Expectation of truncated normal

Question

Suppose I have a bivariate normal $(x,y)\sim \mathcal N(0,\Sigma)$ .

Is there an easy formula for the expectation $\mathrm E(x\mid y>0)$ ?

Answer 1

Let $\phi(\cdot)$ and $\Phi(\cdot)$ be the PDF and CDF of standard normal distribution, as usual.

Suppose $\Sigma=\left[\begin{matrix}\sigma_1^2&\rho \sigma_1\sigma_2\\\rho\sigma_1\sigma_2&\sigma_2^2\end{matrix}\right]$ is the dispersion matrix of $(X,Y)$.

By definition,

\begin{align} E\left[X\mid a<Y<b\right]&=\frac{E\left[XI(a<Y<b)\right]}{P(a<Y<b)} \\&=\frac1{P(a<Y<b)}\iint_{\mathbb R^2} x \mathbf1_{a<y<b}f_{X,Y}(x,y)\,\mathrm{d}x\,\mathrm{d}y \\&=\frac1{P(a<Y<b)}\int_a^b\int_{-\infty}^{\infty} xf_{X,Y}(x,y)\,\mathrm{d}x\,\mathrm{d}y \\&=\frac{1}{P(a<Y<b)}\int_a^b \left\{\int_{-\infty}^{\infty} xf_{X\mid Y}(x\mid y)\,\mathrm{d}x \right\}f_Y(y)\,\mathrm{d}y \\&=\frac{1}{P(a<Y<b)}\int_a^b E\left[X\mid Y=y\right]f_Y(y)\,\mathrm{d}y \end{align}

This of course is same as saying $$E\left[X\mid a<Y<b\right]=\int_a^b E\left[X\mid Y=y\right]f_{Y\mid a<Y<b}(y)\,\mathrm{d}y$$

Now $X$ conditioned on $Y=y$ has a $N\left(\frac{\rho\,\sigma_1}{\sigma_2}y,(1-\rho^2)\sigma_1^2\right)$ distribution, so that

$$E\left[X\mid Y=y\right]=\frac{\rho\,\sigma_1}{\sigma_2}y$$

In this case, we thus have

\begin{align} E\left[X\mid Y>0\right]&=\frac{\rho\,\sigma_1}{\sigma_2}\int_0^\infty \frac{yf_Y(y)}{P(Y>0)}\,\mathrm{d}y \\&=\frac{\rho\,\sigma_1}{\sigma_2}E\left[Y\mid Y>0\right] \end{align}

Finally,

\begin{align} E\left[Y\mid Y>0\right]&=\frac{1}{P(Y>0)}\int_0^\infty \frac{y}{\sigma_2}\phi\left(\frac{y}{\sigma_2}\right)\mathrm{d}y \\&=2\sigma_2\int_0^\infty t\phi(t)\,\mathrm{d}t \\&=2\sigma_2\int_0^\infty (-\phi'(t))\,\mathrm{d}t \\&=2\sigma_2\,\phi(0) \\&=\sqrt{\frac{2}{\pi}}\sigma_2 \end{align}

Hence for $(X,Y)\sim N(\mathbf 0,\Sigma)$, we have the simple formula

$$\boxed{E\left[X\mid Y>0\right]=\sqrt{\frac{2}{\pi}}\rho\,\sigma_1}$$

A general formula where the means of $X$ and $Y$ are not zero, and $Y$ ranging from some $a$ to $b$ can also be found in a similar manner. That formula would involve $\phi$ and $\Phi$, whereas for the current one, we get the values at $\phi$ and $\Phi$ directly.

Answer 2

Let's generalize the question so that a key idea can be revealed.

Let $Y$ and $R$ be independent random variables. Define $$X=f(Y)+R$$ for a specified (measurable) function $f$. For any $a\lt b,$ what is the conditional expectation $$E[X\mid a\le Y \le b]?$$

(This states that $f$ is the regression of $X$ on $Y$ with i.i.d. additive errors.)

