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We've already talked about how we use a probability math function to model the probability associated

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with a discrete random variable.

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We'll use a probability density function to model the probability associated with a continuous random

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variable.

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So if you remember, we looked at the probability mass function for a DAI role, and when we graphed

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that probability mass function, we showed that the probability associated with rolling a one, two,

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three, four, five or six was always one out of six or approximately 0.167.

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And so the sketch of that probability mass function looked like this, where we have a uniform distribution

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because the probability remains uniform, it's always the same.

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The probability is equal for rolling each of these values.

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When we roll a six sided DAI one time, what we notice here is that if we look at the graph of this

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probability mass function, we're only defining probabilities for these exact values here, the exact

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value one, the exact value to the exact value three etc. We don't define a probability for one and

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a half or two and a half or any value between one, two, three, four, five and six.

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And of course that's because with the probability mass function, we're dealing with a discrete random

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variable where we have those discrete countable values and no values in between.

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Remember that a continuous random variable in contrast, can be defined for an infinite set of values,

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not just a discrete set of values like this.

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One, two, three, four, five, six.

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In other words, if we were to sketch the graph of a probability density function for a continuous random

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variable as opposed to a probability mass function for a discrete random variable, we would use a similar

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set of axes like this, the horizontal axis x and the vertical axis indicating probability.

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But the sketch of our probability density function might be a curve that looks something like this.

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And then what we would do is identify values of X.

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So maybe this value here, x one, this value here x two.

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But in theory, for this continuous random variable, we could define an infinite number of values in

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between x one and x two, and we could think about the probability of each of those values.

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And remember, with a continuous random variable, we can always get more and more granular with these

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values as long as we continue measuring out to a larger number of decimal places.

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So really there are an infinite number of values along this horizontal axis for the continuous random

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variable, in contrast with this discrete random variable where we just have this discrete countable

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value set.

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So this is the kind of variable, the kind of scenario we're trying to model with a probability function.

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And in this case we can say that every continuous random variable x has a probability density function

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F of X.

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And what we can say about this probability density function is that the probability that the continuous

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random variable x takes on some value on the interval A to B is given by this expression.

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Over here on the right, where this notation here is called the integral.

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This is the integral from A to B of the function F of x.

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This notation here is like the other side of the integral notation, the integral and the d sandwich,

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whatever is between them and tells us to perform a specific operation with this function F of x which

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we'll talk about.

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Notice here also that now that we're switching to a continuous random variable, when we're calculating

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probability, we have to calculate probability over an interval A to B, whereas with the discrete random

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variable we calculated the probability of a specific value one value at a time.

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So here we have the probability of rolling a one, the probability of rolling it to the probability

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of rolling a three.

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When we transition to the continuous random variable and the probability density function, we are no

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longer going to calculate the probability that X takes on a specific value.

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And the reason for that is because it's essentially impossible for us to measure the probability that

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X takes on an exact value when we're dealing with a continuous random variable.

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So let's say, for instance, that the continuous random variable that the probability density function

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associated with this continuous random variable here, we've sketched it and maybe it model's finishing

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times for a standardized test.

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Let's say here that this is the mean finishing time for the test.

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And that mean finishing time, let's say, is 71.

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Minutes such that let's say here this X to value is a finishing time of we'll call it.

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80 minutes.

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Well, if we want to calculate the probability that a student's finishing time is exactly 71 minutes,

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that's virtually impossible for us to do because they might finish at 71 minutes and 0.0000000 1 seconds.

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In other words, they might be off of 71 minutes by a value of time so small that it's almost immeasurable.

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And because of that theoretical idea where it's basically impossible for us to measure that, they finished

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truly at exactly 71 minutes because remember, we can always divide time into smaller and smaller and

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smaller slices.

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We can't really ever say the probability that a student finished at exactly one minute.

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And so with a continuous random variable and its probability density function, the probability of any

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specific exact value in the range of the continuous random variable, the probability of any exact value

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is actually zero.

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So for a variable like this one, the probability that x is exactly equal to 71 minutes is zero.

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The probability that X is exactly equal to 80 minutes is also zero.

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The probability that the variable takes on any specific exact value is always zero, because in theory,

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for a continuous random variable, we can't pinpoint exactly 71 minutes.

