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How can I compute the conditional expectation C-X in the following formula

E[C-X|X$<$C];

where C is a constant and X a random variable following exponential distribution?

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  • $\begingroup$ Have you tried writing the expression as an integral and solving that? $\endgroup$
    – Björn
    Mar 4 '19 at 20:23
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Suppose that $X \sim \mathcal{E}(\lambda)$, then we have

\begin{align*} \mathbb{E}[C-X \mid X < C] &= \frac{\mathbb{E}[ (C-X)I_{X<C}]}{P(X<C)} \\ &= \frac{CP(X<C) - \mathbb{E}[XI_{X<C}]}{P(X<C)} \\ &= C - \frac{\mathbb{E}[XI_{X<C}]}{1- e^{-\lambda C}} \end{align*}

Where $I_{\cdot}$ is the indicator function.

Now we need to solve the expectation which is basic integration

\begin{align*} \mathbb{E}[XI_{X<C}] &= \int_0^C \lambda x e^{-\lambda x} dx \\ &= \frac{1}{\lambda} - e^{-\lambda C}(C + \frac{1}{\lambda} ) \end{align*}

Thus we finally have

$$ \mathbb{E}[C-X \mid X < C] = C + \frac{ e^{-\lambda C}(C + \frac{1}{\lambda} ) - \frac{1}{\lambda}}{1 - e^{-\lambda C}} $$

Here is a little R code to check if it works with $\lambda = 0.01$ and $C=45$

Ck<-45
X<-rexp(1e7,0.01)
Y<-X[X<Ck]   ## Select only the values of X < Ck
mean(Ck-Y)   ## empirical expectation 
Ck + (Ck*exp(-0.01*Ck) + (exp(-0.01*Ck)-1)/0.01)/(1-exp(-0.01*Ck)) 

More generally when you have an conditional expectation of the type

$$ \mathbb{E}[X \mid A] $$

you can use the relation

$$ \mathbb{E}[X \mid A] = \frac{\mathbb{E}[X I_A]}{P(A)} $$

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  • $\begingroup$ Thank you @winperikle. I had done the sam job but my result is a bit different. In your last equation, in the numerator, it should be $\e^{-\lambda.C}$ (C+1/$\lambda$) I guess, no ? My solution differs from there $\endgroup$
    – Angelıque
    Mar 6 '19 at 12:11

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