Uses of the Probability Density Function versus the Cumulative Distribution Function

[cabrown085]

Hi All,

The probability density function seems to be a constant in most cases. As I found on Quora: A probability density function answers the question: "How common are samples at exactly this value?" I understand that the PDF in a continuous distribution would be equal to 0.

It seems generally that the cumulative distribution function is the more useful equation for solving equations. In total, the max value of the function has to equal to 100% or 1. It looks at the cumulative nature of a variable, that it takes on a value equal to or less than the specific value.

My first question is where would you use the PDF in any instance but finding the CDF?

Finally, how do you take the inverse cumulative distribution function of a cumulative distribution function. In the example: F(x)=x^2/25, the inverse seems to F^-1=5x^(1/2).

I'm not especially familiar with taking the inverse of a function, so any tips would be appreciated.

 

[ami44]

Hi cabrown085,

sorry, you got some stuff mixed up. A pdf for a discrete distribution is almost everywhere zero (and mostly not very useful). pdfs are used with continuous distributions. The advantage of cdfs is indeed, that they can be used for discrete and continuous distributions.

Anyway, pdfs and cdf are both very basic tools of probability theory and they belong together, since the cdf is the integral of the pdf.

I do not fully understand your first question, but the answer to your second question is, that you correctly inverted the function.
F(x) = x^2/25 is only a cdf though, if you restrict it to 0 <= x <= 25, but then the you calculate the inverse like this:
y = F(x) = x^2/25
=> 25 * y = x^2
=> 5 * y^(1/2) = x
=> F^-1(y) = 5 * y^(1/2)

 

[cabrown085]

Hi Ami44, thank you for your time. That helps to explain the inversion of function. Much clearer!

Does anyone have any examples of a PDF and CDF for a discrete distribution? I think a working example would clarify.

Separately, for continuous distributions: What use is a PDF? It seems like its the derivative of the CDF and the only use of the CDF is to take the integral to get the PDF.

Thank you!

 

[ami44]

- Discrete distribution:
There is no pdf.

- Use of pdf:
The pdf is indeed the derivative of the cdf. You need it for example to calculate the expected value of a continuous distribution, which is
E(x) = \int x * f(x) dx
with f(x) being the pdf.

 

[David Harper CFA FRM]

a discrete distribution has a pmf (probability mass function) instead of a prob density function (pdf) which is its continuous analog. An easy example of pmf/CDF is a fair six-sided die:

• the CDF is F(X) = X/6; i.e., the probability of rolling a three or less is 3/6 = 50%
• the pmf is the derivative: if F(X) = 1/6*x, then f(X) = F'(X) = 1/6; ie the pmf of a fair die is f(x) = 1/6
  • if f(x) = ax + b, then f′(x) = a, see elementary power rule which you do want to know: https://en.wikipedia.org/wiki/Differentiation_rules