Please can anyone solve 1(a) and 1(b). I need help. Thank you in advance.

Question

Probability theory tells us that if the average time between events is k, the probability that x, the time between events, is between a and b is given by,

P(a ≤ x ≤ b) = ∫ ^b_a f(x) dx ------ equation (1)

f(x) = 0, x < 0 ------------- equation (2)

= ke^-kx , x ≥ 0

Suppose that at a busy intersection of a city, traffic accidents occur at an average rate of one every three months. After residents complained, changes were made to the traffic lights at the intersection. It has now been eight months since the changes were made and there have been no accidents.

1(a) Find f(x) for accidents are occurring at a rate one of every 3 months.

[Hint: use f(x) from equation (2)]

1(b) Were the changes effective or is the 8-month interval without an accident a result of chance?

[Hint: Compute P(x ≥ 8) by using f(x) from (a) and decide on the obtained

result.]

Answer 1

k is the average number of events in the unit of time and is called rate parameter. Let's say we take month as the unit of time, thus, k is the number of events per month. With the average of one even each three month k is 1/3. Therefore, the probability distribition function f(x) is:

f(x) = 0 for x<0

f(x) = (1/3)*e^(-x/3) for x>0

now, if we assume no changes happened after fixing the traffic lights, the probability of having at least one event after 8 month is the integral of f(x) from 0 to 8 which gives

integral (1/3)*e^(-x/3) from 0 to 8 = -e^(-8/3) + e^(0/3) = 1-e^(-8/3)

But we want the probability of no event which is the complementry of the above, so

Probability of no event = 1 - (1-e^(-8/3)) = e^(-8/3) = 0.069

Or about 7%. Therefore there is a 7% change that nothing changed after they fixed the traffic lights. We usually want to get to less than 1% to be sure. So, if you ask me, you should wait longer to see if the traffic lights fixes had any impact.