Probability- Transformation of variable

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SUMMARY

The discussion centers on the transformation of variables in probability, specifically the cumulative distribution function. The key equation derived is P(X ≤ wY) = lim ∑P(X ≤ wY | v ≤ Y ≤ v + dv)P(v ≤ Y ≤ v + dv), leading to the integral form ∫P(X ≤ wv)dP(Y ≤ v). This transformation illustrates the relationship between joint probabilities and cumulative distributions, clarifying the reasoning behind the integration step.

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  • Understanding of cumulative distribution functions (CDFs)
  • Familiarity with probability theory and integration techniques
  • Knowledge of conditional probability concepts
  • Basic skills in mathematical notation and limits
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  • Study the properties of cumulative distribution functions (CDFs)
  • Learn about conditional probability and its applications in transformations
  • Explore integration techniques in probability, focusing on Lebesgue integration
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jack1234
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Following is the
question
http://tinyurl.com/62uxof
solution
http://tinyurl.com/6m4lcc

The distribution in question means cumulative distribution.

What I do not understand in the solution is the step from P{X_1<=wX_2} to the integration formula that followed immediately. May I know the reasoning for it?
 
Last edited:
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jack1234 said:
What I do not understand in the solution is the step from P{X_1<=wX_2} to the integration formula that followed immediately. May I know the reasoning for it?

Hi jack1234! :smile:

P(X ≤ wY) = lim ∑P(X ≤ wY | v ≤ Y ≤ v + dv)P(v ≤ Y ≤ v + dv)

= ∫P(X ≤ wv)dP(Y ≤ v) :wink:
 

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