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

Convolution Example 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