Conditional Expectation via Integral over Quantile Function

Question

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

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

See here http://www.randomservices.org/random/dist/Conditional.html

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

Answer 1

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 $$ E[X \mid X \leq q_\theta] = E[F^{-1}(U) \mid U \leq \theta] = \frac{1}{\theta}\int_0^\theta F^{-1}(u)\mathrm{d}u. $$