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[10 marks] Parking spaces near the university are taken almost immediately during the day, as they become available. The even

What is the probability that exactly 15 spaces become available in one hour? [2 marks (d) (e) You and your friend are driving

[10 marks] Parking spaces near the university are taken almost immediately during the day, as they become available. The event of a parking space becoming available follows a Poisson process with rate A 1 space per 3 minutes (a) What is the expected value and variance for the number of parking spaces becoming available in an hour? 2 marks What is the expected waiting time for a space to become available? [1 marks] (b) A student has been waiting for a space for 2 minutes already. Find the probability that they (c) will have to wait for at least 10 minutes in total [2 marks
What is the probability that exactly 15 spaces become available in one hour? [2 marks (d) (e) You and your friend are driving to university. Find the probability that you will wait between 4 and 8 minutes for a space, given that your friend found a space in 3 minutes. 3 marks]
math   Statistics-And-Probability   
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Answer #1

The number of parking space becoming available X follows Poisson distribution with A1/3 min

The Poisson PMF of X is P (X x) eIx 0,1,2,3, ...

a)The Poisson parameter for 1 hour is 1 x 60 20 hr A = 3

The expected value is 20 hr

The variance is 20 hr

b)The inter-waiting time follows exponential distribution. The expected waiting time is

= 3 minutes

c) Using the memoryless property of exponetial distribution,

P(T 101T 2) P(T> 10T> 2 P(T 2) P(T 10 P(T 101T 2) =P(T> 2) P(T> 10 T 2) e-(10-2) P(T 1017> 2) e P(T 10|T> 2) e-8/3 P(T 101T 2

d) The probability

P (X 15) e20 2015 15! P (X 15) 0.0516

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