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# 8.5 Probability Applications Question #2 (Medium) 8.5 Probability Applications Question #2 (Medium)

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School
University of Toronto St. George
Department
Mathematics
Course
MAT136H1
Professor
all
Semester
Winter

Description
8.5 Challenging Integral Applications Probability Applications Question #2 (Medium): Forming a Probability Density Function and Its Mean Strategy Mean of a probability density function is given by ( ) ∫ Sample Question Let ( ) 1) For what value of is ( ) a probability density function? 2) For that value of , find( ) 3) Find the mean. Solution 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:
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