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ASSIGNMNETS IE485 – FALL 2006: 1st week: Due 10. 10.
1) It is known that 30 % of mice inoculated with a serum are protected from a certain disease. If 400 mice are inoculated find
the probability that
a) that more than 100 contract the disease.
b) Find the minimal integer x such that P( X ≥ x ) < 0.01, where X denotes the number of mice that contract the disease.
2) Service calls come to a maintenance center according to a Poisson process and on the average 200 calls per hour.
X denotes the number of calls per hour.
Find
a) P( 215 ≥ X ≥ 180 )
b) Find the maximal integer x such that P( X ≤ x ) < 0.03 .
3) In the November 1990 issue of Chemical Engineering Progress a study discussed the percent purity of oxygen from a certain
supplier. Assume that the mean was 99.63 with a standard deviation of 0.08. Assume that the distribution of percent purity was
approximately normal.
a) What percentage of the purity values would you expect to be between 99.6 and 99.7?
b) What percentage of the purity values is more than 0.1 away from the mean?
c) What purity value would you expect to exceed exactly 7% of the population?
d) What purity value is exceeded by exactly 3% of the population?
4) In a biomedical research activity it was determined that the survival time X in weeks of an animal when subjected to a
certain exposure of gamma radiation has a gamma distribution with shape parameter α = 2.2 and scale parameter ß = 11.
a) Find P(X > 35) .
b) Find the number x such that P( X ≥ x ) = 0.05 .
c) Use simulation (samplesize at least 100000) to calcuate E( X | X > 30) .
5) A bookshop is selling a certain monthly magazine for 5$ per piece and buys it from a publishing house at 3$ per piece. Lets
assume that the number of sold journals per month is Poisson distributed with expectation 120. As the bookshop cannot hand
back journals that were not sold, it has to decide about the number of journals that are ordered every month.
Let X denote the demand in a month (this is the total number of journals customers want to buy) and O the number of ordered
journals.
Then clearly the revenue R = 2 O for the case that X ≥ O and R = 5 X -3 O for X < O.
a) Compute the expectation and the variance of the revenue R when ordering 100 journals. (ie: O=100)
b) Compute the expectation and the variance of the revenue R when ordering 120 journals. (ie: O=120)
c) Try to find the number of ordered journals that maximises the expected revenue. Compute the expectation and variance of
the gained money for that number of orders.
d) Comment on the interpretation of the variance in this example. What will the manager of the bookshop try to do if he does
not want to take too much risk.
e) Compute the probability P(R ≤ 180) for O = 100, O = 120 and the O you found in c) .
6) X ~ N( 20; σ = 5.3 ) and Y ~ N( 15; σ = 3.9 ), X and Y independent:
a) Compute the probability P(Y>X).
b) Make a simulation to check your result of a).
c) Compute the probability that P(2X+3Y> 80).
d) Make a simulation to check your result of c).
7) X ~ N( 20; σ = 3 );
a) Compare the probabilities of 2 X > 50 and X + X > 50 using simulation.
b) Compute the probability that the sum of 10 independent realisations of X is bigger than 220.
c) For a random sample of size 100 of the above variate X calculate the probability that the sample-mean is larger than 21.
The files “studentdata0102.txt” and “studentdata0203.txt” (on the course web-page) contain the point results of all students
who have taken both IE255 (probability) and IE256 (statistics).
Use these data and plotting, confidence intervals (for the mean and the difference of two means), the one-sample and two-saple
t-test and linear regression to answer the following questions:
8) What is the average result of IE 255 and of IE256?
9) Which course is more difficult?
10) Is there a relation and what relation is there between the results of IE255 and IE256?
2nd week: due 20.10.
11) Write a R-function that calculates the power of the single sample t-test using simulation. α should be an input to that
function:
the function header should be:
simulpower(nsimul, n, alfa, mu0, side, rdistr, …)
the function should return a single value, the estimated power for the given value of alfa.
