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I started with the following exponential distribution: $$ f_{exp}(x;\lambda) = \lambda\, e^{-x\lambda} \quad \forall\, x \in \mathbb{R}^+ $$ I know from internal courseslides and wikipedia that the expected value $\mathbb E(X)$ of the random variable is supposed to be $\frac{1}{\lambda}$, but I would like to understand how this result was derived.

It is my understanding that in the continuous case the following holds for any distribution: $\mathbb E(X) \equiv \int x\,f(x) dx$ for the range in which $x$ is defined.

So I used this formula to find $\mathbb E(X)$: $$\mathbb E(X) = \int_0^\infty x\,\lambda\,e^{-x\lambda} dx$$ Through partial integration I arrived at the following: $$\mathbb E(X) = x[-e^{-\lambda x}]_0^\infty - [\frac{e^{-\lambda x}}{\lambda}]_0^\infty $$ Which I can "simplify" to this: $$\mathbb E(X) = -x+e^{- \lambda \infty}x-\frac{1}{\lambda}+\frac{e^{- \lambda \infty}}{\lambda} $$

I am completely stumped on how to go from this to $\frac{1}{\lambda}$. Either there are further simplification steps which I don't understand or there is an error in my thinking so far. Could you help me out?

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No $x$ can come out of the integral: $$\begin{align}\mathbb E[X]&=\int_0^\infty \underbrace{x}_u \underbrace{\lambda e^{-\lambda x}dx}_{dv}\rightarrow du=dx, v=-e^{-\lambda x}\\&=\left[xe^{-\lambda x}\right]_0^\infty-\int_0^\infty (-e^{-\lambda x})dx\\&=0 -\left[\frac{e^{-\lambda x}}{\lambda}\right]_0^\infty\\&=\frac{1}{\lambda}\end{align}$$

The first term is $0$ because: $$\left[xe^{-\lambda x}\right]_0^\infty=\lim_{x\rightarrow \infty} {x\over e^{\lambda x}}\overbrace{=}^{\text{L'Hospital}}\lim_{x\rightarrow \infty} \frac{1}{\lambda e^{\lambda x}}=0$$

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  • $\begingroup$ Thankyou! Through your answer I realized what mistake I was making when integrating by parts. Unfortunately I still dont understand how insertin infinity into xe^{-lambda*x) returns 0. Wouldnt that return: undefined/any number possible? $\endgroup$ – Andre Apr 8 at 12:01
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    $\begingroup$ @Andre exponential decays much faster than the increase in $x$, so it'll converge to $0$. I've appended a mathematical explanation to the end of my answer. $\endgroup$ – gunes Apr 8 at 12:27
  • $\begingroup$ Thank you! I get it now. Still quite new to this. $\endgroup$ – Andre Apr 8 at 13:09

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