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This is a very beautiful sample problem from ISI MStat PSB 2008 Problem 2 based on definite integral as the limit of the Riemann sum . Let's give it a try !!

For let

Find .

Integration

Gamma function

Definite integral as the limit of the Riemann sum

, this can be written you may see in details Definite integral as the limit of the Riemann sum .

Therefore , ----(1) , let

Substituting we get ,

**Statistical Insight**

Let i.e X is a standard normal random variable then,

called folded Normal has pdf . (Verify!)

So, from (1) we can say that

( As that a PDF of folded Normal distribution ) .

Find the same when .

This is a very beautiful sample problem from ISI MStat PSB 2008 Problem 2 based on definite integral as the limit of the Riemann sum . Let's give it a try !!

For let

Find .

Integration

Gamma function

Definite integral as the limit of the Riemann sum

, this can be written you may see in details Definite integral as the limit of the Riemann sum .

Therefore , ----(1) , let

Substituting we get ,

**Statistical Insight**

Let i.e X is a standard normal random variable then,

called folded Normal has pdf . (Verify!)

So, from (1) we can say that

( As that a PDF of folded Normal distribution ) .

Find the same when .

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