Question

Let the number restaurant in 1 hour be Poisson distributed with 0, We know, in 40% of the cases is only one person in a car. In 30% 2 person, in 20% 3 person, in 9% 4 person and in 1% 5 person. We also know that the number of persons in a car is independent. of people that are makin g a break in a road house (1) What is the mean (estimated value) and the variance? (2) Let λ-10. Calculate the exact probability, that 2 persons are actually visiting the restaurant. Hint: Represent the number of people as randomised sum and use generating functions.

Answer 1

Answer 1.
Be: X = Number of People in a Car
The probability distribution for X is:

X	1	2	3	4	5
P(X=x)	0,40	0,30	0,20	0,09	0,01

To calculate the mean, the following expression is used:

So:

E (X) = 1 • 0.40 + 2 • 0.30 + 3 • 0.20 + 4 • 0.09 + 5 • 0.01 = 2.01

To calculate the Variance, the following expression is used:

So:

It is concluded then that the Mean is equal to 2.01 and the Variance is equal to 1.0499

Answer 2.

Sea:
Y = The number of people that are making a breack in a road house restaurant in one hour.
According to the statement, it is distributed as a Poisson random variable with parameter λ> 0.
The distribution of probabilities for Y is given by the following expression:

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If λ = 10, you have to:

Thus, the exact probability that 2 people visit the restaurant is 0,002267, assuming that this variable has a continuous interval of 1 hour for the parameter λ = 10 used.