ISI MStat PSB 2009 Problem 3 | Gamma is not abNormal

This is a very simple but beautiful sample problem from ISI MStat PSB 2009 Problem 3. It is based on recognizing density function and then using CLT. Try it !

Problem- ISI MStat PSB 2009 Problem 3

Using and appropriate probability distribution or otherwise show that,

$\lim\limits_{x\to\infty}\int^n_0 \frac{exp(-x)x^{n-1}}{(n-1)!}\,dx =\frac{1}{2}$ .

Prerequisites

Gamma Distribution

Central Limit Theorem

Normal Distribution

Solution :

Here all we need is to recognize the structure of the integrand. Look, that here, the integrand is integrated over the non-negative real numbers. Now, event though here it is not mentioned explicitly that $x$ is a random variable, we can assume $x$ to be some value taken by a random variable $X$ . After all we can find randomness anywhere and everywhere !!

Now observe that the integrand has a structure which is very identical to the density function of gamma random variable with parameters $1$ ande $n$ . So, if we assume that $X$ is a $Gamma(1, n)$ , then our limiting integral transforms to,

$\lim\limits_{x\to\infty}P(X \le n)$ .

Now, we know that if $X \sim Gamma(1,n)$ , then its mean and variance both are $n$ .

So, as $n \uparrow \infty$ , $\frac{X-n}{\sqrt{n}} \to N(0,1)$ , by Central Limit Theorem.

Hence, $\lim\limits_{x\to\infty}P(X \le n)=\lim\limits_{x\to\infty}P(\frac{X-n}{\sqrt{n}} \le 0)=\lim\limits_{x\to\infty}\Phi (0)=\frac{1}{2}$ . [ here $\Phi(z)$ is the cdf of Normal at $z$ .]

Hence proved !!

Food For Thought

Can, you do the proof under the "Otherwise" condition !!

Give it a try !!

Similar Problems and Solutions

ISI MStat PSB 2008 Problem 10 — Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

Related

This is a very simple but beautiful sample problem from ISI MStat PSB 2009 Problem 3. It is based on recognizing density function and then using CLT. Try it !

Problem- ISI MStat PSB 2009 Problem 3

Using and appropriate probability distribution or otherwise show that,

$\lim\limits_{x\to\infty}\int^n_0 \frac{exp(-x)x^{n-1}}{(n-1)!}\,dx =\frac{1}{2}$ .

Prerequisites

Gamma Distribution

Central Limit Theorem

Normal Distribution

Solution :

Here all we need is to recognize the structure of the integrand. Look, that here, the integrand is integrated over the non-negative real numbers. Now, event though here it is not mentioned explicitly that $x$ is a random variable, we can assume $x$ to be some value taken by a random variable $X$ . After all we can find randomness anywhere and everywhere !!

Now observe that the integrand has a structure which is very identical to the density function of gamma random variable with parameters $1$ ande $n$ . So, if we assume that $X$ is a $Gamma(1, n)$ , then our limiting integral transforms to,

$\lim\limits_{x\to\infty}P(X \le n)$ .

Now, we know that if $X \sim Gamma(1,n)$ , then its mean and variance both are $n$ .

So, as $n \uparrow \infty$ , $\frac{X-n}{\sqrt{n}} \to N(0,1)$ , by Central Limit Theorem.

Hence, $\lim\limits_{x\to\infty}P(X \le n)=\lim\limits_{x\to\infty}P(\frac{X-n}{\sqrt{n}} \le 0)=\lim\limits_{x\to\infty}\Phi (0)=\frac{1}{2}$ . [ here $\Phi(z)$ is the cdf of Normal at $z$ .]

Hence proved !!

Food For Thought

Can, you do the proof under the "Otherwise" condition !!

Give it a try !!

Similar Problems and Solutions

ISI MStat PSB 2008 Problem 10 — Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

Related

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