ISCTE – IUL Instituto Universitário de Lisboa
MMST – Master in Management of Services and Technology
Unit: SSO – Simulation of Systems and Operations
Work 1: Probability and Statistics (25% of the final mark)
Deadline: 23/09/2020 (in paper, font size 12, line spacing 1.5, margins 2.5 cm).
Suppose the random variable X is uniformly distributed in the intervals [2, 3] and [6, 9].
Write the probability density function and cumulative distribution function of X.
Calculate E[X] and Var[X].
Consider 8 random samples of equal dimension (n=21), each one drawn from the random variables X1, …,
X8 (please find the data in the attach file Work1_2.xlsx). Suppose all variables follow a common probability
distribution with mean μ and standard deviation σ.
Calculate the sample mean and the sample variance for each of 8 random variables.
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.
What is an approximate distribution of the sum S = X1+…+ X8?
Calculate an approximate 95% equal-tailed probability interval for the random variable S.
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?
Calculate again the 95% equal-tailed probability interval for the random variable S. Comment.
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.
Test the hypothesis μ = 8.5 against the alternative μ > 8.5, for a level of significance of 3%.
Calculate the p-value associated with the test of hypothesis. Comment.
Write a small text (max. 1/2 of A4) about the importance of the Central Limit Theorem in computer