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T-EXERCISES
This file might be adapted during the course.
- Is there an interval [a, b] such that f (x) = sin(x) for a ≤ x ≤ b is a probability density function?
- The edge X of a cube is a random variable with probability density function
fX (x) =
2(3 − x) if 2 < x < 3 , 0 elsewhere.
a) What is the probability that the edge is longer than 2.5? b) What is the expected value of the edge of the cube? c) What is the expected value of the volume of the cube? d) Let W be the volume of the cube. What is the cumulative distribution function of W?
- A random variable X denotes the time between two crashes at the stock exchange and has the following density: fX (x) = (b − 1)/xb^ for 1 ≤ x < ∞, and fX (x) = 0 for x < 1. (We have b > 1).
a) Check that fX is a probability density function. b) Compute the expected value of X. Assume b > 2. c) Consider b = 4 and suppose we have ten independent random variables X 1... , X 10 with probability density function fX. Use the Central Limit Theorem to approxi- mate the probability that the average (X 1 + X 2 +... + X 10 )/10 is larger than 2. You may use the fact that V (Xi) = 3/4. d) Is it possible to approximate this probability if b = 3? Why?
- Correct or wrong? For the life time of a car the exponential distribution is a good model.
- Correct or wrong? The uniform distribution on the interval 0 ≤ x ≤ 1 has the lack of memory property: P (X < t + t 0 |t > t 0 ) = P (X < t).
- Correct or wrong? If X is exponentially distributed, then 2X is also exponentially distributed.
- Philips sells lamps and Philips claims that the life time is at least one year with a probability of more than 0.95. Assume that the lamps do not wear out (but fail because of incidents).
a) Show that the expected life time of such a lamp is at least − 1 / ln(0.95) ' 19. 49 year. b) Now assume that the expected life time is 20 year. Now assume that a lamps burns continuously and that it is immediately replaced when it fails. Is it sufficient to have four lamps to guarantee that one has light for more than 3 years with a probability of at least 0.95?
- Consider a supermarket with pay desks. The probability that one has to wait longer than 5 minutes before paying the bill at a pay desk equals 0.2. A man and a woman go shopping and choose different pay desks at the super market. They arrive at the same time at their pay desk.
a) What is the probability that the man and the woman both have to wait less than 5 minutes? Assume independency.
Assume that the waiting time is exponentially distributed with parameter λ.
b) Show that λ = 0.32 (in two digits). c) The man is waiting two minutes. What is the (conditional) probability that the (total) waiting time (including these two minutes) is more than 5 minutes?
- You call a helpdesk. You hear: there are three people waiting before you. Give a probability distribution that can be used as a model for the waiting time before it is your turn.
- Correct or wrong? If X has an Erlang distribution with parameters (r, λ) and Y is independent from X and has also an Erlang distribution with parameters (r′, λ′), then the sum X + Y is also Erlang distributed.
- The time between two arrivals of patients at a clinic in a hospital is an important variable. So is the treatment time of the patient. Very often it is assumed that both times are exponentially distributed. Assume that the mean time between two arrivals is 4 minutes.
a) What is the probability that the time between two arrivals is less than one minute? b) What is the distribution of the number of arrivals in 3 hour? Give also the para- meter(s) of the distribution. c) Give (using the normal distribution) an approximation for the probability that less than 60 patients arrive in 3 hours.
Assume that the mean treatment time is 5 minutes. Melvin arrives and two people are waiting.
d) What is the expected value of the time Melvin has to wait before it is his turn?
- Consider n uniformly distributed random variables on the interval [0, 1]. The random variables are independent.
a) What is the cumulative distribution function of the maximum of the n random variables? What is the expected value? b) What is the cumulative distribution function of the minimum of the n random variables? What is the expected value?
Application of item a): a production process where a product has to pass n stations on a production line. So, at the same moment n products are on the line (in each station one product). The process can continue if on all the stations the operation has been completed. Let Xi be the operation time in station i. Let T be the time it takes before the production process can continue. Then T = max{X 1 , · · · , Xn}.
