T, the profit for each trip. Each customer pays $5. However, if a customer has a reservation and then cannot get on the bicycle, the company pays for a taxi which costs $7. Find the mean and variance of profit.
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Exercise: Continuous Random Variables
PDFs, CDFs
The pdf for a random variable is
f ( x )=kx 2 −1<x<1
1. Draw f(x)
2. Find P(X<3)
3. Find P(X>-2)
4. What must k equal so that f(x) is a pdf?
5. Find P(X=0.5)
6. Find f(0.5)
7. Find P(X<0.5)
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8. Find P(-0.3<X<0.2)
9. Find P(X ≤ -0.8)
10. Find P(X ≤ -0.2)
11. Find P(X ≤ .0.1)
12. Find P(X ≤ 0.5)
13. Find, in general P(X ≤ x)
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Continuous Random Variables: PDF, CDF, Mean, Variance
Consider the r.v. X with the following pdf: f(x)=(k)x2 -2<x<2
1. Find k such that f(x) is a pdf.
2. Find P (X<-1)
3. Write cdf for X.
4. Find P(X=1.5)
5. Find 75th percentile of this distribution
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Consider the r.v. X with the following pdf: f(x)=(k)x2 -2<x<2
6. Find E(X)
7. Find V(X)
8. The r.v. W=5-3X. Find the mean and variance of W
9. Find the mean and variance of the r.v. Y where Y={ X0 .5 X>0
0 elsewhere .
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Standard Normal Distribution
The dimension of a part is normally distributed with mean 20 and standard deviation 2. Estimate the following based on your knowledge of the normal distribution.
1. Draw the distribution. Define X. Label the axes.
2. Find the probability the dimension is between 16 and 24.
3. Find the probability the dimension is greater than 22.
4. Find the probability the dimension is less than 14.
5. Find 84th percentile.
6. Given dimensions to a potential customer such that 99% of parts will fall within those dimensions.
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Use the standard normal table to find the following: DRAW THE PICTURE
7. P(Z≤1.5)
8. P(Z<-1.82)
9. P(Z>2.34)
10. P(Z>-1.9)
11. P(-2.06<Z<-1.06)
12. Find z value such that 92% of the std normal < z; find Z such that Φ ( z )=0.92
13. Find z value such that 9% of the std normal < z; find Z such that Φ ( z )=0.09
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Exercise: Normal Distribution
Define the random variable, Draw the picture!
The dimension of a part is normally distributed with mean 15 inches and standard deviation 0.3.
1. Find the fraction of parts that are greater than 14.5 inches but less than 15.2 inches.
2. Find the 10th percentile of the dimension
3. What should the mean be set to such that 95 percent of parts have diameter exceeding 14. 7 inches? Assume the sd remains at 0.3.
4. What should the sd be set to such that 95 percent of parts have diameter exceeding 14.7 inches. Assume the mean remains at 15.
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The pounds of a bulk chemical sold per day has the normal density given below:
f ( x )= 13√2 π
e− 1
2 ( x−903 )2
- ∞<x<∞
The cost of the material sold is C=3X+5.
5. Find the probability that the amount sold per day exceeds $93.
6. Find the mean and variance of cost.
7. Can you find the mean and variance of cost if C=3X2?
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Exercise: Exponential and Memoryless Property – Define the r.v.
The lifetime of an electronic component is exponentially distributed with mean 16 months.
1. Define a random variable for the lifetime and write its density function.
2. Find the probability a component has lifetime greater than 16 months.
3. Find the 50th percentile of the distribution which is called the median. Note that the mean and median for an exponential are not equal.
4. Given the lifetime is greater than 12 months, find the probability that the lifetime is greater than 28 months.
5. What is the average number of failures per month. Per week?
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The time between power failures is Exponentially distributed with mean 9.6 days.
6. Find the probability that the time to the next failure exceeds 2 weeks.
7. There has been no power failure for 4 weeks. Find the probability that there will be at least one failure in the next week.
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8. Exercise: Exponential and Poisson
The lifetime of a critical component is exponential with mean 83.33 hours (40 hours operation in a week). When the part fails it is immediately replaced with a new part.
1. Write distribution of time between replacements in weeks.
2. What is the expected number of replacements per week
3. Write distribution of the number of replacements per week.
4. Find the probability of at least1 replacement in a week.
5. Find the probability a part lasts at least 3 weeks (2 ways!)
6. A part has been operating for 200 weeks! Find the probability of at least one failure within the next week.
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The distance between major flaws in a roadway is exponential with mean 2.7 miles.
7. Find the probability of more than 2 flaws in 5 miles.
8. Find the probability that the first flaw occurs between miles 2.5 and 3.5.
The lifetime of a component has the exponential distribution with mean 6.1 months. If the component fails, it is immediately replaced by a new one.
9. Find the probability of at least two failures in a month.
10. I need a year’s supply of the component. If I order 3 components, find the probability that I will have enough components to operate for the whole year.
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Joint Distributions
Use new notation to answer the following questions.
