Old school probability (very optional)
Introduction to Random Variables Introduction to random variables and probability distribution functions.
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- I'll now introduce you to the concept of a random variable.
- And for me this is something that I always had a lot of
- trouble getting my head around, and that's really because
- it's a byproduct of what it's called.
- It's called a variable and we're used to variables as kind
- of an unknown in the equation.
- If I write x plus 3 is equal to 7, the variable was x.
- Maybe you could solve for it or maybe you could have an
- equation y is equal to 3x minus 2.
- And then, here y and x are both variables.
- If you input 1x you could solve for the other variable, y.
- And you can change them.
- And variables were kind of things that could change and
- that you could solve for.
- And they could take on particular values.
- A random variable is kind of the same thing in that it can
- take on multiple values, but it's not something that you
- really ever solve for.
- So just so you get used to the notation, a random variable
- is usually a capital letter.
- Usually a Capital X, Y, or Z.
- Usually a capital X.
- And where it really defers from a traditional variable is that
- it can take on a bunch of different values, like a
- traditional variable but you never solve for it.
- And really, it's a little misleading to call it
- a variable at all.
- It's really a function.
- And it's a function that maps you from the world of random
- processes to an actual number.
- So let's say-- I don't know-- I wanted to somehow
- quantify a random process.
- Is it going to rain or not tomorrow?
- So let's see-- rain tomorrow.
- No rain.
- So you could observe that.
- You could wait until tomorrow and see if it rains or not, but
- then, how do you quantify it?
- Well, we can define a random variable that will quantify it.
- We can say this random variable is going to be equal to 1 if it
- rains tomorrow and it equals 0 if it doesn't rain tomorrow.
- We didn't have to assign 1 and 0; those tend to be a
- little bit more useful.
- They make sense.
- But we could have assigned this as-- I don't know-- we could
- have said that this is 21 and that this is 100.
- It's however you define it.
- So it's important to keep this distinction in mind, that a
- random variable-- it isn't a variable in the traditional
- sense of the world.
- It's more of a function that maps us from the world of a
- random process to a number.
- And then, this number is going to be random.
- Because we don't know if it's going to rain or not tomorrow,
- maybe we have some sense of the probabilities.
- So this variable can take on either value.
- And that leads us-- let me draw a couple of more random
- variable definitions just so you get the intuition that
- these really are functions.
- So we could define a random variable x as equal to--
- well, actually, let me--
- It could just be a very obvious numeric mapping.
- It could just be the number facing up when
- I roll a fair dice.
- That's a random variable.
- I could've also said just the number most facing east when I
- rolled a fair die, although that's a little less useful.
- You could also define a random variable, x is equal to-- you
- could say it's equal to 1 if heads, and I could
- say 0 if tails.
- Or I could have done it the other way around.
- Once again, these heads and tails are outcomes
- of an experiment.
- When I flip a coin that's a random process, each flip is an
- experiment and then the random variable is just quantifying
- that experiment.
- So I know it sounds a little technical with the terminology,
- but what we're really doing is fairly something simple.
- We're just assigning a 1 if we get a heads, a
- 0 if we get a tails.
- And the random variable is just that function mapping.
- Now I'm going into all of this random variable business, which
- I didn't do before when we did probability because it does
- start to become a little bit useful notationally.
- Because we'll start to talk about things like probability
- distributions and expected values, and it really is
- useful to quantify things as a random variable.
- And with that said, let's talk about probability distributions
- and expected values.
- So the first thing, probability-- actually, let
- me define something else.
- There's two types of random variables.
- You can have discrete random variables, or continuous.
- So there's two types.
- Discrete would be like, really all of the definitions
- that we had up until now.
- The flip of a coin, you're either going to get a 1 or a 0.
- There's two discrete cases.
- When I roll a die there's 1 of 6 different
- outcomes I can have.
- Whether it rains tomorrow, there's 1 of
- 2 outcomes: yes or no.
- So in all of these situations you could almost say you
- could have accountable number of outcomes.
- A continuous random variable could take on an infinite
- number of outcomes.
- So a continuous random variable would be the exact-- it could
- be x is equal to the exact amount of rain in
- inches tomorrow.
- Now why is this continuous?
- Well, if you think about it that could take on any of
- an infinite set of values.
- You might have 1 inch of rain tomorrow or you might
- have 1.1 inches of rain.
- You might have 1.111 inches of rain.
- You might have 2.1111 inches of rain.
- As you could see, you can come up with an infinite number
- of combinations of the amount of rain you have.
- To get exact-- well, I'll go into more of this when I
- talk about probability density functions.