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We're always going to be off by some infinitely small fraction of a second.

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However, we can calculate probability over some interval A to B, for instance, we could think about

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the probability that the student's finishing time is somewhere between 71 minutes and 80 minutes.

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And in the same way over here with the probability mass function for a discrete random variable, the

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probability of one specific value is given by this column or this bar with a continuous random variable

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and the probability density function.

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The probability that the discrete random variable takes on a value in some interval is given by the

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area under this probability density curve over that particular interval.

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So if we're interested here in the probability that the student's finishing time is between 71 and 80

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minutes, that probability can be represented by all of the area under this curve, over this interval.

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In other words, if we could geometrically calculate the area under this curve here, we would find

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some value between zero and one, and that value for the area would be equivalent to the probability

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that the variable takes on some value in that interval.

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So to summarize here, instead of using a probability mass function to calculate that a discrete random

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variable takes on an exact value, we'll now use a probability density function to calculate the probability

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that a continuous random variable takes on some value within a specific interval.

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Now, when it comes to the probability density function, there are a couple of things we need to remember.

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So it has to be true that the probability density function f or f of X is greater than or equal to zero

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for all possible values of X that the continuous random variable can take on.

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So that means for all of the values of x between here and here, under the curve of the probability

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density function, all of these values of X on this interval have to produce a positive value for F

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of X.

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And we can think about this curve here as the graph of f of x, which just means that this curve, if

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we were to sketch F of x, this curve, the sketch of f of x has to sit above the horizontal axis anywhere.

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We can't have any negative values.

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This curve can't dip below the horizontal axis at any point in this interval.

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So that's our first requirement.

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And then our second requirement is that the total area under this curve when we add it all up, has

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to be equal to one.

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Because remember, our vertical axis here represents probability.

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So the area under this curve is the total probability for all possible values of our continuous random

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variable.

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And probability always has to add to one, which means that all of the area under this curve has to

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add to one.

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And that's what this means here, this integral notation.

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And then from negative infinity to positive infinity just means everywhere under the curve of F of X,

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to integrate means to find the area under the curve.

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So the area under the curve F of X everywhere underneath it.

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When we add all that together, that has to be equal to one.

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So that's our second requirement.

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So that being said, let's go ahead and look at an example.

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We'll say here that.

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This function f of x is a probability density function as long as we define it over the interval 0 to

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1.

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So under this definition, before we move forward, we need to check to see that we meet both of these

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criteria here.

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That F of X is greater than or equal to zero for all X, and that the total area under the curve is

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equal to one.

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So there's a couple of ways that we can go about this.

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But one easy way to verify this first criteria is to sketch the graph of F.

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So if we use a calculator or any computer algebra system, any graphing software to sketch F of X,

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we see this graph here.

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We're told that we're only looking at the values of X on the interval 0 to 1.

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So here this is x equals zero, this is x equals one.

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And so we're only interested in this section of the graph between these two points and we can see that

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everywhere between zero and one.

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The graph here is at or above the horizontal axis.

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And so we meet this first criteria.

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If we had been given an interval, for instance, of negative one to positive one, we can see that

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over here between negative one and zero, the graph sits below the horizontal axis.

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And so we would not meet this first criteria and we could not call this a probability density function.

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But because the interval has been limited to 0 to 1 and we can see that the graph is at or above the

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horizontal axis, everywhere in that interval we meet this first criteria.

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Now for the second criteria, in order to figure this out by hand, we're going to need to know how

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to evaluate integrals.

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And that only comes up in calculus one or calculus two.

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But that's okay.

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We don't have to know calculus.

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We don't have to know how to evaluate an integral because we can still use calculators or computers

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to help us verify this fact really quickly.

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So all we have to do is substitute here for F of X for x cubed.

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So this is what we would plug into a computer algebra system, software, computer to get this integral

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evaluated for us, we would say here the integral and then instead of from negative infinity to infinity

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in the definition here, the interval has been limited to 0 to 1.

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So we'll say 0 to 1.

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And then instead of F of x here we'll put in for x cubed our F of x function and then D x again, we

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can plug this into any calculator or computer and it will tell us the value of this integral if we don't

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know how to do integration.

