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#1

$Old$ July 2nd, 2009, 12:04 PM

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$Default$ Random Variables, Density function

I have no idea how to approach the following problem:

Random Variable $X$ has density function

$f(x) = \left\{ \begin{array}{ll} ke^{-hx}, & x \geq 0 \\ 0 & x<0 \end{array} \right.$

Find the value of $k$ and determine $E(x)$ .

Last edited by mr fantastic; July 2nd, 2009 at 10:07 PM. Reason: Fixed latex (exponents require ^{} not ^() ) and formatted pdf as an array

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$Old$ July 2nd, 2009, 12:38 PM

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$f(x) = ke^{\text{-}hx}$ for $x \ge 0$

so $k \int_{0}^{\infty} e^{-hx} \ dx = 1$

and after you find k (which is in terms of h)

$E(X) = k \int^{\infty}_{0} xe^{-hx} \ dx$

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$Old$ July 2nd, 2009, 05:30 PM

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This is an exponential density, which is a gamma with $\alpha=1$ and $\beta=h$ .
http://en.wikipedia.org/wiki/Exponential_distribution

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