8.5 Challenging Integral Applications
Question #2 (Medium): Forming a Probability Density Function and Its Mean
Mean of a probability density function is given by ( )
Let ( )
1) For what value of is ( ) a probability density function?
2) For that value of , find( )
3) Find the mean.
1) In order for the function to be a probability density function, when it is integrated over all the
possible values of x, it needs to add up to 1. Thus: ( )
∫ ∫ , because of the function’s symmetrical behavior.
Now disregarding the limit, evaluate the integral. Noting that it is in the form of ∫
( ) . Thus∫ √ [ (√ )] √ (√ ). Plugging back into
the limit: √ (√ ) √ (√ ) √ and this is equal to 1. Thus: