Parameter Estimation - The PDF, CDF and Quantile Function — Count Bayesie
What is the relationship between pdf/cdf and probability (in term of Integral I may be wrong, and that is no problem because I am not teaching youth, neither. Mathematical Relationship: pdf and cdf In general, most problems in reliability engineering deal with quantitative measures, such as the time-to-failure of a. The Reliability Function and related statistical background, this issue's Reliability Basic. The mathematical relationship between the pdf and cdf is given by.
Trying to figure out this missing parameter is referred to as Parameter Estimation. For example suppose I want to know what the probability is that a visitor to this blog will subscribe to the email list do it for science! In marketing terms getting a user to perform a desired event is referred to as the conversion event or simply a conversion and the probability that a user will convert is the conversion rate. The Probability Density Function In this case, let's say for first 40, visitors I get subscribers.
The Beta distribution is very useful for estimating unknown probabilities What does this PDF represent? The area under the curve less than 0.
Probability density functions
We can use our PDF to compare two extremes. The probability that our conversion rate is actually much lower than we have seen is: Introducing the Cumulative Distribution Function!
I know you're thinking: Despite its ubiquity in probability and statistics, the PDF is actually a pretty mediocre way to look at data.
The PDF is only really useful for quickly ascertaining where the peak of a distribution is and getting a rough sense of the width and shape which give a visual understanding of Variance and Skewness. The CDF for our problem looks like this: The Cumulative Distribution Function can be used to quickly estimate precentiles The CDF is so simple it might seem useless, so let's go over a few visual examples of how we can use this amazing tool.
Probability density functions (video) | Khan Academy
First we can easily see the median which can even be challening to compute analytically by visually drawing a line from the point where the cumulative probability is 0. Looking where this intersects the x-axis give us our median! The CDF allows for quick and accurate estimates of the median and other quantiles! If we just need an approximate value we also can save all that integral work we did before for assessing the probability of ranges of values.
To estimage the probability that the conversion rate is between 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. And if that doesn't make sense to you, this is essentially just saying what is the probability that Y is greater than 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. 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. 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.
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.
Then you would go here and if this was 0. 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?
Joint Cumulative Distributive Function| Marginal PMF | CDF
Then your probability won't make any sense. But hopefully if you take this sum it comes to some number. 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.FRM: Terms about distributions: PDF, PMF and CDF
Let me draw one. The sum of all of the probabilities have to be equal to 1.
Connecting the CDF and the PDF
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. Or they don't have to be 0. They have to add to 1.