Ask Questions, Get Answers

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

A continuous random variable $x$ has the p.d.f defined by $f(x) = \left\{ \begin{array}{1 1} ce^{-ax} & \quad 0 < x < \infty \\ 0 & \quad \text{else where} \end{array} \right.$ Find the value of $c$ if $a>0$

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:
The probability density function of a continuous random variable is
$f(x)=\left\{\begin{array}{1 1}ce^{-ax},&0 < x < \infty\\0,&else\;where\end{array}\right.$ where $a > 0$
By definition of a probability density function $\int_{-\infty}^\infty f(x) dx=1$
Step 2:
$\int_{-\infty}^\infty f(x) dx=1$
$\Rightarrow \int_0^{\infty} ce^{-ax}dx=1$
(Since $f(x)=0$ elsewhere)
$\therefore \large\frac{ce^{-ax}}{-a}\bigg]_0^\infty$$=1$
$\Rightarrow [0+\large\frac{c}{a}]$$=1$
$\Rightarrow c=a$
answered Sep 17, 2013 by sreemathi.v

Related questions

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