Question

Ok.\[f(x)=\left(\begin{matrix}ke ^{-2x} \\ 0\end{matrix}\right)\] \[x \ge 0\]otherwise,

Find k

Answer 1

Is this a probability density function?

Answer 2

Yes

Answer 3

Is it piece wise functions?

Answer 4

Just probability density.

Answer 5

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

Answer 6

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

Answer 7

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
\[\large{k=2}\]

Answer 8

did you get it order?

Answer 9

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

Answer 10

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

Answer 11

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

Answer 12

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