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# Probabilities

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Question 1

A restaurant can serve up to 75 meals. Experience shows that 20% of clients who have booked do not turn up.

1. The manager accepts 90 bookings. What is the probability that more than 50 clients turn up?
2. How many bookings should the manager accept in order to have a probability of more than 0.9 that he will serve all the clients who turn up?

Question 2

During a certain time period, the number N of cars going through a toll is a random variable of Poisson distribution with parameter L.

1. Let F and M be the number of female and male driver who pass through this toll, respectively. We have N = F + M. Let p be the probability that the driver of a car is female. Find the probability distributions of F and of M. (proving the answer)

Question 3

Let X and Y be two continuous random variables, which have the following distribution:
f(x, y) = x + y for x in [0;1] and y in [0;1].

1. Find the marginal distributions of X and of Y (proving the answer).
2. Compute E(X+Y), Var(X+Y) and the covariance of X and Y.