Math 230 W8 DQS

Question 1 . answer

For any function to be a probability density function .It must satisfy two conditions

It must be non-negative between its boundaries

The integral of the function between its boundaries must be equal to 1

So , Any function defined between two points can be converted to a probability distribution function by

Making the function non negative in its domain

This can be done by

a ) Adding the greatest negative value of the function at all points

or b ) By taking modulus at each point

Example

sin

(x ) sin (x 1 ?sin (x

Making the function 's integral in its domain to be equal to 1

This can be done by

dividing the function obtained from step 1 by its After doing this two operations any function defined between 2 points a and b can be made a probability density function

The function would first simply to a non negative function as required in condition 1 and the second step would make its This procedure is further illustrated in next answer

Question 2 . answer

Between the points 0 to 2 , the function f (x - x2 2x looks like As we see that the function is always non negative in its domain ,it obviates step 1 (- x2 2x .dx between limits 0 to 2 [- x3 /3 x2 ] between limits 0 to 2

The Dividing the function by 4 /3 gives g (x (- x2 2x 3 /4

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h ?A ?A

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V 'h] 'hgd

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n negative and its Hence g (x (- x2 2x 3 /4 is the required function PAGE

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