Combination of Probability Density Functions

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SUMMARY

The discussion focuses on calculating the probability density function (PDF) f(Y) for the function Y = A + 2B, where A and B have known individual PDFs f(A) and f(B). The correct approach involves two steps: first, scaling the PDF of B by a factor of 2 to obtain C = 2B, and then applying convolution to combine the PDFs of A and C to derive f(Y). This method effectively utilizes the properties of probability density functions to achieve the desired result.

PREREQUISITES
  • Understanding of probability density functions (PDFs)
  • Knowledge of convolution of functions
  • Familiarity with scaling transformations in probability
  • Basic statistics concepts related to random variables
NEXT STEPS
  • Study the convolution theorem in probability theory
  • Learn about scaling transformations of probability density functions
  • Explore examples of combining multiple PDFs in statistical analysis
  • Investigate the properties of linear combinations of random variables
USEFUL FOR

Statisticians, data analysts, and anyone involved in probability theory who needs to combine probability density functions for random variables.

FrankDrebon
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Hi all,

I need to calculate the probability density function [itex]f\left( Y \right)[/itex] of a function [itex]Y[/itex] of two variables [itex]A[/itex] and [itex]B[/itex] with known individual probability density functions [itex]f\left( A \right)[/itex] and [itex]f\left( B \right)[/itex]. What is the correct way to combine the PDF's?

Specifically, I have a variable Y dependent on A and B by:

[itex]Y = A + 2B[/itex]

I know [itex]f\left( A \right)[/itex] and [itex]f\left( B \right)[/itex], how do I write [itex]f\left( Y \right)[/itex] in terms of [itex]f\left( A \right)[/itex] and [itex]f\left( B \right)[/itex]?

Stats isn't my strong point so apologies if this is trivial!

F
 
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The most direct way is to do it in two steps.
Step 1: C = 2B (scale density function).
Step 2: Y = A + C (convolution of density functions of A and C results in density function for Y).
 

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