2022W 040132 UE Statistics 1 Kalix, Klockmann, Köstenberger.
2022W 040132 UE Statistics 1 Kalix, Klockmann, Köstenberger.
2022W 040132 UE Statistics 1 Kalix, Klockmann, Köstenberger.
Exercise sheet 4
Multivariate distributions and confidence intervals
01.12.2022
Task 1. The following table shows the common probability function of two discrete random variables, X and Y:
a
aaaaaya x
1
2
3
1
1/9
1/9
1/9
2
1/9
1/9
1/9
3
1/9
1/9
1/9
Determine the common distribution function.
Determine the one-dimensional boundary distributions.
Are X and Y stochastically independent?
Determine covariance and correlation of X and Y .
Task 2. For the returns X, Y and Z of three securities: E(X) = −30, E(Y ) = 25, E(Z) = 40, V ar(X) = 100, V ar(Y ) = 10, V ar(Z ) = 20, and σ X,Y = Cov(X,Y ) = −10, σ X,Z = 14 and σ Y,Z = −0. 5. What is (a) the expected return, and
b) the variance of return
of the portfolio, whichis composed of three parts from the first, four parts from the second and five parts from the third security?
Task 3. In a fictional city, a ticket for the metro costs 2.5€. If you are caught driving in the dark, you pay 100€. On the i-th day of the week (i = 1,… 7) is the chance of being caught driving in the black p i ∈ (0.1). They want to take the subway every day of the week, and make trips on the first day of the week. They wonder if they should risk driving black this week, or rather buy tickets.
Model the (random)cost difference D between the strategy of always going black this week and the strategy of always buying tickets.
Note: A Bernoulli variable X ∼ Bern(p i) is equal to 1 with probability p i and 0 with probability 1 − p i.
Calculate the expected cost difference E(D).
Task 4. You invest in a security P with a current value of 200. This security is traded 100 times in a row over the next hour by differenttraders who know nothing about each other (i.e. independently of each other) at the same volume. The dealers each carry out a normally distributed price change X i (i = 1,…,100). That is, the first trader buys the security for 200€ and sells it for 200 + X 1€, the second trader buys the security for 200 + X 1€ and sells it for 200 + X 1 + X 2€ etc. where all X i i.i.d.∼ N(0. 05.0. 1) are (i.i.d. means independent, identically distributed, i.e. they are independent random variablesthat all have the same distribution, namely N(0. 05.0. 1).
Specify a 95% confidence interval for the value of the security after one hour.
Task 5. In a pack of 1000 pencils, the probability that a pen has a broken lead is 0. 005%.
Calculate the expectation and variance of the number of pins with broken lead.
Specify a 98% confidence interval for the number of pins with broken refill. What assumptions do you use to justify the use of this confidence interval?
Task 6: You observe 100 British leaders and conclude that the political lifespan of a top British politician(in days) is approximately poisson distributed with parameter λ = 100. Provide a 95% confidence interval for the length of a British leader’s political lifespan. Is the assumption of a Poisson distribution justified? What are the problems with confidence intervals? Instead, what does it look like assuming that the political lifetime in months is poisson distributed with parameter λ = 3?
Task 7. Ten randomly selected graduates of a course of study are asked 5 years after graduation what salary in euros they receive. The answers are:
28252,34600,23672,122492,78214,37352,33921,56920,40427,28541.
Construct a 98% confidence interval for a graduate’s median salary after 5years.
Task 8. You compare the prices of laptops over a year and get
Price 2021
Price 2022
750
780
700
800
1223
1345
1344
1500
1100
1200
1120
1250
900
980
1800
2200
1560
1700
1430
1550
Calculate a 95% confidence interval for the median percentage price increase of a laptop over the last year.
1
1
1