PLEASE CAN ANYONE SLOVE 1(A) ONLY. THANK YOU

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

Well, the main part of the problem is realizing that the rate is 1/3, since an accident occurs once every 3 months. So as a result, f(x)= 1/3*e^(-1/3*x).