Joint PDF of Chi-Square & Normal Distribution

Asked Oct 16 '14 at 21:27

Active Jul 07 '18 at 14:02

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Let the independent random variables X1 and X2 be N(0,1) and $\chi^2(r)$, respectively. Let $Y_1$ = $X_1/sqrt(X_2/r)$ and $Y_2$ = $X_2$

a) Find the joint pdf of $Y_1$ and $Y_2$.

b) Determine the marginal pdf of $Y_1$ and show that $Y_1$ has a t distribution.

Thoughts: So beyond confused I don't even know where to start. I assume there must be a trick because creating a joint pdf with a chi-square distribution and a normal distribution using the traditional method seems really strenuous.

edited Oct 17 '14 at 02:06

asked Oct 16 '14 at 21:27

Frank

2

you forgot defining $Y_2$ – drhab Oct 16 '14 at 21:33
what is your definition of the t distribution? mine is the result of the exercise. – mookid Oct 16 '14 at 22:44
@mookid you are correct that the t distribution should come from this exercise, but I have no idea how that happens – Frank Oct 17 '14 at 12:09
I think you can use copula. It will definitely solve your problem. – Akhil Singh Jul 07 '18 at 14:02

Joint PDF of Chi-Square & Normal Distribution

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