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The value of this integral is in fact one.

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And so we meet this criteria.

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If we wanted to evaluate this integral by hand, the way that we would do it is by adding one to this

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exponent here to get X to the fourth instead of x cubed.

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So we would have X to the fourth and then we would divide by this new exponent of four.

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So when we divide this function by four, it'll cancel this coefficient of four.

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And so the integral of four x cubed is x to the fourth.

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And then we would evaluate that result, that integrated result on the given interval 0 to 1.

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And when we do, when we evaluate over this interval, we always plug in this upper limit here of one

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first.

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So we get one to the four.

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And then we always subtract whatever we get when we plug in this lower limit of integration.

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So when we plug in zero, we get zero to the fourth here.

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And of course, when we evaluate that, we get one -0 or one.

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And so we see that the value of that integral over this interval 0 to 1 is in fact one.

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And so we can double check that we do in fact meet this requirement.

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But again, we don't need to know integration.

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If we haven't taken calculus, we can use a computer to help us calculate any integral, the integral

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of any function F of x over any interval, just to verify that this is one and therefore that we have

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a probability density function.

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So once we've verified that we have a probability density function, we can answer probability questions

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about the continuous random variable, like what is the probability that the continuous random variable

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x takes on some value between, let's say, one half and three fourths?

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So looking at this curve here, that's this interval here, this is one half and this is three over

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four.

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So that interval, if we want to calculate that probability, what we're looking for is this area here,

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geometrically this physical area.

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If we can calculate this area, that will give us the probability that X takes on a value between one

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half and three fourths.

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And so here's what that would look like if we actually plugged into this formula here, we would say

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that the probability that X is between A and B, we said one half and three fourths.

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So one half less than or equal to X less than or equal to three over four is equal to the integral from

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one half to three fourths of the function F of x, which we were already told was for x cubed and then

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D x.

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And again, this is the same thing we just did here.

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We can use any computer or calculator to help us find this value if we don't know how to integrate or

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if we want to integrate.

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Remember we said that the integral of four x cubed was x to the fourth.

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So the integral here is x to the fourth, and then we just evaluate that integral over this interval,

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one half to three quarters, we plug in our upper limit of integration first three over four, so we

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get three over four.

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To the fourth power, and then we subtract whatever we get when we plug in our lower limit of integration,

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which is one half.

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So we get one half.

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To the fourth power.

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And then if we do this math by hand here, we'll get three to the four is 81, and four to the four

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is 256, minus one to the four is one, and two to the four is 16.

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If we find a common denominator by multiplying this second fraction by 16 over 16, then we can call

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this fraction here 16 over 256 instead of this.

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And the result then there is 81 -16 is 65.

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So 65 over 256, which when we convert to a decimal and round, is approximately 25.4%.

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And so what that tells us then is that the probability that our continuous random variable will have

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a value between one half and three fourths is just over 25%.

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Given that this is the probability density function that models our continuous random variable over

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this interval 0 to 1.

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So again, the biggest takeaway here is that we use probability mass functions to model discrete random

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variables, and we calculate probability of exact values.

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When we have a continuous random variable, we instead use a probability density function and we only

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calculate probabilities over intervals.

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We look for the probability that the continuous random variable falls between A and B.

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Instead of that, the random variable takes on a specific value like A or B, And in order to calculate

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that probability, we first check that we meet these two conditions to verify that we do in fact have

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a probability density function.

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And then we calculate this integral in order to find the probability over that interval.

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If we've taken calculus one and calculus two and we're comfortable with integration, we can, of course

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do that by hand, or if not, no problem at all.

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We can use any calculator, online calculator, computer algebra system, etc. to calculate this integral

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for us.

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The result that we get when we integrate the function f of x over the interval A to B should always

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be a value between zero and one.

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We might get a fraction, but even if we do, we can convert that to a decimal value.

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In this case, we got approximately 0.254.

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We can also convert that to a percentage to say there's an approximately 25.4% chance.

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Or we could just say the probability is about 0.254 and make the conclusion that that is the probability

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that the continuous random variable takes on a value in the interval that we defined.

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In this particular example, the interval one half to three fourths.