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 larger than some
value little x falls exponentially
with that value x.
We are now told that a certain light bulb has already been
operating for t time units without failing.
So we know that the value of the random variable capital T
is larger than little t.
We are interested in how much longer the light bulb will be
operating, and so we look at capital X, which is the
remaining lifetime from the current time until the light
bulb burns out.
So capital X is this particular random variable
here, and it is equal to capital T minus little t.
Let us now calculate the probability that the light
bulb lasts for another little x time units.
That is, that this random variable, capital X, is at
least as large as some little x.
That is, that the light bulb remains alive
until time t plus x.
We use the definition of conditional probabilities to
write this expression as the probability that capital X is
bigger than little x.
On the other hand, capital X is T minus t, so we
write it this way--
T minus t is bigger than little x, and also that T is
bigger than little t, divided by the probability of the
conditioning event.
Just write this event in a cleaner form, capital T being
larger than little t plus x and being larger than little
t, again divided by the probability of the
conditioning event.
And now notice that capital T will be greater than little t
and also greater than little t plus x, that is, capital T is
larger than this number and this number, if and only if it
is larger than this second number here.
So in other words, the intersection of these two
events is just this event here, that capital T is larger
than little t plus x.
Now, we can use the formula for the probability that
capital T is larger than something.
We apply this formula, except that instead of little x, we
have t plus x.
And so here we have e to the minus lambda t plus x divided
by the probability that capital T is bigger than t.
So we use this formula, but with little t in the place of
little x, and we obtain e to the minus lambda t.
We have a cancellation, and we're left with e to the minus
lambda x, which is a final answer in this calculation.
What do we observe here?
The probability that the used light bulb will live for
another x time units is exactly the same as the
corresponding probability that the new light bulb will be
alive for another x time units.
So new and used light bulbs are described by the same
probabilities, and they're probabilistically
identical, the same.
Differently said, the used light bulb does not remember,
and it is not affected by how long it has been running.
And this is the memorylessness property of
exponential random variables.
Let us now build some additional insights on
exponential random variables.
We have a formula for the density, the PDF.
And from this, we can calculate the probability that
T lies in a small interval.
For example, for a small delta, this probability here
is going to be approximately equal to the density of T
evaluated at 0 times delta, which is lambda times e to the
0, which is 1, times delta.
What if we are told that the light bulb has been alive for
t time units?
What is the probability that it burns out during the next
delta times units?
Since a used but still alive light bulb is
probabilistically identical to a new one, this conditional
probability is the same as this probability here that a
new light bulb burns out in the next delta times units.
And so this is also approximately
equal to lambda delta.
So we see that independently of how long a light bulb has
been alive, during the next delta time units it will have
a lambda delta probability of failing.
One way of thinking about this situation is that the time
interval is split into little intervals of length delta.
And as long as the light bulb is alive, if it is alive at
this point, it will have probability lambda delta of
burning out during the next interval of length delta.
This is like flipping a coin.
Once every delta time steps, there is a probability lambda
delta that there is a success in that coin flip, where
success corresponds to having the light bulb actually burn
down, and the exponential random variable corresponds to
the total time elapsed until the first success.
In this sense, the exponential random variable is a close
analog of the geometric random variable, which was the time
until the first success in a discrete time setting.
This analogy turns out to be the foundation behind the
Poisson process that we will be studying
later in this course.