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Given probability density function find its cumulative distribution function


New member
Dec 20, 2018
Hi :) Here's my problem along with what I've done.

Here is the problem:


That is the p.d.f. of a random variable X.

I have to find the cdf.

I don't know which I should do so I tried it two ways. First:

$\int_{-1}^{1} \ \frac{2}{\pi(1+x^{2})} dx = {{\frac{2}{\pi} arctan(x)]}^{1}}_{-1}=1$


$\int_{-1}^{x} \ \frac{2}{\pi(1+t^{2})} dt = {{\frac{2}{\pi} arctan(x)]}^{x}}_{-1}=\frac{2(arctan(x)+\frac{\pi}{4}}{\pi}$

Which one is the required CDF for X?


Well-known member
MHB Math Scholar
Jan 30, 2012
Hi, and welcome to the forum!

Which one is the required CDF for X?
The second one, except for the missing closing parenthesis. That is, the CDF is $\dfrac{2}{\pi}\arctan x+\dfrac12$.