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Lecture

MAT136H1 Lecture Notes - Radian


Department
Mathematics
Course Code
MAT136H1
Professor
all

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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:
  
, then  
.
2)      relates to
 

 



  . Remember to
calculate the tangent inverse in the radian measure, because degree mode returns the answer
in degree mode, which is not compatible with . Therefore, for the given pdf,     

3) In order to find the mean,

 
 

 

  
 
  
 
. Now, evaluate the integral.
Since it is in in the form of chain rule:
 
 
  .
Likewise,
 
 
 
 . Plugging into limit:

 
  . Therefore, the mean is   .
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