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Let $X$ have the pdf $f(x)=1 /\left[\pi\left(1+x^{2}\right)\right]$ for $-\infty< x<\infty($ a central Cauchy distribution), and show that $Y=1 / X$ has the same distribution. [Hint: Consider $P(1 Y | \leq y),$ the cdf of $|Y|,$ then obtain its pdf and show it is identical to the pdf of $|X| . ]$

Intro Stats / AP Statistics

Chapter 3

Continuous Random Variables and Probability Distributions

Section 9

Supplementary Exercises

Continuous Random Variables

Piedmont College

University of St. Thomas

Idaho State University

Boston College

Lectures

02:44

In Exercise 127 suppose in…

02:40

Let $X$ have the pdf $f_{X…

07:23

Normal distribution An imp…

02:07

Let $X_{1}, \ldots, X_{n}$…

02:57

Suppose $f$ is a probabili…

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Let $X$ be a nonnegative c…

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Show that the density func…

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Verify that the functions …

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Let $X \sim$ Unif[0, 1]. F…

04:00

If $X$ is distributed as $…

02:22

Show that the given functi…

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02:01

02:36

06:02

Find the expected value, t…

07:45

Show that if the random va…

01:56

01:19

If the pdf of a measuremen…

04:08

Use Fubini's Theorem …

02:37

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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