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ISI MStat PSB 2008 Problem 2 | Definite integral as the limit of the Riemann sum

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 !!

Problem- ISI MStat PSB 2008 Problem 2


For k \geq 1, let a_{k}=\lim {n \rightarrow \infty} \frac{1}{n} \sum_{m=1}^{kn} \exp \left(-\frac{1}{2} \frac{m^{2}}{n^{2}}\right)

Find \lim_{k \rightarrow \infty} a_{k} .

Prerequisites


Integration

Gamma function

Definite integral as the limit of the Riemann sum

Solution :

a_{k}=\lim {n \rightarrow \infty} \frac{1}{n} \sum_{m=1}^{kn} \exp \left(-\frac{1}{2} \frac{m^{2}}{n^{2}}\right) = \int_{0}^{k} e^{\frac{-y^2}{2}} dy , this can be written you may see in details Definite integral as the limit of the Riemann sum .

Therefore , lim_{k \to \infty} a_{k}= \int_{0}^{ \infty} e^{\frac{-y^2}{2}} dy ----(1) , let \frac{y^2}{2}=z \Rightarrow dy= \frac{dz}{\sqrt{2z}}

Substituting we get , \int_{0}^{ \infty} z^{\frac{1}{2} -1} e^{z} \frac{1}{\sqrt{2}} dz =\frac{  \gamma(\frac{1}{2}) }{\sqrt{2}} = \sqrt{\frac{\pi}{2}}

Statistical Insight

Let X \sim N(0,1) i.e X is a standard normal random variable then,

Y=|X| called folded Normal has pdf f_{Y}(y)= \begin{cases} \frac{2}{\sqrt{2 \pi }} e^{\frac{-x^2}{2}} & , y>0 \\  0 &, otherwise \end{cases} . (Verify!)

So, from (1) we can say that \int_{0}^{ \infty} e^{\frac{-y^2}{2}} dy = \frac{\sqrt{2 \pi }}{2} \int_{0}^{ \infty}\frac{2}{\sqrt{2 \pi }} f_{Y}(y) dy

=\frac{\sqrt{2 \pi }}{2} \times 1 ( As that a PDF of folded Normal distribution ) .


Food For Thought

Find the same when a_{k}=\lim {n \rightarrow \infty} \frac{1}{n} \sum_{m=1}^{kn}  {(\frac{m}{n})}^{5} \exp \left(-\frac{1}{2} \frac{m}{n}\right).


Similar Problems and Solutions



ISI MStat PSB 2008 Problem 10
Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

Subscribe to Cheenta at Youtube


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 !!

Problem- ISI MStat PSB 2008 Problem 2


For k \geq 1, let a_{k}=\lim {n \rightarrow \infty} \frac{1}{n} \sum_{m=1}^{kn} \exp \left(-\frac{1}{2} \frac{m^{2}}{n^{2}}\right)

Find \lim_{k \rightarrow \infty} a_{k} .

Prerequisites


Integration

Gamma function

Definite integral as the limit of the Riemann sum

Solution :

a_{k}=\lim {n \rightarrow \infty} \frac{1}{n} \sum_{m=1}^{kn} \exp \left(-\frac{1}{2} \frac{m^{2}}{n^{2}}\right) = \int_{0}^{k} e^{\frac{-y^2}{2}} dy , this can be written you may see in details Definite integral as the limit of the Riemann sum .

Therefore , lim_{k \to \infty} a_{k}= \int_{0}^{ \infty} e^{\frac{-y^2}{2}} dy ----(1) , let \frac{y^2}{2}=z \Rightarrow dy= \frac{dz}{\sqrt{2z}}

Substituting we get , \int_{0}^{ \infty} z^{\frac{1}{2} -1} e^{z} \frac{1}{\sqrt{2}} dz =\frac{  \gamma(\frac{1}{2}) }{\sqrt{2}} = \sqrt{\frac{\pi}{2}}

Statistical Insight

Let X \sim N(0,1) i.e X is a standard normal random variable then,

Y=|X| called folded Normal has pdf f_{Y}(y)= \begin{cases} \frac{2}{\sqrt{2 \pi }} e^{\frac{-x^2}{2}} & , y>0 \\  0 &, otherwise \end{cases} . (Verify!)

So, from (1) we can say that \int_{0}^{ \infty} e^{\frac{-y^2}{2}} dy = \frac{\sqrt{2 \pi }}{2} \int_{0}^{ \infty}\frac{2}{\sqrt{2 \pi }} f_{Y}(y) dy

=\frac{\sqrt{2 \pi }}{2} \times 1 ( As that a PDF of folded Normal distribution ) .


Food For Thought

Find the same when a_{k}=\lim {n \rightarrow \infty} \frac{1}{n} \sum_{m=1}^{kn}  {(\frac{m}{n})}^{5} \exp \left(-\frac{1}{2} \frac{m}{n}\right).


Similar Problems and Solutions



ISI MStat PSB 2008 Problem 10
Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

Subscribe to Cheenta at Youtube


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