- But in general, even though you might say, oh, well, I can't
- imagine having an infinite amount of rain because we're
- not going to have 100 gallons of rain tomorrow.
- But if you think about it, you could take on any value between
- 1 and 2 inches of rain.
- There's actually an infinite number of
- values between 0 and 1.
- For any value you could find a number that's a little--
- just to think about it.
- Between 0 and 1, 1/2 is in between those.
- And you can always find a number that's halfway between
- those and then a number that's halfway between those.
- There's an infinite number of numbers between
- any two numbers.
- While, with the discrete random variables, it just took on
- a finite number of values.
- Those ones up here, it could only take on 1 or 0.
- It didn't take on 1.1.
- Or it couldn't take on infinity.
- And I make that difference because actually, how we look
- at it in terms of their probability distributions
- are a little different.
- But they're very related.
- So a discreet random variable like the ones that we had
- defined at the beginning of these videos, they have a
- probability distribution.
- Let's do it for-- let's say x is equal to the number
- facing up on a fair dice.
- So we already know that to get each outcome--
- there's 6 outcomes.
- You either can get a 1, 2, 3, 4, 5, or 6 facing up.
- So let me draw the probability distribution.
- So here are all the outcomes.
- I'll draw that on the horizontal axis.
- So you get one outcome is 1, 2, 3, 4, 5, 6.
- And then, in the y-axis or in the vertical, you plot what is
- the probability of each of those outcomes occurring?
- What's the probability?
- I'll switch colors.
- So what's the probability of getting a 1?
- Well, it's 1/6.
- So if I draw 1/6 here.
- So the probability is of each of these occurring-- oh, I
- didn't want to do it like that-- are 1/6.
- It's essentially like a histogram or a bar chart.
- My intention is to draw them all the exact same height.
- So this is telling us that each of these there's a 1/6
- chance of it occurring.
- If I were to draw a probability distribution for-- if I were to
- find the random variable x is equal to 1 if
- heads, 0 if tails.
- This has a very simple probability distribution.
- There's only 2 outcomes.
- You're either going to get a 1 or a 0.
- And then in the y-axis you'll have the probabilities.
- And each of these, if it's a fair dice, it's going
- to be a 1/2 probability.
- And you're going to draw a little box like that that tells
- you each of them are 1/2.
- Now let's say that you had another-- I don't know--
- probably distribution function that looks something like this.
- Those are kind of very run of the mill examples, but what if
- I had something like this?
- Let's see.
- Where I had some dice-- let's say we had some weird
- weightings on it where it had an outcome of 1-- I had a
- 1/6 chance like that.
- Let's say for some reason a 2 could never happen.
- Maybe there is no 2.
- Let's say 3 has a 1/6 chance, a 4 has a 1/6 chance,
- a 5 has a 1/6 chance.
- And let's say there's two 6's.
- Like we erased the 2 and we put a 6 where the 2 is,
- so there's two 6's.
- So the 6 has a 2/6 chance.
- So the 6 would be a 2/6 chance.
- So now I think you can see where the probability
- distribution function starts to become a little
- bit more interesting.
- In the other ones it was kind of what we would call a
- uniform distribution.
- All of the outcomes are equally likely.
- That's a uniform distribution.
- Once again, this is a uniform distribution.
- This one, all of a sudden, is not a uniform distribution.
- 1 is just as likely as 3, 4, 5.
- 2 is impossible to happen, but 6 is twice as likely to happen
- as everything else-- is 2/6.
- If someone gives you a probability density function or
- if they give you a little chart like this, you can immediately
- say, what's the probability of different events occurring?
- So if you said that x is equal to the number on unfair dice
- that's described by this probability
- distribution function.
- Then if I were to ask you the probability that x is equal to
- 6, you'd say, oh, the probability x equals 6
- is equal to 2/6 or 1/3.
- If I were to say what's the probability that x, the random
- variable x as given by this definition, is greater than 5,
- you would take-- well, let's say greater than or equal to 5.
- Because you want to include 5, and at least, I did.
- So it would include this situation and this situation.
- So it'd be 1/6 plus 2/6.
- And so you could say that's 3/6 or 1/2.
- So it gets pretty interesting once you don't have a
- completely uniform distribution.
- And I'm doing all of this because in the next couple of
- videos-- well, one, we'll move away from discreet and actually
- study continuous probability distributions, which are called
- probability density functions.
- And then we'll go to study kind of a lot of distributions
- that show up in nature.
- The uniform distribution is one of them.
- There's also the binomial distribution and the normal
- distribution, which is what some people call the
- Gaussian or the bell curve.
- Which is kind of a very common thing.
- But anyway, I'm all out of time and I'll see
- you in the next video.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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