Question

A little boy plays outside in the yard. On his own he would come back inside at a random time Y uniformly distributed on the interval (0, 1) (Let us take the units to be hours.); that is, the pdf of Y is given by f(y) = 1 for y ∈ (0, 1), f(y) = 0 otherwise. However, if the boy is not back inside in 45 minutes, his mother brings him in. Let X be the time when the boy comes back inside. Find the cumulative distribution function F of X.

Hint: You need to compute the probabilities of the form P(X ≤ a) based on the distribution of Y . An observation: if Y = 0.5, then X = 0.5; if Y = 5/6, then X = 3/4. Thus, you might need to consider several cases.

Answer 1