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Test of probability theory

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Test of probability theory

Exercise 1

The number of customers in a queueing systems takes the following values:

• 0 with probability 1/4,

• 1 with probability 1/8,

• 2 with probability 1/2,

• 3 with probability 1/8.

Compute the expected number of customers in the queueing system.

Exercise 2

A machine is idle with probability 0.3 and busy with probability 0.7. The probability of a breakdown is 0.05 when the machine is idle, and 0.4 when the machine is busy. Compute the prior probability of a breakdown of the machine.

Exercise 3

Consider two independent continuous random variables X and Y , with X following a uniform distribution over the interval [2, 6], and Y following an exponential distribution with the same ex- pected value as X. Compute the probability P(X + Y ≤ 6).

Variant of Exercise 3: Compute the probability P(X + Y ≤ t) for all t ∈ R.

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