L09.4 Memorylessness of the Exponential PDF
We now revisit the exponential random variable that we introduced earlier and develop some intuition about what it represents. We do this by establishing a memorylessness property, similar to the one that we established earlier in the discrete case for the geometric PMF. Suppose that it is known that light bulbs have a lifetime until they burn out, which is an exponential random variable. You go to a store, and you are given two choices, to buy a new light bulb, or to buy a used light bulb that has been working for some time and has not yet burned out. Which one should you take? We want to approach this question mathematically. So let us denote by capital T the lifetime of the bulb. So time starts at time 0, and then at some random time that we denote by capital T, the light bulb will burn out. And we assume that this random variable is exponential with some given parameter lambda. In one of our earlier calculations, we have shown that the probability that capital T is la...