Categories

# 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 !

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 !!

## Subscribe to Cheenta at Youtube

This site uses Akismet to reduce spam. Learn how your comment data is processed.
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://www.cheenta.com/isi-mstat-psb-2009-problem-3-gamma-is-not-abnormal/", "fetch_time": 1606342895000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2020-50/segments/1606141184870.26/warc/CC-MAIN-20201125213038-20201126003038-00057.warc.gz", "warc_record_offset": 620271974, "warc_record_length": 48097, "token_count": 550, "char_count": 1917, "score": 4.0625, "int_score": 4, "crawl": "CC-MAIN-2020-50", "snapshot_type": "longest", "language": "en", "language_score": 0.7801178097724915, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00012-of-00064.parquet"}