In contrast to the probability mass function, which assigns probabilities to specific values for discrete random variables is the probability density function, which is associated with continuous random variables. Just like a probability mass function has specific rules that it has to follow, a probability density function has specific rules that it has to follow. Specifically, to be a valid probability density function, a function must satisfy being larger than or equal to zero everywhere. And, then the total area under it must be one. And, then here is the basic rule of a probability density function. Areas under probability density functions correspond to probabilities for that random variable. So for example, if I say intelligence quotients are normally distributed with a mean of 100 and a standard deviation of 15, that implies that the population follows a specific bell shaped looking curve. And I am assuming that the probability that I draw sam-, that the probability, if I were to draw a sample, the probability that a person from that sample has an IQ between 100 and say, a hundred, 115, is this area right here. Note again, our probability density function is a statement about the population of intelligence quotients in this case. It's not a statement about the data itself. Again, we're going to use the data to evaluate that assumption and to evaluate statements about the population probability. But inherently whenever you say the word probability, you are talking about a population quantity. Interestingly note when we model continuous probabilities like this, or probabilities for continuous random variables like this, the probability that it takes any specific value is in fact zero because the area of a line is zero. This isn't a problem. This is just a quirk from modeling random variables as if they have infinite precision, but it doesn't gum up the works in any way at all. It turns out that the bell shaped curve is a rather difficult density to work with. At least until you learn how, which we're going to be doing in a separate lecture. So let's work with a much simpler density right now, specifically one that looks just like a right triangle. So f of x equals 2x for x between 0 and 1, and 0 otherwise. And let's give some context around it. Let's say it's the proportion of help calls that get addressed in a random day by a help line is given by this. So what does this mean? This means that the probability that between 20% and 60% of the calls get addressed that day is given by this area. Let's ask the question, is this a mathematically valid density? I plotted the probability density function right here. It looks like a right triangle. Notice the R code is given here, and is given in the lecture slides itself because the slides are created using the slidify format. So it's important to actually use git to pull the course repository. And you can actually look in the rmd, or R markdown file, and get all of the code for all of the examples for all of the lectures. So at any rate, we need to discuss whether or not this probability density function is a valid probability density function. Notice, it's always bigger than or equal to zero. And then secondly, let's calculate its area. Well, that's not too hard because it's a right triangle. One half the area of the base, right, which is one half, times the height, which is 2. So one half times 2 is 1. So the area is 1. So in fact, this is a valid probability density function. Let's go through an example of working with this density. What's the probability that 75% or fewer calls get addressed in a randomly sampled day from this population? Well, it turns out quite nice that this is just another right triangle that we need to figure out. So the height at this point is 1.5, because remember the function is just 2 times x, so at the point, point se, 0.75, the height is 1.5. And then of course, the value of the, the base is 0.75. But then we divide it by 2 because it's one-half the area of the base times the height. And that works out to be 56%, as shown right here. It turns out that this density is actually a special case of a known density, called the beta distribution. And I give you the R code here for directly getting this probability from the beta distribution. Of course, we don't need it, because it's just working with triangles. However, in more complicated settings, we're going to need these functions. I want to add that the word, the letter p, in front of a function asks for probabilities. So pbeta is going to ask for the probability from a beta density of being less than 0.75. Here the 2 and the 1 are the specific parameters that turn it into the exact triangle that we're using right here, and you see that you get the same number, 56%. Certain areas of the density are so useful we give them names. For example, the cumulative distribution function of a random variable x, returns the probability that a random variable is less than or equal to the value x. So it returns the probability capital X is less than or equal to little x. And this definition applies whether x is discrete or continuous. So remember in our beta distribution that we just looked at, the pbeta function in R always returns whatever the first argument is, the probability of being less than or equal to that. So in fact whenever you do pdensity name in R, it is actually just returning the cumulative distribution function. The survival function is sometimes interesting to work with instead of the distribution function, and it's just 1 minus the distribution function. So, instead of the probability of being less than, it's the probability of being greater than. So imagine if we wanted the cumulative distribution function of the density considered before. Say for example, we wanted to know what's the probability that 40% or fewer of the calls get answered in a given day? What's the probability that 50% or fewer of the calls get answered in a given day? And what's the probability that 60% or fewer of the calls get answered in a given day. Given the particular model that we have right now, which is this right triangle as a population, probability density function. Well, for any of those numbers, just looking at the picture, it's going to be very similar to 0.7, how we figured it out for 0.75. It's going to be a right triangle, so it's going to be one-half the area of the base times the height. That's always going to be one-half x, whatever x you're plugging in, times 2x, the height at that point, which works