Random variables and probability distributions
Probability Density Functions Probability density functions for continuous random variables.
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- In the last video, I introduced you to the notion of-- well,
- really we started with the random variable.
- And then we moved on to the two types of random variables.
- You had discrete, that took on a finite number of values.
- And the these, I was going to say that they tend to be
- integers, but they don't always have to be integers.
- You have discrete, so finite meaning you can't have an
- infinite number of values for a discrete random variable.
- And then we have the continuous, which can take
- on an infinite number.
- And the example I gave for continuous is, let's
- say random variable x.
- And people do tend to use-- let me change it a little bit, just
- so you can see it can be something other than an x.
- Let's have the random variable capital Y.
- They do tend to be capital letters.
- Is equal to the exact amount of rain tomorrow.
- And I say rain because I'm in northern California.
- It's actually raining quite hard right now.
- We're short right now, so that's a positive.
- We've been having a drought, so that's a good thing.
- But the exact amount of rain tomorrow.
- And let's say I don't know what the actual probability
- distribution function for this is, but I'll draw one and
- then we'll interpret it.
- Just so you can kind of think about how you can think about
- continuous random variables.
- So let me draw a probability distribution, or they call
- it its probability density function.
- And we draw like this.
- And let's say that there is-- it looks something like this.
- Like that.
- All right, and then I don't know what this height is.
- So the x-axis here is the amount of rain.
- Where this is 0 inches, this is 1 inch, this is 2 inches,
- this is 3 inches, 4 inches.
- And then this is some height.
- Let's say it peaks out here at, I don't know,
- let's say this 0.5.
- So the way to think about it, if you were to look at this and
- I were to ask you, what is the probability that Y-- because
- that's our random variable-- that Y is exactly
- equal to 2 inches?
- That Y is exactly equal to two inches.
- What's the probability of that happening?
- Well, based on how we thought about the probability
- distribution functions for the discrete random variable,
- you'd say OK, let's see.
- 2 inches, that's the case we care about right now.
- Let me go up here.
- You'd say it looks like it's about 0.5.
- And you'd say, I don't know, is it a 0.5 chance?
- And I would say no, it is not a 0.5 chance.
- And before we even think about how we would interpret it
- visually, let's just think about it logically.
- What is the probability that tomorrow we have exactly
- 2 inches of rain?
- Not 2.01 inches of rain, not 1.99 inches of rain.
- Not 1.99999 inches of rain, not 2.000001 inches of rain.
- Exactly 2 inches of rain.
- I mean, there's not a single extra atom, water molecule
- above the 2 inch mark.
- And not as single water molecule below the 2 inch mark.
- It's essentially 0, right?
- It might not be obvious to you, because you've probably heard,
- oh, we had 2 inches of rain last night.
- But think about it, exactly 2 inches, right?
- Normally if it's 2.01 people will say that's 2.
- But we're saying no, this does not count.
- It can't be 2 inches.
- We want exactly 2.
- 1.99 does not count.
- Normally our measurements, we don't even have tools that
- can tell us whether it is exactly 2 inches.
- No ruler you can even say is exactly 2 inches long.
- At some point, just the way we manufacture things, there's
- going to be an extra atom on it here or there.
- So the odds of actually anything being exactly a
- certain measurement to the exact infinite decimal
- point is actually 0.
- The way you would think about a continuous random variable,
- you could say what is the probability that Y is almost 2?
- So if we said that the absolute value of Y minus is 2 is
- less than some tolerance?
- Is less than 0.1.
- And if that doesn't make sense to you, this is essentially
- just saying what is the probability that Y is greater
- than 1.9 and less than 2.1?
- These two statements are equivalent.
- I'll let you think about it a little bit.
- But now this starts to make a little bit of sense.
- Now we have an interval here.
- So we want all Y's between 1.9 and 2.1.
- So we are now talking about this whole area.
- And area is key.
- So if you want to know the probability of this occurring,
- you actually want the area under this curve from this
- point to this point.
- And for those of you who have studied your calculus, that
- would essentially be the definite integral of this
- probability density function from this point to this point.
- So from-- let me see, I've run out of space down here.
- So let's say if this graph-- let me draw it
- in a different color.
- If this line was defined by, I'll call it f of x.
- I could call it p of x or something.
- The probability of this happening would be equal to the
- integral, for those of you who've studied calculus, from
- 1.9 to 2.1 of f of x dx.
- Assuming this is the x-axis.
- So it's a very important thing to realize.
- Because when a random variable can take on an infinite number
- of values, or it can take on any value between an interval,
- to get an exact value, to get exactly 1.999, the
- probability is actually 0.
- It's like asking you what is the area under a
- curve on just this line.
- Or even more specifically, it's like asking you
- what's the area of a line?
- An area of a line, if you were to just draw a line,
- you'd say well, area is height times base.
- Well the height has some dimension, but the base,
- what's the width the a line?
- As far as the way we've defined a line, a line has no with,
- and therefore no area.
- And it should make intuitive sense.
- That the probability of a very super-exact thing happening
- is pretty much 0.
- That you really have to say, OK what's the probably
- that we'll get close to 2?
- And then you can define an area.
- And if you said oh, what's the probability that we get
- someplace between 1 and 3 inches of rain, then of course
- the probability is much higher.
- The probability is much higher.
- It would be all of this kind of stuff.
- You could also say what's the probability we have
- less than 0.1 of rain?
- Then you would go here and if this was 0.1, you would
- calculate this area.
- And you could say what's the probability that we have more
- than 4 inches of rain tomorrow?
- Then you would start here and you'd calculate the area in the
- curve all the way to infinity, if the curve has area all
- the way to infinity.
- And hopefully that's not an infinite number, right?
- Then your probability won't make any sense.
- But hopefully if you take this sum it comes to some number.
- And we'll say there's only a 10% chance that you have more
- than 4 inches tomorrow.
- And all of this should immediately lead to one light
- bulb in your head, is that the probability of all of the
- events that might occur can't be more than 100%.
- Right?
- All the events combined-- there's a probability of 1 that
- one of these events will occur.
- So essentially, the whole area under this curve
- has to be equal to 1.
- So if we took the integral of f of x from 0 to infinity, this
- thing, at least as I've drawn it, dx should be equal to 1.
- For those of you who've studied calculus.
- For those of you who haven't, an integral is just the
- area under a curve.
- And you can watch the calculus videos if you want to learn a
- little bit more about how to do them.
- And this also applies to the discrete probability
- distributions.
- Let me draw one.
- The sum of all of the probabilities have
- to be equal to 1.
- And that example with the dice-- or let's say, since it's
- faster to draw, the coin-- the two probabilities have
- to be equal to 1.
- So this is 1, 0, where x is equal to 1 if we're heads
- or 0 if we're tails.
- Each of these have to be 0.5.
- Or they don't have to be 0.5, but if one was 0.6, the
- other would have to be 0.4.
- They have to add to 1.
- If one of these was-- you can't have a 60% probability of
- getting a heads and then a 60% probability of getting
- a tails as well.
- Because then you would have essentially 120% probability
- of either of the outcomes happening, which makes
- no sense at all.
- So it's important to realize that a probability distribution
- function, in this case for a discrete random variable, they
- all have to add up to 1.
- So 0.5 plus 0.5.
- And in this case the area under the probability
- density function also has to be equal to 1.
- Anyway, I'm all the time for now.
- In the next video I'll introduce you to the idea
- of an expected value.
- See you soon.
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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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