- A company sells bags of 25 kg special sand used for making cement. The company purchases these bags for 1.50 euro per sack. The company sells the bags for 2.50, but gives some discount if a customer buys more bags. See the table below.
Number of bags Price Profit Profit per the customer buys per bag per bag customer i w(i) y 1 2. 50 1. 00 1. 00 2 2. 45 0. 95 1. 90 3 2. 40 0. 90 2. 70 4 2. 36 0. 86 3. 44 5 2. 32 0. 82 4. 10 6 2. 28 0. 78 4. 68 7 2. 25 0. 75 5. 25 8 2. 22 0. 72 5. 76 9 2. 19 0. 69 6. 21 10 or more 2. 16 0. 66 i · 0. 66
The company wants to know what the expected profit per customer is. A random sample of size 76 is taken and one makes a list of the number of bags each customer buys.
Number of Number bags of customers i a(i) 1 13 2 18 3 9 4 6 5 9 6 7 7 6 8 2 9 3 10 2 12 1
Use the empirical distribution function to answer the questions below.
a) Estimate the expected value of the number of bags that a customer buys. b) The answer for a) is 304/76 = 4. Somebody argues: ’now we also know an estimate for the expected profit per customer’; it equals 4 · 0 .86 = 3.44’, where 0.86 is the profit per bag if a customer buys 4 bags. Why is this not correct? c) Give a proper estimate for the expected profit per customer.
- Consider the density fX (x) = (b − 1)/xb^ for 1 ≤ x < ∞, and fX (x) = 0 for x < 1. Assume b > 1.
a) Verify that fX is a density. b) Compute the expected value of X. Assume b > 2. c) Now assume b = 4 and and consider 10 independent random variables X 1... , X 10 with density fX. Use the Central Limit Theorem to approximate the probability that the average (X 1 + X 2 +... + X 10 )/10 is larger than 2. Use the fact that V (Xi) = 3/4.
- The number of telephone calls at call center A follow a Poisson-process with expected value μ per hour. The number of calls at call center B follow a Poisson-process with expected value 2μ. One has n observations at call center A (X 1 , X 2 , · · · , Xn) and n observations at call center B (Y 1 , Y 2 , · · · , Yn). All observations are independent. Consider the following estimators for μ
W 1 = (X + Y )/ 2 , W 2 = (2X + Y )/ 4.
a) Are W 1 and W 2 unbiased estimators for μ? b) Which of two estimators is better with respect to the MSE?
- Let X 1 , X 2 , · · · , Xn a random sample from a uniform distribution on the interval [0, θ] with θ unknown. The density of the distribution is
f (x) =
θ
(0 ≤ x ≤ θ).
- In a lake one investigates the mercury (Dutch: kwik) contamination using a random sample of fish with size 53. A normal probability plot gives
a) Can the normal distribution be used to find a confidence interval on μ (the popu- lation mean)? Give an argument. A statistical package gives the following output.
Summary Statistics for mercury (kwik)
Count = 53 Average = 0. Variance = 0. Standard deviation = 0. Minimum = 0. Maximum = 1. Range = 1.
b) Give a 95% confidence interval for μ. c) Is the following statement correct? If one has many observations, then 95% of the observations will be in interval from b). Give an argument.
- The waiting time X in minutes at a service desk is exponentially distributed with unknown parameter λ > 0. So the expected value μ equals 1/λ. A random sample of size 20 was selected and the mean sample waiting time was 10 minutes. A good estimator for the expected value μ is X, the average of the sample. For the variance of X we have
V (X) =
nλ^2
μ^2 n
a) Construct a 95% two-sided confidence interval for the expected value μ of the waiting time using the normal approximation.
It is also possible to construct an exact confidence interval using the χ^2 -distribution. It is known that the quantity
2 nX μ
has a χ^2 -distribution with 2n degrees of freedom.
b) Construct the 95% confidence interval by use of (1). Hint: first find values l and u such that
P
l < 2 nX μ
< u
- A manufacturer wants to know the mean of the lifetime of the batteries he produces. He wants to have a 95% confidence interval on the expected lifetime. It is known that the lifetime has a normal distribution with standard deviation 10 hour.
a) What should the sample size be if the width of the interval should not exceed 2?