out to be x squared. So the function x squared takes the number that you're want to evaluate and gives the probability of that percent of calls or fewer getting answered on a randomly sampled day. Okay. So we see that here when plug in pbeta, which is our cumulative distribution function in R for these three values. The 2 and the 1 make it so that we're evaluating the specific beta density. And that works out to 16%, 25% and 36%. So, on 16, so the probability that 40% of the calls or fewer get answered on a given day is 16%. The probability that 50% of the calls or fewer get answered on a given day is 25%. And the probability that 60% of the calls or fewer on a given day get answered is 36%. If you wanted the survivor function, it's 1 minus the cumulative distribution function, so it's 1 minus x squared. In the future we're going to have, work with more complicated density functions, but it'll actually be easier, because we'll just be relying, for example, on the pnorm and pbeta function, like this, instead of directly figuring them out. You've already heard of sample quantiles. For example, if you score into the 95th percentile, which is the 0.95th quantile on an, on an exam, know that 95% of the students scored worse than you and 5% scored better. These are the so called sample quantiles. We're going to define their population analogs. So remember in the sample quantile, if you want to define the 95th percentile or the 0.95 quantile, what you would do is line the observations from least to greatest. And you would find the exam score or the point, such that 95% of the observation lie below it. The alphath quantile of the distribution function with distribution function F is the point x sub alpha such that F of x of alpha equals alpha. That's quite a tongue twister, so let's see if we can draw a picture. If we were to draw a density, F of x, the distribution function evaluated at x, is the area below the point x. Which is the probability that a random variable from this population is less than or equal to x. So you might think of it as a population of test scores. And it's a infinite population of students. And here, this would be the probability of getting a score for a randomly drawn student from this population of x or lower. The alphath quantile is we move this line around until we find the point x sub alpha, so that exactly alpha probability lies below it. That is exactly what we are doing with our data when we find an empirical quantile. We keep finding the data points such that, for example, 95% of the test scores lie below it. That would be the sample 95th percentile. Now we move the x point along until we find the point such the probability of lying below it is 95% in this population distribution. Again, the percentile is simply a quantile with alpha expressed as a percent instead of a proportion. And the median is the per, perhaps the most well known quantile, is the 50th percentile. We're going to use quantiles a lot, so it's worth reinforcing this point. So the 95th percentile of a population distribution is the point, such that the probability a random variable is drawn from that population is less than that value is 95%. And the probability that a random variable drawn from that population is more is 5%. Let's work through our previous example. What is the median of the distribution that we were working with before? Remember that the distribution looked like a right triangle. In the distribution function, for example if wanted to find the probability that x proportion of calls got answered on a given day or less, that F of x, that distribution function, worked out to simply be x squared. Where x has to be a value between 0 and 1 for it to make sense. In this case, we want to solve 0.5 equals F of x, which is equal to x squared. Resulting in the solution square root 0.5. This is 0.7. So what this means is that on about 50% of the days, 70% of the phone calls, or fewer get answered. And on about 50% of the days, about 70% of the phone calls or more get answered. We work with quantiles a lot, especially quantiles from the normal distribution. We almost never go through this process of working directly with the densities to calculate quantiles, because the distributions we work with are common and this has already been done for us. In R, there's an easy tri-, trick. Basically, q in front of the function name, function density name, gives the quantiles. So in this case, we know that this is a beta density. Well, we don't know. I'm telling you that this is a beta density. And so qbeta gives us the relevant quantile. Here we plug in 0.5. And remember R takes the argument of the quantile as a proportion. So, if you plug in 0.5, it will work. If you plug 50 for 50%, it will not work. Okay, and the 2 and the 1 are the parameters that we haven't really fully described. But that you are going to have to just take my word for it that those are the parameters that yield the specific data density that we're looking at. And when we plug this in, we get 0.7, 0.71, exactly like we got before. You might be wondering, at this point, I've heard of a median before, but it wasn't as complicated. I just ordered my observations from least to greatest and took the middle, or the average of the two middle observations if I had an even number of observations. There you had a sample quantity. It's an estimator. In this class, we're going to build up not just estimators but the targets of estimation or the estimand. So in this case, your sample median is estimating your population median. So for example, a sample median from our previous example is if we were to sample a couple of days, calculate the percentage of calls that got answered on that, of those days,. Line up those percentages, and take the middle one, that would be the median. We would think of that as a target as a, as an estimator of a population. Sort of a true median percentage of calls that get answered. And there's, of course, a lot of assumptions that we're going to need to make to connect that sample to the population. However we're going to be wide eyed about those assumptions and we're going to formally develop them. So at any rate, in this class, to every estimator there will be an estimand. The sample mean will estimate the population mean. The sample median will estimate the population median. The sample standard deviation will estimate the population standard deviation. And so on. This is the formal process of statistical inference, linking your sample to a population.