MATH105 April 2011
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Is there any value of for which the function below is a probability density function?
If yes, find all such values of If there is no such , explain why.
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A probability density function integrates to .
Use partial fractions.
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Notice from the definition of that if then is always 0, which cannot be a probability density function.
The same is true if . Therefore we only need to consider positive values of as candidates.
We require that
To deal with the integral, we use partial fractions. We write
Now we multiply the equation through by to find .
To find and we can choose ``convenient" values for .
Setting yields so .
Setting yields so .
Note this is independent of and it will never be . Thus no such exist!
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