# Derivation of P.D.F. from distribution function

• A
• WMDhamnekar
In summary, the author computed ##f_{X_k}(x)## using the formula ##f_{X_k} (x) =\frac{n!}{(k-1)!(n-k)!}f(x) F^{k-1}(x)(1-F(x))^{n-k}## and the member asking the question was able to understand the computations after receiving an explanation.
WMDhamnekar
MHB
TL;DR Summary
If ## F_{X_k}(x) = p(X \leq x) = \displaystyle\sum_{j=k}^n \binom{n}{j} F^j(x)(1-F(x))^{n-j}, -\infty
< x < \infty ## then how to prove ##f_{X_k} (x) =\frac{n!}{(k-1)!(n-k)!}f(x) F^{k-1}(x)(1-F(x))^{n-k}##
Author computed ##f_{X_k}(x)## as follows but I don't understand it. Would any member explain me the following computations?

WMDhamnekar said:
Summary: If ## F_{X_k}(x) = p(X \leq x) = \displaystyle\sum_{j=k}^n \binom{n}{j} F^j(x)(1-F(x))^{n-j}, -\infty
< x < \infty ## then how to prove ##f_{X_k} (x) =\frac{n!}{(k-1)!(n-k)!}f(x) F^{k-1}(x)(1-F(x))^{n-k}##

Author computed ##f_{X_k}(x)## as follows but I don't understand it. Would any member explain me the following computations?
I tag this question as "SOLVED". I understood all the computations. Thanks.

Last edited:

## 1. What is a P.D.F. and a distribution function?

A P.D.F. (Probability Density Function) is a mathematical function that describes the probability of a continuous random variable taking on a certain value. A distribution function, also known as a cumulative distribution function, shows the probability that a random variable is less than or equal to a given value.

## 2. How is a P.D.F. derived from a distribution function?

To derive a P.D.F. from a distribution function, we take the derivative of the distribution function. This results in the probability density function, which can then be used to calculate the probability of a random variable falling within a certain range of values.

## 3. What is the importance of deriving a P.D.F. from a distribution function?

Deriving a P.D.F. from a distribution function allows us to understand the probability distribution of a continuous random variable. This information is crucial in many fields, such as statistics, economics, and engineering, as it helps us make predictions and make informed decisions based on data.

## 4. Are there any limitations to deriving a P.D.F. from a distribution function?

One limitation is that the P.D.F. and distribution function only apply to continuous random variables. They cannot be used for discrete random variables. Additionally, the P.D.F. may not exist for all types of distributions, such as those with infinite variance.

## 5. Can a P.D.F. be used to calculate the probability of a specific value?

No, a P.D.F. only gives the probability of a random variable falling within a range of values. To calculate the probability of a specific value, we need to use the distribution function or integrate the P.D.F. over that specific value.

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