## anonymous 4 years ago Ok.$f(x)=\left(\begin{matrix}ke ^{-2x} \\ 0\end{matrix}\right)$ $x \ge 0$otherwise, Find k

1. ash2326

Is this a probability density function?

2. anonymous

Yes

3. anonymous

Is it piece wise functions?

4. anonymous

Just probability density.

5. ash2326

integrate it between 0 to infinity, it's value is 1, you'll be able to find k

6. anonymous

I'm not sure how to integrate it from infinity... that's the problem I'm having.

7. ash2326

we know for a probability density function $\int_{-\infty}^{\infty} f(x)=1$ now f(x)=ke^-2x for x$$\ge$$0 $\int_{0}^{\infty} f(x)=1$ so our integral becomes now f(x)=ke^-2x now $\int_{0}^{\infty} ke^{-2x}=1$ integral of e^x=e^x so $(\frac{-1}{2}*k*e^{-2x})=1$ now insert the limits e^(-2x) as x--->$$\infty$$ is 0 so we get $(0-(-1/2*k*e^0)=1$ or $k/2=1$ or $\large{k=2}$

8. ash2326

did you get it order?

9. anonymous

Yes... Just one question, How did (0-(-1/2*k*e^0)=1 become 2?

10. ash2326

see we have, (0-(-1/2*k*e^0)=1 this is an equation now e^0=1 so (0+1/2k*1)=1 so k/2=1 or k=2

11. anonymous

Ah OK. I just figured that out :) Thanks!

12. anonymous

A computer ink cartridge has a life of X hours. The variable X is modelled by the probability density function $f(x)=\left(\begin{matrix}kx^{-2} \\ 0\end{matrix}\right)$$x \ge 0 -otherwise -$ (a) Find K