Finding densities of |K| and K² from standard normal K

[Bachelier]

Let K be a standard normal random variable. Find the densities of each of the following random variables:

X= |K|

Y = K²

I get:

f_X(x) = √(2/π) e^-x2/2

and

f_Y(y) = 1/√(2*π) 1/√y e^-y2/2

 

[Bachelier]

it's correct

 

[chiro]

For the Y random variable, it is definitely a chi-squared distribution with 1 degree of freedom.

 

[Bachelier]

For the Y random variable, it is definitely a chi-squared distribution with 1 degree of freedom.

We skipped the Chi-Squared dsn. I think I should read about it on Wikipedia. Where is it mostly used in?

 

[chiro]

We skipped the Chi-Squared dsn. I think I should read about it on Wikipedia. Where is it mostly used in?

Chi-squared distributions are used in a variety of cases.

One application is what is called goodness of fit. This is used to test how an observed set of frequencies are fitted to some expected set of frequencies.

Another application is for representing the sampling distribution of the ratio of the sample variance to the true variance. Given your degrees of freedom, you get a distribution that allows you to calculate a confidence interval for the ratio of sample variance to true variance, which effectively allows you to get an interval for your variance since you can calculate your sample variance from your data.

These uses are for classical frequentist statistics where these rely on asymptotic results.

 

[Bachelier]

Thanks Chiro