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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 !!

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