Let X denote an exponential random variable with unknown parameter λ>0 . Let Y=I(X>5) , the indicator that X is larger than 5 .

Question

Recall the definition of the indicator function here is

I(X>5)={1ifX>50ifX≤5.

We think of Y as a censored version of the Exponential random variable X : we cannot directly observe X , but we are able to gather some information about it (in this case, whether or not X is larger than 5 .)

Observe that Y is a Bernoulli random variable. Thus, the statistical model for Y can be written ({0,1},{Ber(f(λ))}λ>0) for some function f of λ . What is f(λ) ?

Answer 1

f(λ) = P(X > 5) = 1 - P(X ≤ 5) = 1 - e^(-λ*5).