Question

The length of a game in minutes is a continuous random variable Z, with pmf f(t) = e^-t for t > 0 (exponential random variable). You have already sat through t minutes of the game, and are interested in whether the game is about to end immediately or not (hazard rate h(t)).

h(t) = f(t)/ (1- F(t)).

f(t) = Game ends between time t and time t + dt. (1- F(t)) = Probability the game does not end before time t.

It's a conditional probability that the game ends between t and t + dt given that it did not end before t.

What's the hazard rate for Z

Answer 1

Random variable z follows the following pdf

So, the cdf will be

The hazard rate will be

Hence, the hazard rate of Z is 1, which is constant and it independent of random variable Z.