How can I solve this?

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- Nov 19th 2006, 07:13 PMjamster114calc2 problem
How can I solve this?

http://i15.tinypic.com/2ymzyop.jpg - Nov 19th 2006, 07:28 PMThePerfectHacker
First,

$\displaystyle p(x)\geq 0$.

Next we need to find,

$\displaystyle \int_{-\infty}^{\infty} p(x)dx$

Subdivide the interval as,

$\displaystyle \int_{-\infty}^0p(x)dx+\int_0^{\infty}p(x)dx$

Since the integral is continous accept at possibly one point we can chose to ignore that fact.

Now,

$\displaystyle \int_{-\infty}^0 p(x)dx=0$ because $\displaystyle p(x)=0$ for $\displaystyle x<0$.

And,

$\displaystyle \int_0^{\infty} p(x)dx=\int_0^{\infty} \lambda e^{-\lambda x}dx$

The anti-derivative is,

$\displaystyle e^{-\lambda x} | ^{\infty}_0 =1$

So it is indeed a probability density function.