Question

The time till a blood moon appears is exponentially distributed with an average of 3 years.

a) What's the probability that the next blood moon occurs within the next 3 years?

b) Suppose 5 years have already gone without a blood moon.

Conditioned on the above information, what's the probability that an additional year will go by without a blood moon?

c) Find the median time of arrival of the next blood moon. Remember the median is defined as 50% of events occur above and below this value.

Answer 1

We are given the waiting time distribution here as:

as mean for exponential distribution is reciprocal of its parameter.

a) The probability that the next blood moon occurs within the next 3 years is computed here as:

Therefore 0.6321 is the required probability here.

b) Exponential distribution follows memoryless property, therefore it does not matter how long has the waiting time already passed, the probability here would be computed as:

Therefore 0.7165 is the required probability here.

c) Let the median time be M.

Then, we have here:
P(X > M) = 0.5

Therefore 2.0794 is the required median value here.