<s>[INST] Calculation of an "unconstrained" normal distribution (starting from a censored one) Assume that two r.v. $W$ and $Y|W=w$ with
(1) $W \sim \text{N}(\mu_w,\sigma_w^2)$ (iid)
(2) $Y|W=w \sim \text{N}(w,\sigma_y^2)$ (iid)
Further we only observe $Y$ if $Y$ is less then $W$, i.e., 
(3) $Y|Y\le W$
Goal: Find the pdf of the censored observations, i.e., of $Y|Y\le W$ and from that deduce the uncensored pdf and the first two moments (so i.m.h.o. we have to find$f_Y(y)$). The first two moments of this uncensored pdf are supposed to depend upon $E(Y|Y\le W)$ and $Var(Y|Y\le W)$.

By definition of conditional pdf we have that:
(4) $f_{Y|W}(y|W = w)= \frac{f_{Y,W}(y,w)}{f_W(w)}$ 
Next, the definition of a truncated density gives for a abitrary value of $W$:
(5) $ f_{Y|Y\le W}(y|y\le w) = \frac{f_Y(y)}{P(Y\le W)}$

I would simply rewrite (4) to
$f_{Y|W}(y|W = w)f_W(w) = f_{Y,W}(y,w)$
then integration over $f_{Y,W}(y,w)$ w.r.t $w$ should yield $f_Y(y)$, i.e.,
(a) $\int_{-\infty}^{\infty} f_{Y,W}(y,w) dw = \int_{-\infty}^{\infty} f_Y(y|W = w)f_W(w) dw = f_Y(y)$
Plugin in $f_Y(y)$ into (5), ($P(Y\le W)$ will also be given by $f_Y(y)$) I will se how the moments of $f_{Y|Y\le W}(y|y\le w)$ will look and how the moments of $f_Y(y)$ depend upon them.
So (a) will look like
$f_Y(y) = \int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi\sigma^2_y}}\text{exp}\big(-\frac{(y-w)^2}{2\sigma_y^2}\big)\frac{1}{\sqrt{2\pi\sigma^2_w}}\text{exp}\big(-\frac{(w-\mu_w)^2}{2\sigma_w^2}\big)dw$
Except for the $w$ in the first $\text{exp}$, this looks very easy but since there is a $w$ im a little bit stuck how to solve this... 
 [/INST] Ok. Let's do this, for CV's shake.
First compact by setting $C=\frac{1}{\sqrt{2\pi\sigma^2_y}}\frac{1}{\sqrt{2\pi\sigma^2_w}} = \frac{1}{2\pi\sigma_y\sigma_w}$, so 
$$f_Y(y) =C \int_{-\infty}^{\infty}\exp\left\{-\frac{(y-w)^2}{2\sigma_y^2}\right\}\exp\left\{-\frac{(w-\mu_w)^2}{2\sigma_w^2}\right\}dw$$
We have 
$$exp\left\{-\frac{(y-w)^2}{2\sigma_y^2}\right\}\exp\left\{-\frac{(w-\mu_w)^2}{2\sigma_w^2}\right\} = 
\exp\left\{-\frac{y^2-2yw+w^2}{2\sigma_y^2}\right\}\exp\left\{-\frac{w^2-2w\mu_w+\mu_w^2}{2\sigma_w^2}\right\} 
=\exp\left\{-\frac{y^2}{2\sigma_y^2}-\frac{\mu_w^2}{2\sigma_w^2}\right\} \exp\left\{-\frac{w^2}{2\sigma_y^2}-\frac{w^2}{2\sigma_w^2}\right\}\exp\left\{\frac{2yw}{2\sigma_y^2}+\frac{2w\mu_w}{2\sigma_w^2}\right\}$$
Setting $s^2\equiv \sigma_y^2+\sigma_w^2$ we arrive at
$$=\exp\left\{-\frac{y^2}{2\sigma_y^2}-\frac{\mu_w^2}{2\sigma_w^2}\right\} \exp\left\{-\frac{s^2}{2\sigma_y^2\sigma_w^2}w^2\right\}\exp\left\{\frac{\sigma_w^2y+\sigma_y^2\mu_w}{\sigma_y^2\sigma_w^2}w\right\}$$
Include the first $\exp$ in the constant, $C^*=C \exp\left\{-\frac{y^2}{2\sigma_y^2}-\frac{\mu_w^2}{2\sigma_w^2}\right\}$.
Set
$$\beta\equiv \frac{s^2}{2\sigma_y^2\sigma_w^2},\qquad \alpha\equiv \frac{\sigma_w^2y+\sigma_y^2\mu_w}{\sigma_y^2\sigma_w^2}$$ to obtain
$$f_Y(y) =C^* \int_{-\infty}^{\infty}e^{-\beta w^2+\alpha w}dw=C^*\left[ \int_{-\infty}^{0}e^{-\beta w^2+\alpha w}dw + \int_{0}^{\infty}e^{-\beta w^2+\alpha w}dw\right]$$
$$=C^* \int_{0}^{\infty}e^{-\beta w^2}\left[e^{-\alpha w}+e^{\alpha w}\right]dw =2C^* \int_{0}^{\infty}e^{-\beta w^2}\operatorname{cosh}(\alpha w)dw$$
where $\operatorname{cosh}$ is the hyperbolic cosine.
Using a formula provided in Gradshteyn & Ryzhik (2007), "Table of Integrals, Series and Products", 7th ed., p. 384, eq. 3.546(2) we have
$$f_Y(y)=2C^*\frac 12 \sqrt {\frac {\pi}{\beta}} \exp\left\{\frac {\alpha^2}{4\beta}\right\}$$
Now $$\frac {\alpha^2}{4\beta} = \frac {\left(\frac{\sigma_w^2y+\sigma_y^2\mu_w}{\sigma_y^2\sigma_w^2}\right)^2}{4\frac{s^2}{2\sigma_y^2\sigma_w^2}} = \frac {(\sigma_w^2y+\sigma_y^2\mu_w)^2}{2\sigma_y^2\sigma_w^2s^2}$$ 
and bringing back in $C^*$ (and $\beta$) in all its glory we have
$$f_Y(y)=\frac{1}{2\pi\sigma_y\sigma_w}\exp\left\{-\frac{y^2}{2\sigma_y^2}-\frac{2\mu_w^2}{\sigma_w^2}\right\}\sqrt{\pi} \left(\sqrt {\frac{s^2}{2\sigma_y^2\sigma_w^2}}\right)^{-1} \exp\left\{\frac {(\sigma_w^2y+\sigma_y^2\mu_w)^2}{2\sigma_y^2\sigma_w^2s^2}\right\} $$
The constant terms simplify to 
$$\frac{1}{2\pi\sigma_y\sigma_w}\sqrt{\pi} \left(\sqrt {\frac{s^2}{2\sigma_y^2\sigma_w^2}}\right)^{-1} = \frac{1}{s\sqrt{2\pi}} $$
and, the exponentials end up in the normal exponential. So in the end
$$f_Y(y) = \frac{1}{s\sqrt{2\pi}}\exp\left\{-\frac{(y-\mu_w)^2}{2s^2}\right\}= N(\mu_w, s^2),\qquad s^2\equiv \sigma_y^2+\sigma_w^2$$
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