University of Alberta Guide/STAT/222/Combining Continuous Random Variables

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Convolution[edit | edit source]

Example[edit | edit source]

  • Start by converting the pdf's to indicator functions
      • Now is defined only when and is defined only when
  • Use the convolution formula above to write out the integral
  • Factor out any constants, in this case, a multiplier
  • Factor out the indicator function for into the integral bounds
      • Note that
  • Now that have isolated the indicator for z, we can combine the entire integral for that indicator
  • Finally, split the integral into the separate cases based on the remaining indicator function
      • When the integral has no bounds since so the upper bound would be less than which would be .
      • When the integral is bound between and since will be at least but less than
      • As you can see there is a pattern here, it goes as follows:
        • Given you will have