# Conditional Expectation via Integral over Quantile Function

Following this thread "Does a univariate random variable's mean always equal the integral of its quantile function?" I tried to do a similar thing for a conditional expectation. It seems like my stochastic skills are a bit rusty. For a continuous r.v. with support on the real line I think that it holds

$E[X|X<q_\theta] = \int_{-\infty}^\infty x f(x|x<q_\theta)dx = ... = \frac{1}{F(q_\theta)} \int_{-\infty}^{q_\theta} x f(x)dx = \frac{1}{\theta} \int_{0}^{\theta}F^{-1} (p) dp$

where $q_\theta$ is the $\theta$ quantile and $f(x)$ is the density, $F(x)$ is the cdf and $F^{-1}(x)$ is the quantile function.

EDIT: My solution so far is

$E[X|X<q_\theta]=\int xf(x|x<q_\theta)dx = \int x \frac{f(x)P(x<q_\theta|X=x)}{\int f(u)P(u<q_\theta|X=u)du}dx = \frac{1}{\int f(u) 1{(u<q_\theta)}du} \int x f(x) 1{(x<q_\theta)}dx= \frac{1}{F(q_\theta)} \int_{-\infty}^{q_\theta} xf(x)dx = \frac{1}{\theta} \int_{0}^{\theta}F^{-1} (p) dp$

using the relationsip

$f(x|B)=\frac{f(x)P(B|X=x)}{\int f(x)P(B|X=x)dx}$

Is this alright? Thanks a lot in advance for any suggestions!

• I believe that if you consult your favorite definition of conditional distribution, this question will resolve itself easily. In particular I am thinking of the relationship $\Pr(A|B)\Pr(B)=\Pr(A)$ where $A$ is the event $X\le x$ and $B$ is the event $X\le q_\theta$.
– whuber
Jun 19, 2014 at 19:02
• Two notes. Firstly, $P(X=x)$ for any random variable $X$ which has a continuous distribution is 0. Secondly, I think you need the denominator in your first step. Specifically $E\left[X\vert X<q_\theta\right] = \frac{\int_{-\infty}^\infty xf(x\vert x < q_\theta)dx}{\int_{-\infty}^\infty f(x\vert x < q_\theta)dx}$ The expectation is only over the viable area. Jun 19, 2014 at 19:25
• See edit in the original question. It would be nice if someone could confirm the correctness of the solution. Thanks a lot for your help so far!!
– Joz
Jun 20, 2014 at 19:19

The random variable $X$ has the same law as $F^{-1}(U)$ where $U$ has the uniform distribution on $[0,1]$. Thus we can assume $X=F^{-1}(U)$, and the event $\{X \leq q_\theta\}$ equals $\{U \leq \theta\}$. Conditionally to this event, $U$ has density $\frac{1}{\theta}{\boldsymbol 1}_{[0,\theta]}(u)$. Then $$E[X \mid X \leq q_\theta] = E[F^{-1}(U) \mid U \leq \theta] = \frac{1}{\theta}\int_0^\theta F^{-1}(u)\mathrm{d}u.$$