Poisson Distribution Question?

Question

I can't find the equations to use for this type of problem... if someone can just give me the equations that's great, if you can walk me through it that's even better.

Imperfections in an optical fiber are distributed according to a Poisson process such that the distance between imperfections in meters has an exponential distribution with paramter L = 2m^(-1).

a)What is the expected distance between imperfections?
b)What is the probability that the distance between the imperfections is longer than 1 meter?
c)What is the distribution of the number of imperfections in a 3-meter stretch of fiber?
d)What is the probability that a 3-meter stretch of fiber has no more than four imperfections?

Answer 1

The formula for the Poisson distribution is
P(x=k) = exp(-L) * [L^k / k!]

a) The expected value of x is L.

b) P(x>a) = exp(-L*a)

c) F(t) = P(x<=t) = 1 - P(x>t) = 1 - exp(-L*t)

d) P(N(y)=z) = exp(-L*y) * [(L*y)^z / z!]
so P(N(3)<=4) = P(N(3)=0) + P(N(3)=1) + P(N(3)=2) + P(N(3)=3) + P(N(3)=4)