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&gt;\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.