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**preview**shows half of the first page. to view the full**1 pages of the document.**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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