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If X and Y are independent random variables from the standard normal distribution, then show that t(1).Remark that the distribution with degree of freedom is also c...

Question

If X and Y are independent random variables from the standard normal distribution, then show that t(1).Remark that the distribution with degree of freedom is also classed Cauchy distribution, where it is well-known that the mean of the Cauchy distribution does NOT exist.

If X and Y are independent random variables from the standard normal distribution, then show that t(1). Remark that the distribution with degree of freedom is also classed Cauchy distribution, where it is well-known that the mean of the Cauchy distribution does NOT exist.



Answers

Let $X_{1}$ and $X_{2}$ be independent random variables, each having a standard normal distribution.
Show that the pdf of the ratio $Y=X_{1} / X_{2}$ is given by $f(y)=1 /\left[\pi\left(1+y^{2}\right)\right]$ for $-\infty<y<\infty$ (This is called the standard Cauchy distribution; its density curve is bell-shaped, but the tails are so heavy that $\mu$ does not exist.)

To prove this. We're going to utilize definition 73. and in definition 7 3, we have W one and W two are independent, chi squared distributed random variables with visa of one and visa to degrees of freedom respect actively. Then why would be equal to W one over V one divided by W two over V two has an F. Distribution. Mhm. With V one numerator and V two denominator degrees of freedom. Mhm. So therefore if we think about this setup as being W one over V one over W two over V two now when it comes time to talk about you U. is equal to one over why? Which means reciprocal of why? Yeah. So then you Would be W two over V two over W one Over V one. Mhm. Mhm. And that fits the same model as definition 7 3. So we could say that you has an f distribution With the two numerator and the one denominator degrees of freedom. So we have proved what we set out to prove. Yeah.

Okay, so we know that the function why has a point of discontinuity at X is equal to zero. Therefore, you can split the function domain into the two separate intervals Negative infinity to zero on day zero to positive infinity Thanks belongs in the first interval. Then why is going to decrease from zero zero to negative infinity? And if it actually belongs to a second interval done, why decreases from infinity? Tuju? It's a topic in general, use the general roof for transferring the random variable on therefore the pdf off the transformed random variable is a function Why that shit Thanks Times a job Why I'm sketch press Why and for the caution distribution we have is if each X you go to one over pi Why squared over one plus lightsquared Negative infinity That's why I just is young on the pdf will have the same form of the second triple A swell conservative The inverse one show of the wider y variable is h prime of y physical to negative one over escape. So therefore, pdf off the transform variable which is the Koshi distribution simply f wide. She was won over by Y squared That's why squaring so ice cream, which is simply equal. Hi. Over times one one y squared plus one. Okay, just cancel it this way, school.


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