###### 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).

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I

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 the attach file Work1_2.xlsx). Suppose all variables follow a common probability

distribution with mean μ and standard deviation σ.

- 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.

- 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.