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Let $X$ be an exponential random variable with parameter $\lambda$. For any fixed $s>0$, I would like to compute $E(X\vert X>s)$. I have read in multiple places that this conditional expectation is equal to $1/\lambda +s$, however, the definition of a conditional expectation implies that $E(E(X\vert X>s))=E(X)=1/\lambda\ne E(1/\lambda +s)=1/\lambda +s$. (For any $\sigma$-field $\mathcal{F}$, it is a fact that $E(E(X\vert\mathcal{F}))=E(X)$, so here we just have $\mathcal{F}=\sigma(\{X>s\})=\{\emptyset, \Omega, \{X>s\}, \{X\le s\}\}.)$

Why is there a discrepancy here? There shouldn't be...I'm probably missing something, but I don't know what.

EDIT: I suppose the discrepancy could be if $E[\cdot \vert X>s]$ was meant to be the expectation of $X$ with respect to the probability measure $P(\cdot \vert X>s)$. However, does one know what the conditional expectation $E[X\vert X>s]$ should be as a RV?

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  • $\begingroup$ Why is $\mathbb{E}(\mathbb{E}(X|X>s))=\mathbb{E}(X)$? $\endgroup$ – Gerhard S. Sep 29 '17 at 14:39
  • $\begingroup$ @GerhardS. I could go into a proof, but it's just immediate from the definition of a conditional expectation. See my edit. $\endgroup$ – Satana Sep 29 '17 at 15:02
  • $\begingroup$ $E(X|A)=E(X;A)/P(A)$, for (non-zero probability) events $A$, right? So it is not a RV, right? (The notation $E(X;A)$ means $E(XI_A)$ where $I_A$ is the indicator function). $\endgroup$ – Nap D. Lover Sep 29 '17 at 20:00
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    $\begingroup$ @LoveTooNap29 Thanks, I'll check that formula and see if I get the result I want! But in general, for $E(X\vert A)$ I mean the conditional expectation of the RV $X$ conditioned on the $\sigma$-field generated by the event $A$. This is $\textit{always}$ a RV. Perhaps, as you suggested, the RV in this case (when you condition on a single event) is almost surely constant, so that it's essentially not random. I'll check it out. $\endgroup$ – Satana Oct 1 '17 at 2:51
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    $\begingroup$ Satana: Sorry but no. If $P(A)\ne0$, then $E(X\mid A)$ is the number $E(X\mathbf 1_A)/P(A)$, not the random variable $E(X\mid\sigma(A))$. $\endgroup$ – Did Oct 16 '17 at 20:36
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Recall that the exponential distribution is the unique continuous probability distribution with the memoryless property. That is, if $X\sim\mathrm{Exp}(\lambda)$ then for $s,t>0$, $$ \mathbb P(X>t+s\mid X>s) = \mathbb P(X>t). $$ (This follows immediately from $\mathbb P(X>t)=e^{-\lambda t}$ and the definition of conditional probability.) Therefore we may compute \begin{align} \mathbb E[X\mid X>s] &= \int_0^\infty tf_{X\mid X>s}(t)\ \mathsf dt\\ &= \int_0^\infty tf_X(t-s)\ \mathsf dt\\ &= \int_0^\infty (s+t)f_X(t)\ \mathsf dt\\ &= s\int_0^\infty \lambda e^{-\lambda t}\ \mathsf dt + \int_0^\infty t\lambda e^{-\lambda t}\ \mathsf dt\\ &= s + \frac1\lambda. \end{align}

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