The joint distribution of X1 and X2 is given below:
X2=0 X2=1 X2=2X1=0 0.2 0.3 0X1=1 0.3 0.1 0.1
Given E(X1)=0.5, V(X1) =0.25, E(X2)=0.4, V(X1) =0.24
1. Write the marginal distribution of X2.
2. Find P(X2≤1¿.
3. What is f x1x2(0,1)
4. Write the conditional distribution of X2 given X1=1.
5. What is E(X2 / X1=1).
6. What is V(X2 / X1=1)
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X2=0 X2=1 X2=2X1=0 0.2 0.3 0X1=1 0.3 0.1 0.1
Given E(X1)=0.5, V(X1) =0.25, E(X2)=0.4, V(X1) =0.24
7. What is E(XY)
8. What is E(3X+2Y)
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Mean & Variance of a Linear Function of Independent Random Variables
1. The number of calories I consume for lunch is normally distributed with mean 500 and sd 50. The number of calories I use during the day is normally distributed with mean 500 and sd 20. Find the mean and sd of the net amount of calories per day.
2. A material is composed of 5 layers of laminate, each with mean 2 and standard deviation 0.2. Find the probability that the total thickness exceeds 10.5. What assumptions are you making?
3. A truck will carry 4 cartons of soda bottles. Each carton consists of 12 bottles, each with weight normally distributed with mean 16.1 ounces and standard deviation 0.2. Each carton has weight that is normally distributed with mean 8 ounces and standard deviation 0.5 ounces. Find the distribution, mean, and standard deviation of the the total weight.
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4. A process uses two raw materials, A and B. The amount of A used per week has mean 100 pounds and standard deviation 5 with a cost of $6 per pound. The amount of B used per week has mean 50 pounds and standard deviation 3 with a cost of $8 per pound. Find the mean and standard deviation of the cost of raw material in one year (52 weeks).
5. One piece of PVC pipe is to be inserted inside another piece. The overlap should be normally dist with mean 1 inch and sd 0.1. The lengths of the pieces are independent and normally distributed: for piece 1 the mean is 20 and sd 0.5 inches; for piece 2 the mean is 15 and sd 0.4 inches. Find the probability that the total length after insertion is between 34.5 and 35 inches.
6. The width of a drawer has mean 36 inches and standard deviation 0.5. The width of the frame that it fits into has mean 36.1 and standard deviation 0.7 The distributions are normal. what percent of randomly selected drawers & cabinets won’t fit?
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Mean & Variance of a Linear Function of Independent Random VariablesClearance, Averages, Profits, Candies
Based on historical data, the weight filled by a filling machine is normally distributed with mean 50 and standard deviation 0.75.
1. Find the probability that a single container weight exceeds 51.5.2. Find the probability that the average of nine containers exceeds 51.5.
Each section of fencing is normally distributed with mean length equal to 6 feet and standard deviation 0.3 feet.
3. If 10 sections of fencing are installed, end to end, find the mean and standard deviation of the total length.
Each layer of a laminate has thickness normally distributed with mean 1 cm and standard deviation 0.1 cm. I create a composite with 3 layers.
4. Find the mean and sd of the composite in cm.5. Find the mean and sd of the composite in inches.
1cm=0.4 inches or 1 inch=2.5 cm6.
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Central Limit Theorem
The weight on a bridge is estimated to have the uniform distribution between 3 and 5 tons. At 16 random times I observe the bridge, record the vehicles on the bridge, and figure out the weight. I then compute the average.
Give the mean and standard deviation of the sample average. Find the probability that the sample average exceeds 4.5 tons. In your calculation, show where independence is necessary and why you are
willing to assume independence. Show where you invoke the Central Limit Thm. Why do you need it?
The joint distribution of X1 and X2 is given below:
X2=0 X2=1X1=0 0.2 0.3X1=1 0.4 0.1
Given E(X1)=0.5, V(X1) =0.25, E(X2)=0.4, V(X1) =0.24
1. Write the marginal distribution of X2.2. Write the conditional distribution of X2 given X1=1.3. Find P(X2≤0¿4. What is f x1x2
(0,1)5. What is E(X2 / X1=1).6. What is V(X2 / X1=1).7. Guess: is the correlation of X1 and X2 negative or positive?8. Compute the covariance of X1 and X2 .9. Compute the correlation of X1 and X2 .10. Compute the mean and variance of Z=5X1X2
2 11. Compute the mean and variance of W=5 + 3X1 + 4X2 (Watch out for
independence!!)
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The time to assemble a unit is exponentially distributed with mean 2 minutes. I have 9 units to assemble.
7. Find the mean and sd of the total time to make the assemblies.8. Find the probability that the total time exceeds 27 minutes.
Confidence Intervals
1. What is z.03? z.97? z.01?2. The standard deviation of a dimension is known to be 0.8. For a random sample
of 5 parts the dimensions are: 20.2, 20.4, 19.9, 19.5, 18.9. (Confirm that the sample average is 19.78 and the sample standard deviation, S, is 0.59.) Create a 95% and 94% confidence interval for the mean dimension.
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