Contents of this Probability theory episode:

The forever young property, Exponential distribution, Probability, Average, Density function, Distribution function, Expected value, Standard deviation.

Text of slideshow

The useful lifetime of a smartphone follows the exponential distribution, with an expected life of 4 years.

a) What is the probability that it works at least for 3 years?

b) What is the probability that it works longer than 3 years, but less than 5?

c) What is the probability that if it has been working for 3 years, it will die in the next 2 years?

The last question will be funny.

Let's try to figure out how it is different from the previous one.

For this, let's do some drawing.

We know it has been working for 3 years,

so it will die somewhere here.

But within 5 years.

So far, this is very much like the previous question.

To understand the difference between the two questions better, take Bob.

We try to predict the probability that Bob will die between his 70th and 71st birthdays.

The question is whether this probability is high or low. Well, it depends.

If we make the prediction at the moment of Bob's birth for the probability of him dying between his 70th and 71st birthdays, then the probability will be low.

It is low, because many things can happen to Bob until then, for example he gets hit by the bus at age 5, or gets a heart attack at age 60...

On the other hand, if Bob just turned 70, and as a present for his birthday we predict his chances of dying in the next year, then we can reassure him that this probability is pretty high.

Well, this is the difference between the two types of questions.

Both cases are about the phone dying between its 3rd and 5th year,

but in one case we ask at the moment of its birth,

in the other case we ask after 3 years of operation.

this was the first question

if we still remember

Finally, there is one more funny thing.

The exponential distribution has a strange feature.

This is a feature that Bob, unfortunately, does not have.

This feature is being forever young.

In the case of Bob, who does not have this trait, if we want to know the probability of him dying within a year, then we need to know how old he is.

His chances of dying within a year are not the same at the age of 10 or 60 or even 102. As time passes, Bob indeed has an increasing chance of dying, because he is not ageless.

However, the exponential distribution is just that.

It means it doesn't matter whether those three years have passed.

We could even lie about it.

As if the 3 years in the conditions didn't even exist.

# The "forever young" property of the exponential distribution

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The forever young property, Exponential distribution, Probability, Average, Density function, Distribution function, Expected value, Standard deviation.

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