MAT136H1 Lecture Notes - Radian
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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
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
. Plugging back into
and this is equal to 1. Thus:
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:
. Plugging into limit:
. Therefore, the mean is .
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