Density, distribution and derivative relationship (stats)

In summary, the probability density function and the distribution function are two different things and the result that is given in the course book is an expression that reflects the definition of the probability density in a formula.
  • #1
Andrea94
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8
I am currently enrolled in a statistics course, and the following is stated in my course book with no attempt at an explanation:

Suppose that f is the probability density function for the random variable (X,Y), and that F is the distribution function. Then,

[tex]f_{X,Y}(x,y)=\frac{\partial^{2} F_{X,Y}(x,y)}{\partial x \partial x}[/tex]

I have repeatedly tried to find an explanation for why this is so, but all I keep finding is documents from various university statistics courses that just flat out give this result with no attempt at an explanation.

My question is, can you show or explain why the above result is true?
 
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  • #2
Hi Andrea,

You call it a result, but perhaps if you study the accompanying text you might find that it's in fact just an expression that reflects the definition of probability density in a formula. (or the other way around: the definition of the cumulative distribution function)

Can you find the equivalent in the single-stochastic variable (univariate, e.g. ##x##) case ? Do you have difficulty with that one ?
 
  • #3
Hi and thanks for the reply,

In the single-variable case, I think of it in terms of the fundamental theorem of calculus as follows:

[tex] F(x) = \int_{-\infty}^{x} f(t)dt
\\\\
F'(x) = f(x)[/tex]

Even though the definition for the distribution function looks like an obvious extension,

[tex]F_{X,Y}(x,y)= \int_{-\infty}^{x}\int_{-\infty}^{y} f(u,v)dudv[/tex]

I still don't see it as an immediately obvious result from the definition. Is there anything you could provide in this regard that might help?
 
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  • #4
Is this correct?

[tex] \frac{\partial}{\partial y}\frac{\partial}{\partial x}\int_{-\infty}^{x}\int_{-\infty}^{y} f(u,v)dudv = \frac{\partial}{\partial y}\int_{-\infty}^{y} f(x,v)dv = f(x,y) [/tex]

If so, I think I got it
 
  • #5
Andrea94 said:
Is this correct?

[tex] \frac{\partial}{\partial y}\frac{\partial}{\partial x}\int_{-\infty}^{x}\int_{-\infty}^{y} f(u,v)dudv = \frac{\partial}{\partial y}\int_{-\infty}^{y} f(x,v)dv = f(x,y) [/tex]

If so, I think I got it
This is easily shown to be wrong: if ##f## only depends on one of the two arguments, you get back zero, so not ##f## itself...
Basically a probability for an interval is an integral of probability density over that interval. If the interval size is infinitesimally small then that becomes the probability density times the interval size. (or area if it's in two variables). That's really all there is to it.

https://en.wikipedia.org/wiki/Probability_density_function in particular:
https://en.wikipedia.org/wiki/Proba...bsolutely_continuous_univariate_distributions and
https://en.wikipedia.org/wiki/Cumulative_distribution_function

but I guess you looked there already ?
 
  • #6
BvU said:
This is easily shown to be wrong: if ##f## only depends on one of the two arguments, you get back zero, so not ##f## itself...
Basically a probability for an interval is an integral of probability density over that interval. If the interval size is infinitesimally small then that becomes the probability density times the interval size. (or area if it's in two variables). That's really all there is to it.

https://en.wikipedia.org/wiki/Probability_density_function in particular:
https://en.wikipedia.org/wiki/Proba...bsolutely_continuous_univariate_distributions and
https://en.wikipedia.org/wiki/Cumulative_distribution_function

but I guess you looked there already ?

Yes I have checked those links before.

I understand how the probability (in the case of a continuous random variable) is an integral of probability density over an interval (or area if it's in 2 variables), I just cannot make the connection between the density function and the distribution function in terms of the mixed partial derivative. I suspect this has more to do with my calculus knowledge than my probability and statistics knowledge.
 
  • #7
BvU said:
This is easily shown to be wrong: if ##f## only depends on one of the two arguments, you get back zero, so not ##f## itself...

If a joint density ##f(x,y)## depends only on ##x## then what constant value can a number like ##f(7,y)## have on an interval ##[-\infty, y] ## besides zero ?
 
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  • #8
BvU said:
This is easily shown to be wrong: if ##f## only depends on one of the two arguments, you get back zero, so not ##f## itself...
I think what @Stephen Tashi is saying in post #7 is: there is no density on the plane that depends only one argument/variable in rectangular coordinates. In polar coordinates you can sometimes get away with one argument.
 
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  • #9
Andrea94 said:
Is this correct?

[tex] \frac{\partial}{\partial y}\frac{\partial}{\partial x}\int_{-\infty}^{x}\int_{-\infty}^{y} f(u,v)dudv = \frac{\partial}{\partial y}\int_{-\infty}^{y} f(x,v)dv = f(x,y) [/tex]

If so, I think I got it
If f is continuous it is correct and should be in any calculus text treating several variables. E.g. Courant Vol II, page 239.
There is no one natural way to define a distribution function for f on the plane as there almost is on the line.
 
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  • #10
Zafa Pi said:
If f is continuous it is correct and should be in any calculus text treating several variables. E.g. Courant Vol II, page 239.
There is no one natural way to define a distribution function for f on the plane as there almost is on the line.

Fantastic, thank you so much for the reference! I found the book on Google and looked at page 239. Interestingly, I never saw this anywhere in my multivariable calculus course.
 
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1. What is density in statistics?

Density in statistics refers to the measure of how many values in a data set fall within a specific range. It is typically represented as a graph or a histogram, with the height of the bars representing the frequency of values within that range.

2. How is distribution related to density in statistics?

Distribution in statistics refers to the spread or pattern of the data. Density and distribution are closely related, as the shape of the distribution curve can be seen in the density graph. A symmetric distribution, for example, will have a bell-shaped density curve.

3. What is the derivative relationship in statistics?

The derivative relationship in statistics refers to the relationship between the original data and its derivative, which is a new data set created by taking the rate of change of the original data. This relationship can be used to understand the behavior and trends of the original data over time.

4. How is the derivative related to density and distribution?

The derivative of a density curve is called the probability density function (PDF). It represents the probability of a random variable falling within a specific range of values. The shape and characteristics of the PDF can help determine the distribution of the data.

5. What are some practical applications of understanding density, distribution, and derivative relationship in statistics?

Understanding these concepts can help in various fields such as finance, economics, and science. For example, in finance, understanding the density and distribution of stock prices can help in making investment decisions. In science, the derivative relationship can be used to analyze changes in data over time, such as tracking the growth of a population or the rate of reaction in a chemical experiment.

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