Calculating the Central Density Function for a Continuous Random Variable

[MaxManus]

Homework Statement

-infinity<x<infinity
x> theta
f(x) = $\frac{\lambda}{2}e^{-\lambda (x-\theta)}$

F(x) = $\int_{-\infty}^x f(x) dx$

Homework Equations

The Attempt at a Solution

Homework Statement

$\int \frac{\lambda}{2}e^{-\lambda (x-\theta)} dx$
= $-\frac{1}{2}e^{-\lambda(x-\theta)}$

Insert the limits:
$-\frac{1}{2}e^{-\lambda(x-\theta)} + \frac{1}{2}e^{-\lambda(-\infty-\theta)}$

= infinity.

The last part should not be infinity so can anyone see where I go wrong?

 

[LCKurtz]

You have defined f(x) for $x>\theta$. Is it zero elsewhere? Should your lower limit be $\theta$?

 

[MaxManus]

Yes it should! Thanks

Insert the limits:
$-\frac{1}{2}e^{-\lambda(x-\theta)} + \frac{1}{2}e^{-\lambda(\theta-\theta)}$
=
$1/2 -\frac{1}{2}e^{-\lambda(x-\theta)}$

 

[HallsofIvy]

You mean $e^{-\lambda(\theta- \theta)}$, not $e^{-\lambda(-\theta- \theta)}$

 

[MaxManus]

Thanks, corrected it now.