[PIANO MUSIC PLAYING]
HOST: Welcome to The Random Fish.
We hope you have a continuously lovely evening.
Here is our menu.
NORMA: What do you recommend?
HOST: Each of our dishes has its own special flavor and story.
Our most popular fish is the Normal fish.
But if you'd like something more exotic, you can try the Cauchy fish.
It has an amazingly thick, juicy tail.
Or the Expo fish.
It will make your hunger decay exponentially.
And now for tonight's entertainment, this is our star, Randi.
She loves swimming all around her tank randomly.
Your challenge, if you choose to accept it,
is to predict her horizontal position in the tank
and answer questions about Randi's probabilities.
JOE BLITZSTEIN: Randi and her tank are two dimensional.
The area of the water is 1.
Each time Randi pauses, her position is uniformly random in the tank.
This means the probability of her being in a particular region
is the area of that region.
For example, the probability that Randi is in this triangle
is half the probability that she is in this square, whose area is double
the triangle's area.
And the probability of being in the square
is the same as the probability of being in this blob with the same area.
HOST: OK everyone.
What's the probability that Randi is at x equals 1.5?
We are looking at her horizontal position in the tank
based on where her eye is.
BERNIE: That's easy.
I'll just measure the height of the tank at x equals 1.5.
OK-- 0.33.
NORMA: No, the probability is 0.
BERNIE: Wait, that's crazy.
Doesn't that mean we would never see Randi at 1.5?
Look, she's there right now.
NORMA: Randi is approximately at 1.5, but do you really think it's exact?
1.50000...
forever?
BERNIE: Well, she's at 1.5 as far as we can tell.
Even with the best ruler in the world, there's a limit to how precise
our measurements can be.
NORMA: So when you say the probability of being at 1.5 is 0.33,
do you mean the probability of being at 1.50, 1.500, or what?
BERNIE: Um, I don't know.
NORMA: Randi's position is a continuous random variable,
even if we can't measure it to infinite precision.
The probability is 0 that her position is 1.50000 forever.
BERNIE: I know that's what's taught in Continuopolis,
but it doesn't seem very practical just to say the probability is 0.
NORMA: Can we talk about something other than fish?
I'm just not sure we're compatible.
I admire how you handle challenges so straightforwardly
and with such discretion, but we live in such different spaces.
BERNIE: Norma, I love you.
I know I'm not the smoothest guy in the world,
but we actually have a lot in common.
And together, combining discrete and continuous perspectives,
we can achieve so much more than we could separately.
NORMA: I love you, too.
We do speak the same language at least, the language
of randomness and uncertainty.
Let's continue trying to understand each other then.
BERNIE: OK.
I agree that Randi's position is continuous, not discrete.
She's a fish swimming in the water, not a frog
jumping from one lily pad to another.
But then how can the probability of being at 1.5 be 0?
Doesn't that mean it's impossible for Randi to be at 1.5?
But to get from the left half of the tank
to the right half of the tank she must pass through 1.5.
NORMA: Good point.
To swim from x equals 1 to x equals 2, Randi
must cross through every real number from 1 to 2.
Ancient Statland philosophers were puzzled
about how that could be possible.
How can Randi pass by infinitely many intermediate points
in a finite amount of time?
Anyway, if we observe Randi over a long time interval,
it is very likely that Randi will visit x equals 1.5 at least once.
But here, we are checking on Randi at one time point.
BERNIE: So the probability is 0 that Randi is exactly
at 1.5 at that one time point.
But as a practical matter, what's the probability that Randi is close to 1.5?
NORMA: Well, then we should specify an error tolerance.
How close is close?
If say, we want Randi to be within 0.05 of 1.5 we can look at the probability
that she is between 1.45 and 1.55.
BERNIE: And how would you compute that?
NORMA: Just find the area under the curve from 1.45 to 1.55.
JOE BLITZSTEIN: The curve formed by the tank
is a probability density function, also known as a PDF
(not to be confused with Portable Document Format.)
A PDF is a non-negative function such that the area under the curve is 1.
For a continuous random variable x, the probability
that x is in a particular interval is the area
under the PDF curve over that interval.
Note that the PDF of Randi's horizontal position
is the curve formed by the tank since the probability of her being
in a region is the area of that region.
HOST: Well, that's great.
Now time for a new question.
What's the probability that Randi is between x equals 1 and x equals 2?
NORMA: Same idea.
We just need the area under the curve from x equals 1 to x equals 2.
Seems messy to compute though.
BERNIE: But wait, there's a sweet symmetry here.
Randi is equally likely to be in the intervals 0 to 1, 1 to 2, and 2 to 3.
So by symmetry, the probably is one third of being between 1 and 2.
NORMA: You're right.
That's beautiful.
I didn't realize that breaking the problem up into discrete chunks
would help so much.
HOST: Well done, everyone.
Now what's the probability that Randi is between 1.5 and 2?
NORMA: We need the area under the curve from 1.5 to 2.
We can't invoke symmetry here, so finding this area requires calculus.
BERNIE: As an approximation though, how about making a grid in the tank
and using a discrete random variable?
NORMA: Good idea.
That's like how we can approximate the area of a region
by adding up the areas of lots of narrow rectangles.
IN UNISON: 0.14.
HOST: Our final round for the evening will be a face-off between the two
tables that have done the best so far.
Norma and Bernie and Gamma and Gumbel.
Instead of having a probability for Randi, let's up the ante
and give a prediction.
Where do you think she is?
The table with the answer that's closer to the truth
will win a free trip to the Statland Games in Averagemont.
BERNIE: What if there's a tie?
HOST: Ha, well, that may be common where you come from,
but here in Continuopolis we don't worry about ties.
The probability is 0 that two independent continuous random variables
will be equal.
BERNIE: But I'm not sure the guesses will
be specified to infinite precision.
And two tables could follow the same strategy
and arrive at exactly the same guess.
HOST: Well, in that unlikely event I will choose the winner randomly.
BERNIE: OK.
So half the time Randi is to the left of 1.42
and half the time she is to the right of 1.42.
So how about guessing 1.42?
NORMA: That's the median.
But here is the mean--
the center of mass.
So how about guessing 1.46?
BERNIE: How can we decide which form of average
to use for our prediction, the mean or the median?
NORMA: Last time I visited Averagemont I learned
that if we are predicting a quantity x and want to minimize the expected
absolute difference between x and our prediction, we should guess the median.
But if we want to minimize the expected squared difference between x
and our prediction, we should guess the mean.
BERNIE: But the closest guess in terms of absolute difference
will also be the closest guess in terms of squared difference.
So it's still not clear whether we should choose the mean or the median.
NORMA: A-ha.
Let's keep in mind that we are trying to do better than the other table,
not trying to optimize some measure of long-run performance
if we play this game many times.
If we guess the median and the other table guesses a different value, C,
we have more than a 50% chance that our guess will be closer.
If C is greater than the median, we win for all the cases
to the left of the median plus everything
up to halfway between the guesses.
Similarly, if C is less than the median, we win for all cases
to the right of the median plus everything up to halfway between.
BERNIE: Beautiful.
So lets guess the median--
1.42.
HOST: There's Randi at 1.37.
Norma and Bernie guessed 1.42, while Gamma and Gumbel guessed 1.46.
Congratulations, Norma and Bernie.
Your outstanding thinking about averages has earned you a free trip
to the Statland Games in Averagemont!
JOE BLITZSTEIN: Bernie and Norma have realized
that discrete and continuous thinking are complementary.
Each can give useful approximations and insights for the other.
Together, discrete and continuous approaches form a powerful framework
for modeling the world.
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