Proving of Y=g(X) as a continuous random variable

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SUMMARY

If X is a continuous random variable and g is a continuous function defined on X (Ω), then Y = g(X) is indeed a continuous random variable. This conclusion is supported by the fact that the cumulative distribution function (CDF) of Y, denoted as F_Y(y), can be expressed in terms of the CDF of X, F_X, using the relationship F_Y(y) = P{g(X) < y} = F_X(g^{-1}(y)). The continuity of both g and F_X ensures that F_Y is continuous as well.

PREREQUISITES
  • Understanding of continuous random variables
  • Knowledge of cumulative distribution functions (CDF)
  • Familiarity with continuous functions and their properties
  • Basic concepts of probability theory
NEXT STEPS
  • Study the properties of continuous random variables in depth
  • Learn about the implications of transformations of random variables
  • Explore the concept of inverse functions in the context of probability
  • Investigate the relationship between CDFs and probability density functions (PDFs)
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Students and professionals in statistics, mathematicians, and anyone studying probability theory who seeks to understand the behavior of transformed continuous random variables.

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If X is a continuous random variable and g is a continuous function
defined on X (Ω), then Y = g(X ) is a continuous random variable.
Prove or disprove it.
 
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bl00d said:
If X is a continuous random variable and g is a continuous function
defined on X (Ω), then Y = g(X ) is a continuous random variable.
Prove or disprove it.

If X is a continuous r.v. then $F_{X} (x) = P \{ X < x\}$ is continous. Now if $y=g(x)$ is continuous then $x=g^{-1} (y)$ is also continuous and the same is for... $$F_{Y} (y) = P \{g(X) < y\} = P \{X < g^{-1} (Y)\} = F_{X} (g^{-1} (y))$$Kind regards $\chi$ $\sigma$
 

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