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Content
0.59 -> What I want to discuss a
little bit in this video
3.31 -> is the idea of a
random variable.
8.97 -> And random variables at first
can be a little bit confusing
11.93 -> because we will want to think
of them as traditional variables
15 -> that you were first exposed
to in algebra class.
17.38 -> And that's not quite what
random variables are.
20.16 -> Random variables are
really ways to map outcomes
23.28 -> of random processes to numbers.
25.49 -> So if you have a random process,
like you're flipping a coin
29.32 -> or you're rolling dice or you
are measuring the rain that
34.09 -> might fall tomorrow,
so random process,
37.24 -> you're really just mapping
outcomes of that to numbers.
44.735 -> You are quantifying
the outcomes.
49.34 -> So what's an example
of a random variable?
51.64 -> Well, let's define
one right over here.
53.27 -> So I'm going to define
random variable capital X.
55.94 -> And they tend to be
denoted by capital letters.
59.29 -> So random variable capital
X, I will define it
61.73 -> as-- It is going
to be equal to 1
65.41 -> if my fair die rolls heads--
let me write it this way--
71.1 -> if heads.
76.09 -> And it's going to be
equal to 0 if tails.
80.464 -> I could have defined
this any way I wanted to.
82.38 -> This is actually a
fairly typical way
83.921 -> of defining a random variable,
especially for a coin flip.
86.69 -> But I could have
defined this as 100.
88.67 -> And I could have
defined this as 703.
91.45 -> And this would still be a
legitimate random variable.
94.71 -> It might not be as pure a
way of thinking about it
98.03 -> as defining 1 as
heads and 0 as tails.
100.5 -> But that would have
been a random variable.
102.84 -> Notice we have taken this
random process, flipping a coin,
107 -> and we've mapped the outcomes
of that random process.
109.727 -> And we've quantified them.
110.81 -> 1 if heads, 0 if tails.
113.75 -> We can define another random
variable capital Y as equal to,
121 -> let's say, the sum of
rolls of let's say 7 dice.
130.569 -> And when we talk
about the sum, we're
132.11 -> talking about the
sum of the 7-- let
134.47 -> me write this-- the
sum of the upward face
146.52 -> after rolling 7 dice.
152.96 -> Once again, we are quantifying
an outcome for a random process
157.73 -> where the random process
is rolling these 7 dice
161.03 -> and seeing what
sides show up on top.
163.82 -> And then we are taking those
and we're taking the sum
166.26 -> and we are defining a
random variable in that way.
171.71 -> So the natural
question you might ask
173.49 -> is, why are we doing this?
175.1 -> What's so useful about defining
random variables like this?
178.68 -> It will become
more apparent as we
180.31 -> get a little bit
deeper in probability.
182.206 -> But the simple way
of thinking about it
183.83 -> is as soon as you
quantify outcomes,
186.71 -> you can start to do a little
bit more math on the outcomes.
190.529 -> And you can start
to use a little bit
192.07 -> more mathematical
notation on the outcome.
195.85 -> So for example, if you
cared about the probability
198.38 -> that the sum of the upward
faces after rolling seven
201.43 -> dice-- if you cared
about the probability
203.2 -> that that sum is less than
or equal to 30, the old way
207.532 -> that you would have
to have written
208.99 -> it is the probability
that the sum
211.35 -> of-- and you would have to write
all of what I just wrote here--
219.15 -> is less than or equal to 30.
223.6 -> You would have had to
write that big thing.
226.09 -> And then you would try
to figure it out somehow
228.38 -> if you had some information.
229.91 -> But now we can just
write the probability
233.01 -> that capital Y is less
than or equal to 30.
237.33 -> It's a little bit
cleaner notation.
239.37 -> And if someone else cares
about the probability
243.13 -> that this sum of the upward
face after rolling seven dice--
246.6 -> if they say, hey, what's the
probability that that's even,
249.112 -> instead of having to
write all that over,
250.82 -> they can say, well, what's the
probability that Y is even?
257.839 -> Now the one thing that
I do want to emphasize
259.89 -> is how these are different
than traditional variables,
262.46 -> traditional variables that
you see in your algebra class
264.86 -> like x plus 5 is equal
to 6, usually denoted
267.42 -> by lowercase variables.
269.18 -> y is equal to x plus 7.
272.33 -> These variables, you can
essentially assign values.
275.44 -> You either can solve for
them-- so in this case,
278.35 -> x is an unknown.
279.36 -> You could subtract 5 from
both sides and solve for x.
282.12 -> Say that x is going
to be equal to 1.
284.99 -> In this case, you could say,
well, x is going to vary.
288.29 -> We can assign a
value to x and see
290.69 -> how y varies as a function of x.
293.39 -> You can either
assign a variable,
294.77 -> you can assign values to them.
296.51 -> Or you can solve for them.
298.95 -> You could say, hey x is
going to be 1 in this case.
302.78 -> That's not going to be the
case with a random variable.
305.36 -> A random variable can take on
many, many, many, many, many,
307.93 -> many different values with
different probabilities.
310.66 -> And it makes much
more sense to talk
312.32 -> about the probability of
a random variable equaling
315.48 -> a value, or the probability
that it is less than or greater
318.43 -> than something,
or the probability
320.14 -> that it has some property.
321.99 -> And you see that in
either of these cases.
324.669 -> In the next video, we'll
continue this discussion
326.71 -> and we'll talk a little
bit about the types
