Is the Derivation of the Relationship Between CDF and PDF Correct?

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SUMMARY

The discussion centers on the relationship between the Cumulative Distribution Function (CDF) and the Probability Density Function (PDF) in probability theory. It establishes that CDF(Z) = Prob(Z < z) and CDF(Y) = Prob(Y < y) where y = f(z). The derivation shows that PDF(Z) can be expressed as the derivative of CDF(Z) with respect to z, leading to the conclusion that PDF(z)dz = PDF(f(z)) df(z). This implies that CDF(z) can be represented as CDF(f(z)) plus a constant, challenging the reader to find a function f(z) that demonstrates a non-constant difference between the two CDFs.

PREREQUISITES
  • Understanding of Cumulative Distribution Functions (CDF)
  • Familiarity with Probability Density Functions (PDF)
  • Knowledge of random variables and their properties
  • Basic calculus, specifically differentiation and integration
NEXT STEPS
  • Research the properties of CDF and PDF in probability theory
  • Explore the implications of transformations of random variables
  • Study examples of functions f(z) that affect CDF calculations
  • Learn about the applications of CDF and PDF in statistical analysis
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Statisticians, data scientists, and students of probability theory seeking to deepen their understanding of the relationship between CDF and PDF, as well as those interested in the implications of random variable transformations.

RRraskolnikov
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No, this is not for a homework. Please don't delete the thread.

CDF(Z) = Prob(Z &lt; z)
CDF(Y) = Prob(Y &lt; y) where y = f(z)
PDF(Z) = \frac{d(CDF(Z))}{dz}
PDF(Y) = \frac{d(CDF(Y))}{df(z)}

Now, it is known from various internet sources and wikipedia that:
E(f(z))= \int_{-\infty}^{\infty}{f(z) PDF(z) }dz - (1)

Also, since z is a random variable, f(z) is also a random variable, hence:
E(f(z))= \int_{-\infty}^{\infty}{f(z) PDF(f(z)) }df(z) - (2)

From (1) and (2),
PDF(z)dz = PDF(f(z)) df(z)

From this doesn't it follow that:

CDF(z) = CDF(f(z)) + const.
 
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OK - check by seeing if you can find some function f(z) where the CDF of f(z) differs from CDF of z by something other than a constant.
 

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