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.