9.1.2 Expected Lifetimes: Discrete Approximation

Now we return to the question of the expected life of a bulb. Intuitively, this is the same question as asking what the average lifetime would be if we burned all `100` bulbs until they failed — or, for that matter, what the average lifetime was for the `100` bulbs. Later we will move on to related questions, such as: Now that we know the average lifetime, if we pick a similar bulb "at random," how close to average is it likely to be?

Finding the average lifetime or expected performance is a challenging problem for two reasons. First, lifetime is not a discrete quantity: A bulb can burn out at any instant over a long continuous time interval. Second, the time interval is unbounded. In principle, an exceptional light bulb could burn "forever" — or at least for a very long time relative to the usual lifetimes of light bulbs. (In the case of the Centennial Light, "very long time" is not just a principle, it's a fact.) We will pay more attention in the next section to the problem of averaging over an "infinitely long" time interval.

As we have done with a number of other problems, we start by considering a discrete approximation to our continuous problem.

Activity 2

1. Suppose we have a collection of (idealized) bulbs, all mixed up in one box, of exactly two types:

   • Type A burns out in exactly `35` days, and we have `50` of these.

   • Type B burns out in exactly `63` days, and we have `100` of these.

   If we pick one bulb at random out of the box, we have a one-third probability of getting a bulb that will burn out in exactly `35` days and a two-thirds probability of getting a bulb that burns out in `63` days. What is the expected (or average) life of a bulb drawn at random from this collection?

2. Suppose now that we have a collection of three types of bulbs, each with an exact lifetime and a probability of occurrence given in Table 2. What is the expected life of a bulb drawn at random from this collection?

   Table 2   Bulb lifetimes and probabilities

   Type	Probability	Exact lifetime
   1	`p_1 = 1/4`	`t_1 = 23` days
   2	`p_2 = 1/2`	`t_2 = 47` days
   3	`p_3 = 1/4`	`t_3 = 65` days

3. If we have `n` types of bulbs, the `k`th occurring with probability `p_k` and having a lifetime of exactly `t_k`, for `k=1, 2, ..., n`, what is the expected life of a bulb taken at random from this collection?

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