Modelling lifetime of a lightbulb

Question

I have a question regarding how to model the lifetime of a lightbulb, when give the probability that it will fail during a $5$-hour shift.

The problem states that in a building all the lightbulbs will work perfectly before they fail and then must be replaced. The probability that a lightbulb fails during any $5$-hour shift is $0.15$.

What is the probability that a lightbulb will be operational for at most $4$ shifts (i.e. fewer than $5$ shifts) before it fails?

How do we know which kind of distribution we should use to model the lifetime of a lightbulb? Once I know the distribution, I know i have to use the failure rate function lamba(t), but i need to know the F(t) and f(t) for the variable in order to do so.

Thanks.

Answer 1

How do we know which kind of distribution we should use to model the lifetime of a lightbulb?

When you were introduced to the common distributions, you should have been taught what is measured by their random variables.   Revise and memorise!   You just have to match the pattern.

The problem states that in a building all the lightbulbs will work perfectly before they fail and then must be replaced. The probability that a lightbulb fails during any 5-hour shift is 0.15.

It is suggested that each shift is an independent Bernoulli trial with identical failure rate.

The lifetime of a lightbulb is: the count of shifts until failure given a failure rate of $0.15$ each shift.   This is a discrete random variable; can you see which kind?

Answer 2

I agree wholeheartedly. You need a model. (For that matter, you aren't told that the failure is the first jump in a compound Poisson process and so can be described in terms of a failure rate $\lambda(t).$

That said, the lack of specification in the problem is probably implies they want you to assume the simplest model, that the failure is memoryless and constant rate. This reduces the problem to assuming that each shift is an independent trial with failure probability $0.15.$ Whether you complete the problem by remembering which distribution describes the time of the first failure and looking up its CDF or by figuring it out from scratch is up to you. (Hint: working for at most four shifts means it isn't the case that it worked for all of the first five shifts.)