12) a) Use the function to estimate the power of a t-test with H0 = 10, if n =100 and the real distribution is normal with mu =
11 the real sigma = 5
b) same as a) for n= 10, n=200 and n=1000
13) 14) see below
15) We generate a uniform distribution instead of a normal distribution. Select both distributions such that they have the same
mean and variance
a) What sample size is necessary that the power of the chi-squared test gest larger than 90 %?
b) As a) but for the Kolmogorov-Smirnov Test.
3rd week: due 3.11.
13) Corrected:
For a Gamma(2) distributed parent population and
H0: mu = 2 and H1: mu > 2 we observed the T-statistic of
a) 4.5 for n=5
b) 1.7 for n = 20
c) 2.3 for n = 50.
Use simulation to find the P-value for the above results of the t-test
14) corrected
Compare your results with the results when using the t-test. Should we expect that the results are similar?
16) Use the stock – data of our web-page.
Check if the standard model with iid. Normal log returns is close to realistic for the GE (General electric) stock prices.
Comment on the outliers (if there are any).
17) Calculate the VaR in percent for α = 0.05 and the horizon 2 weeks (= 10 days) for the GE stock.
18) Is it possible to use the volume data of the last day to draw conclusions on the log return. Try to find a regression model
that can help to forecast the direction of the movement of the stock using yesterdays volume.
19) Assume that you can invest either in a GE stock or in a riskless investment with a risk free rate of r = 0.05.
Calcuate the VaR for a portfolio where you invest 50 $ in GE stocks and 50$ in the riskless investment.
4th week: due 10.11
20) Using the NY data from our webpage, the “multinormal log-return” model and simulation calculate:
a) The VaR( α = 0.05 , horizon 1 day) for all weights equal.
b) The ES associated to the above VaR
21) Same as 20) but horizon one year.
22) Calculate the approximate results for 20) and 21) without simulation. Compare the results with the simulation results. For
which case do you expect that the simulation is better?
23) Calculate the result of 20) using resampling (historic simulation) instead of the “multinormal log-return model.
24) What changes in the results of 22 if you change the length of the history or remove outliers from the data?
25) Try to find a portfolio that has a clearly smaller VaR than the “equal weights portfolio” we used for question 20 to 24..
5th week: due 17.11
26) Calcuate the Cholesky factor for a general 2x2 correlation matrix R. Compare your result with the result of R for different
values of ρ .
Hint use the property of the Cholesky factor together with the fact that it is a lower triangular matrix to find a system of linear
equations.
27) Using the result of 26) formulate a direct simulation formula for a multi normal vector of length 2 with correlation ρ and
general mean values and standard deviations. Assume that two iid. standard normal variates Z 1 and Z2 are available.
28) For a portfolio with w=(0.3, 0.7) and joint normal distribution of log-returns with yearly values ρ=0.5, σ1=0.2 and σ2=0.3
calculate the VaR(0.01) for horizon 1day (1 week, 1 month, 1 year) using the approximate formula.
29) For the portfolio of 28) calculate the VaR(0.01) for the different horizons using simulation. Repeat the simulations often
enough to calculate confidence intervals.
30) Compare the results of 28) and 29). What can we see?
6 th week: due 24.11
31) Take one of the stocks of our stock-data. Use density estimation to plot the density of the log-returns of these stocks.
a) Use the normal kernel and different bandwidths. Comment which one are under smoothing and which one are over
smoothing the data. Compare with the optimal bandwidth.
b) The same as a) but using a constant kernel on (-1,1).
c) Compare your estimated density plots with the Normal distribution.
32) Use the data “wood” from the web-page. Make a regression to explain density by stiffness. Try to find a good model where
the residual plot indicates a good fit.
33) Use the data “fuel” from our webpage.
a) Make a regression for MPG including all variables. Interpret the results and the residual plot.
b) Check if the fit of the model is better if you change the y-variable MPG (miles per gallon) into the transformed y-variable
GPM (gallon per miles).
c) Try to find a good and simple multiple regression model to forecast the fuel consumption.