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

AI Thread Summary
The discussion revolves around the derivation of the probability density function (PDF) from the distribution function. The author initially presents the cumulative distribution function (CDF) and seeks clarification on proving the relationship between the CDF and the PDF. The key computation involves demonstrating that the PDF can be expressed as a function of the CDF and its derivatives. Ultimately, the author confirms understanding of the computations, indicating that the question has been resolved. The thread highlights the mathematical relationship between the CDF and PDF in probability theory.
WMDhamnekar
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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?
1655626767982.png
 
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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?
1655659126417.png
I tag this question as "SOLVED". I understood all the computations. Thanks.
 
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