Random variables and probability distributions
Probability density functions Probability density functions for continuous random variables.
Probability density functions
- 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%.
- 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
- 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.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
Have something that's not a question about this content?
This discussion area is not meant for answering homework questions.
Share a tip
When naming a variable, it is okay to use most letters, but some are reserved, like 'e', which represents the value 2.7831...
Thank the author
This is great, I finally understand quadratic functions!
Have something that's not a tip or thanks about this content?
This discussion area is not meant for answering homework questions.
At 2:33, Sal said "single bonds" but meant "covalent bonds."
For general discussions about Khan Academy, visit our Reddit discussion page.
Here are posts to avoid making. If you do encounter them, flag them for attention from our Guardians.
- disrespectful or offensive
- an advertisement
- low quality
- not about the video topic
- soliciting votes or seeking badges
- a homework question
- a duplicate answer
- repeatedly making the same post
- a tip or thanks in Questions
- a question in Tips & Thanks
- an answer that should be its own question