EXAMINATION: Tentamensskrivning i Matematisk Statistik (TMS061)

Time: Wednesday 16 January 2008

Jour: Anastassia Baxevani, mobile: 0702972910

Aid: You are allowed to use a scientific calculator and a half page (both sides) of hand written notes

Lab: Depending on the performance in the lab 0-5 points will be added to your test score to provide with your final score.

Grade: You need 21 points for 5, 16 points for 4 and 11 points for 3.

Motivate all your answers. Good Luck!

1) Suppose the random variable X has a normal distribution with mean 3 and variance 9. Let Y = \frac{1}{3}X − 1.

a) What are the mean and variance of Y? (2p)

b) What is the probability that Y is at least 1? (1p)

2) a) Suppose that A and B are two events such that: P (A) = 0.6 and P (B) = 0.8. Are A and B disjoint? Explain. (1.5p)

b) True or false: If A and B are events, then: P (A∪B) ≥ P (A)+P (B). Justify your answer. (1.5p)

3) State in your own words the Central Limit Theorem. (2p)

4) a) Someone is recording the number of clients that arrive at a shop between 3 and 4 every Saturday afternoon for three months. Which distribution best describes the recordings? (1p)

b) What are the expected value and the variance of a Poisson random variable X for which P (X = 2) = P (X = 3)? (1p)

5) 500 observations from a random variable X have given 35 zeros, 140 ones, 158 twos, 121 threes and 46 fours. Test using a χ^{2} test the hypothesis that the random variable X is binomial with n = 4 and p = 1/2. (3p)

6) a) The random variable Z is Poisson with mean value 2.4. Compute the probability P (Z > 2).(1p)

b) The random variable Y is normally distributed with mean value μ = 3 and standard deviation σ = 0.8. Compute P (Y > 2).(1p)

7) For the random variable X with probability density function

f(x) = \frac{λ^{3}x^{2}}{2} e^{−λx}, x > 0,

find the maximum likelihood estimator of λ. (3p)

8) Let X be the random variable that measures the content of a bottle of a specific perfume (in ml). A sample of size 16 has been taken from the this perfume and gave x̄ = 476.4 and s = 0.7 ml. Assume that X is normally distributed and

a) Compute P (X ≤ 475). (1p)

b) Construct a confidence interval for the true mean μ for α = 0.95. (1p).

9) Let the random variable X have the probability density function P (X = x) = 0.1 + 0.05x, x = 0, 1, 2, 3, 4.

a) Compute E(X) and V ar(X). (1p)

b) What is the probability P (X_{1}+X_{2} > 5) if X_{1} and X_{2} are independent random variables distributed like X? (2p)

10) For a engineering study we have recorded the time it takes two different machines A and B to warm up (in min.). The results are:

A : 6.7 7.2 5.9 6.9 7.0 6.7 5.9
B : 5.4 5.8 6.3 6.2 5.6 5.5

Assume that the above observations are independent samples from a normal distribution with the same variance. Test the hypothesis that the means of the two distributions are also the same with alternative hypothesis that are different. α = 0.01 (3p)