On conditional probability of an exponential random variable

[Postante]

You are given a random exponential variable X: f(x) = λ exp(-λ x).
Suppose that X = Y + Z, where Y is the integral part of X and Z is the fractional part of X:
Y = IP(X), Z = FP(X).
Which is the following conditional probability:
P(Z < z | Y = n) for 0 ≤ z < 1 and n = 0, 1, … ?

 

[haruspex]

You are given a random exponential variable X: f(x) = λ exp(-λ x).
Suppose that X = Y + Z, where Y is the integral part of X and Z is the fractional part of X:
Y = IP(X), Z = FP(X).
Which is the following conditional probability:
P(Z < z | Y = n) for 0 ≤ z < 1 and n = 0, 1, … ?

P(Z < z | Y = n) = P(X < n+z | Y = n) = P(X < n+z | n <= X < n+1)
= (F(n+z)-F(n))/(F(n+1) - F(n))
= (e^-λn - e^-λ(n+z))/(e^-λn - e^-λ(n+1))
= (1-e^-λz)/(1-e^-λ)