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 !
Using and appropriate probability distribution or otherwise show that,
.
Gamma Distribution
Central Limit Theorem
Normal Distribution
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 is a random variable, we can assume
to be some value taken by a random variable
. 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 ande
. So, if we assume that
is a
, then our limiting integral transforms to,
.
Now, we know that if , then its mean and variance both are
.
So, as ,
, by Central Limit Theorem.
Hence, . [ here
is the cdf of Normal at
.]
Hence proved !!
Can, you do the proof under the "Otherwise" condition !!
Give it a try !!
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 !
Using and appropriate probability distribution or otherwise show that,
.
Gamma Distribution
Central Limit Theorem
Normal Distribution
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 is a random variable, we can assume
to be some value taken by a random variable
. 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 ande
. So, if we assume that
is a
, then our limiting integral transforms to,
.
Now, we know that if , then its mean and variance both are
.
So, as ,
, by Central Limit Theorem.
Hence, . [ here
is the cdf of Normal at
.]
Hence proved !!
Can, you do the proof under the "Otherwise" condition !!
Give it a try !!