A random sample of size 20 gives as results
∑^20
i=
x^2 i = 47399. 2
∑^20
i=
xi = 956.
b) Give a 95-% lower-confidence bound (of the form (l, ∞) for the expected lifetime. Assume that the standard deviation is unknown.
Assume that the interval in b) equals (47, ∞). The manufacturer claims ’If you buy a battery at our company, the probability that the lifetime is more than 47.0 hour is 0 .95’.
c) Do you support this claim?
a) What is the probability distribution of the test statistic (give also the parameter(s) of the distribution)? b) What is the expected number of men with a weight less or equal than 60 kg? c) The value of the test statistic is 3.52. Is the null hypothesis rejected (α = 0.05)?
- A company sells vegetables in cans. Some of the cans do not meet the specifications. There are several reasons for this: a stain(1), a dint(2), the eye to open it is on the wrong place(3), there is no eye to open it (4), other (5). We call these deviations. One takes a random sample and classifies the wrong scans.
(1) (2) (3) (4) (5) Sum 89 145 58 54 29 375
One wants to investigate if the number of deviations for the 5 classes follow the pattern 2:3:2:2:1.
a) Formulate the null hypothesis in terms of probabilities and compute the number of expected cans for each class when the null hypothesis is true. b) The value of the test statistic is 23.7. Is the null hypothesis rejected (α = 0.05)? Why? c) Give a 95% two-sided confidence interval for the proportion of tins with a dint.
- Consider crates with 20 bottles with beer. Define X as the number of bottles that contain not enough beer. The number of bottles with not enough beer is counted for 50 crates and the results are given below.
Value of X 0 1 2 3 Number of crates 15 22 11 2
a) One wants to investigate if the binomial model is a good model for the number of bottles with not enough beer. Estimate the expected number of crates in class 0 (so all bottles contain enough beer) if the binomial model is a good model. b) The value of the test statistic is 2.09. Give the critical value for a confidence level of α = 0.05? Is the null hypothesis rejected? Why (not)?
- A company operates four machines in three shifts. From production records the follo- wing data on the number of breakdowns are collected.
Machines Shift A B C D 1 41 20 12 16 2 31 11 9 14 3 15 17 16 10
One wants to test the null hypothesis that the breakdowns are independent of the shift.
a) Compute the expected number of breakdowns for machine B for shift 2 if the hypothesis is true.
b) The value of the test statistic is 11.65. Is the null hypothesis rejected? Why (not)? Use α = 0.05.
- One adds a certain ingredient to an laundry detergent to try to improve the effect of the detergent. At random 10 dirty overalls are chosen and cut in two parts. One part is washed with the detergent with the ingredient and the other part is washed with the detergent without the ingredient. One measures in some unit how clean the (parts of) the overalls are. The observations are
Overall 1 2 3 4 5 6 7 8 9 10 Detergent without (’OUD’) 5 5 12 29 10 33 33 17 2 8 Detergent with (’NIEUW’) 7 9 17 36 8 40 29 27 5 19
With a statistical package two analyses are done (one for paired observations and one for independent observations). The results are given below
a1) Which of the two methods (paired or independent) should be applied here for the analysis? Explain your choice. a2) Explain the difference in p-values for the two methods. b) Test ONE-SIDED the null hypothesis that the new detergent (with the ingredient) is better. Use α = 0.05.
- Corrosion of iron in reinforced concrete is a problem for its sustainability. For a number of constructions the strength y (in MPa) is measured and also the so called ’depth of carbonification’ x (in mm) which is an important variable for corrosion. A statistical package gives as result
Regression Analysis - Linear model: Y = a + b*X
Dependent variable: y Independent variable: x
Standard T Parameter Estimate Error Statistic P-Value
Intercept 27.1829 1.65135 16.4611 0. Slope -0.297561 0.0411642 ??????? ??????
