Ask Questions, Get Answers

Want to ask us a question? Click here
Browse Questions
0 votes

Verify that the following are probability density functions.$f(x)=\large\frac{1}{\pi}\frac{1}{(1+x^{2})'}$$-\infty$$<$$x$$<\infty $

Can you answer this question?

1 Answer

0 votes
  • The probability density function (continuous probability function $f(x)$ satisfies the following properties :
  • (i) $P(a\leq x\leq b)=\int_a^b f(x) dx$
  • (ii) $f(x)$ is non-negative for all real $x$
  • (iii) $\int_{-\infty}^\infty f(x) dx=1$
  • Also $P(x=a)=0$
  • $P(a\leq x\leq b)=P(a\leq x\leq b)$=P(a < x < b)
Step 1:
We verify whether $f(x)$ satisfies the properties $f(x)\geq 0,-\infty < x <\infty$ and $\int_{-\infty}^\infty f(x) dx=1$
Step 2:
$f(x)=\large\frac{1}{\pi}.\frac{1}{1+x^2}$$ >0$ for all $x$.
Since $1+x^2 >0$ for all $x$
Step 3:
$\int_{-\infty}^\infty f(x) dx=\int_{\infty}^{\infty}\large\frac{1}{\pi}\frac{1}{1+x^2}$$dx$
Step 4:
From the above,$f(x)$ is a probability density function.
answered Sep 16, 2013 by sreemathi.v

Related questions

Ask Question
student study plans
JEE MAIN, CBSE, AIPMT Mobile and Tablet App
The ultimate mobile app to help you crack your examinations
Get the Android App