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Given probability density function find its cumulative distribution function

sofanglom

New member
Dec 20, 2018
2
Hi :) Here's my problem along with what I've done.

Here is the problem:

save.PNG

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:

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

Second:

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

Which one is the required CDF for X?
 

Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,490
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 $\dfrac{2}{\pi}\arctan x+\dfrac12$.