Suppose the random variable X is uniformly distributed in the intervals [2, 3] and [6, 9].

a) Write the probability density function and cumulative distribution function of X.

b) Calculate E[X] and Var[X].

II

Consider 8 random samples of equal dimension (n=21), each one drawn from the random variables X1, …, X8 (please find the data in attach file Work1_2.xlsx). Suppose all variables follow a common probability distribution with mean μ and standard deviation σ.

1.

a) Calculate the sample mean and the sample variance for each of 8 random variables.

b) Calculate the mean of the means and the pooled variance for these variables. In what follows, assume that the values obtained are the true values of μ and σ2, respectively.

c) What is an approximate distribution of the sum S = X1+…+ X8?

d) Calculate an approximate 95% equal-tailed probability interval for the random variable S.

2.

a) Suppose now that the 8 random variables follow an exponential distribution with parameter λ=2 (mean= 0.5). What is the exact distribution of S?

b) Calculate again the 95% equal-tailed probability interval for the random variable S. Comment.

III

Suppose that the fuel consumption of a car in a city follows a Gaussian distribution with unknown mean μ and standard deviation σ = 1. The last five records of the fuel consumed by the car are: 9.3, 10.2, 8.7, 7.6 and 11.3 litres per 100 km.

a) Test the hypothesis μ = 8.5 against the alternative μ > 8.5, for a level of significance of 3%.

b) Calculate the p-value associated with the test of hypothesis. Comment.

IV

Write a small text (max. 1/2 of A4) about the importance of the Central Limit Theorem in computer simulation.