Can the pdf be determined numerically for a given data set?

[rohitashwa]

I have just been learning probability density function (pdf) and there is something I need to ask. I understand the idea that for any value v, the pdf (f(v)) gives the probability that a value picked from a data set is less than v. It seems ok to find mean,variance, skewness etc. when f(v) is known. However, how is the expression for f(v) arrived at.

If you have a data set given, can the pdf be found numerically?

Thank you.

 

[mathman]

If you have a data set given, can the pdf be found numerically?

Yes. Actually you can approximate the density function (pdf derivative) by setting a bin structure (x intervals) and sort the data into the intervals. Cumulative sums (normalized by dividing by the total number of items) will be an approximation to the pdf.

 

[rohitashwa]

Thanks. But, could you tell me why the cumulative sums approximate the pdf?

 

[mathman]

Thanks. But, could you tell me why the cumulative sums approximate the pdf?

The underlying assumption is that the given data was generated from the pdf. All statistical analysis is based on this assumption, i.e. given enough sample data, the probability distribution can be approximated by the sample distribution.

 

[rohitashwa]

Thank you. That was very helpful.

 

[chiro]

I have just been learning probability density function (pdf) and there is something I need to ask. I understand the idea that for any value v, the pdf (f(v)) gives the probability that a value picked from a data set is less than v. It seems ok to find mean,variance, skewness etc. when f(v) is known. However, how is the expression for f(v) arrived at.

If you have a data set given, can the pdf be found numerically?

Thank you.

Theres a couple of ways to get the pdf. In univariate distributions you could "fit" the results you get to a standard distribution (like say gaussian, lognormal, uniform etc) or you could use numerical analysis to come up with a distribution based on interpolation and other techniques.

If the data happened to fit a "stock" standard distribution then analyzing it would be a lot easier than analyzing a distribution based on numerical analysis since the assumptions of the stock standard distributions are easier understood.