Analysis of Variance
Source Sum of Squares Df Mean Square F-Ratio P-Value
Model 428.615 1 ??????? ????? ????? Residual 131.242 16 ???????
Total (Corr.) 559.858 17
Correlation Coefficient = -0. R-squared = 76.5579 percent R-squared (adjusted for d.f.) = 75.0928 percent Standard Error of Est. = 2.
a) What is the number of observations n? b) Compute, using the above results, the sum of squares Sxx =
(xi − x)^2. c) Give the 95%-confidence interval for the intercept. d) Test the null hypothesis β 1 = 0. Use α = 0.05.
- The height of a tree y (in cm) is measured during a period of 10 year. The years x are coded with 1, 2 , 3 , · · · , 10. The table is (partially) given here.
yi(hoogte) 150 161 167 ?? ?? ?? ?? ?? ?? 208 xi(jaar) 1 2 3 4 5 6 7 8 9 10
Regression analysis (based on this 10 observations) gives the following results (’hoogte’=height and ’jaar’=year).
a) Give a two sided 95%-confidence interval for σ^2. b) Give an estimate of the height of the tree in the next year (year 11). c) Give a 95% prediction interval for the height of the tree in the next year (year 11).
The next picture shows the residuals against the number of the year.
d) Use the picture to comment on the model assumptions. What are the consequences for the prediction of the height in the next year (11)?
Analysis of Variance
Source Sum of Squares Df Mean Square F-Ratio P-Value
Model 111.54 5 22.3081 16.51 ?????? Residual 43.248 32 1.
Total (Corr. 154.788 37
And for a model with only three variables
Multiple Regression Analysis
Dependent variable: Quality
Standard T Parameter Estimate Error Statistic P-Value
CONSTANT 6.46719 1.33279 4.85238 0. Aroma 0.58012 0.262185 2.21264 0. Flavor 1.19969 0.274881 4.36441 0. Oakiness -0.602325 0.264401 -2.27807 0.
Analysis of Variance
Source Sum of Squares Df Mean Square F-Ratio P-Value
Model 108.935 3 36.3117 26.92 0. Residual 45.8534 34 1.
Total (Corr.) 154.788 37
a) Test the hypothesis β 1 = β 2 = β 3 = β 4 = β 5 = 0. b) Test the hypothesis β 1 = β 3 = 0.
See next page
All possible 32 models (using the 5 regressors) are considered. This gives the following output
Regression Model Selection
Dependent variable: Quality Independent variables: A=Clarity B=Aroma C=Body D=Flavor E=Oakiness
Number of models fit: 32 Model Results
Adjusted Included MSE R-Squared R-Squared Variables
4.18347 0.0 0. 4.18347 2.7027 0.0 A 2.14852 50.0308 48.6427 B 3.00516 30.1074 28.1659 C 1.61593 62.4174 61.3735 D 4.18347 2.7027 0.0 E 2.20886 50.0544 47.2004 AB 2.9001 34.4245 30.6773 AC 1.62128 63.3404 61.2456 AD 4.18347 5.40541 0.0 AE 2.04696 53.7151 51.0703 BC 1.51006 65.8552 63.904 BD 2.04406 53.7807 51.1396 BE 1.65123 62.6632 60.5296 CD 3.01392 31.8508 27.9566 CE 1.49874 66.1112 64.1747 DE 2.08516 54.1984 50.1571 ABC 1.53649 66.2504 63.2725 ABD 2.10258 53.8158 49.7407 ABE 1.63487 64.0893 60.9207 ACD 2.82341 37.9826 32.5105 ACE 1.45631 68.0116 65.1891 ADE 1.55192 65.9114 62.9036 BCD 1.91841 57.8612 54.143 BCE 1.34863 70.3767 67.7629 BDE 1.52707 66.4572 63.4975 CDE 1.57108 66.5054 62.4454 ABCD 1.91353 59.2046 54.2597 ABCE 1.33818 71.4708 68.0128 ABDE 1.43902 69.3209 65.6022 ACDE 1.38502 70.4721 66.893 BCDE 1.3515 72.0599 67.6943 ABCDE