The independence assumption makes this an easy question to answer, because the random variables $f(Y)$ and $R$ will still be independent, whence (by "taking out what is known")

$$E[X\mid a\le Y \le b] = E[f(Y)+R\mid a\le Y \le b] = E[f(Y)\mid a \le Y \le b] + E[R].\tag{*}$$

This is very general. Let's specialize to the case $E[R]=0$ and $f$ is a linear transformation $$f(y) = \alpha\, y$$ for some constant number $\alpha.$ In this case $(*)$ simplifies to

$$E[X\mid a\le Y \le b] = \alpha\, E[Y\mid a \le Y \le b].\tag{**}$$

The right hand side has a direct expression when the distribution of $Y,$ $F_Y,$ has a density $f_y,$ for then (by the definitions of conditional probability and expectation)

$$E[Y\mid a \le Y \le b] = \frac{1}{F(b)-F(a)}\int_a^b y f_y(y) \mathrm{d}y.$$

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The result is now immediate and obvious to anyone who has studied bivariate regression. (A good non-mathematical reference with all the necessary details is Freedman, Pisani, and Purves, Statistics [any edition].) The next section of this post is a review for those who might not have encountered this theory.

In the case of the question, where $(x,y)$ is bivariate Normal with mean $(0,0),$ standard deviations $\sigma_x$ and $\sigma_y,$ and correlation $\rho,$ we know that the distribution of $(x,y)$ is the same as the distribution of $(X,Y)$ constructed as above where $Y$ has a Normal$(0,\sigma_y^2)$ distribution and $R$ independently has a Normal$(0,\tau^2)$ distribution (with $\tau$ to be found). The proof of this lies in the observations that

1. Because $Y$ and $R$ are independent, the variance of $X = \rho\,Y+R$ equals $$\rho^2 \operatorname{Var}(Y) +\operatorname{Var}(R) =\rho^2\sigma^2_y + \tau^2.$$ Consequently, if we set $$\tau^2 = \sigma_x^2 - \rho^2\sigma_y^2,$$ the variance of $X$ will equal the variance of $x.$

2. The covariance $(X,Y)$ is $$\operatorname{Cov}(X,Y) = \operatorname{Cov}(\alpha\,Y+R,Y) = \alpha\operatorname{Var}(Y) = \alpha\,\sigma_y^2.$$ If we set $$\alpha = \rho\frac{\sigma_x}{\sigma_y}$$ then the correlation of $X$ and $Y$ will be $$\rho(X,Y) = \frac{\operatorname{Cov}(X,Y)}{\sigma_x\sigma_y} = \frac{\alpha\,\sigma_y^2}{\sigma_x\sigma_y} = \frac{\rho\frac{\sigma_x}{\sigma_y} \sigma_y^2}{\sigma_x\sigma_y} = \rho.$$

Observations (1) and (2) establish that $(X,Y)$ and $(x,y)$ have the same moments through second order, which means they have identical bivariate Normal distributions.

Applying the main result $(**)$ gives

$$E[X\mid a\le Y \le b] = \alpha\, E[Y\mid a \le Y \le b] = \rho\frac{\sigma_x}{\sigma_y}E[Y\mid a \le Y \le b].$$

We can go a little further towards simplification by recognizing that $Y/\sigma_y$ is a standard Normal variate. Thus,

$$E[X\mid a\le Y \le b] = \rho\,\sigma_x E\left[\frac{Y}{\sigma_y}\mid \frac{a}{\sigma_y} \le \frac{Y}{\sigma_y} \le \frac{b}{\sigma_y} \right].$$

This shows that for fixed $a,b,$ the answer is directly proportional to $\rho\, \sigma_x.$ Evaluating the constant is a matter of manipulating integrals of a standard Normal variable, but doing so is primarily an exercise in Calculus rather than of statistical interest.

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I will conclude with the expression that I actually began with when solving this problem. Notice that the answer can also be written $$E\left[\frac{X}{\sigma_x}\mid a\le Y \le b\right] = \rho\, E\left[\frac{Y}{\sigma_y}\mid \frac{a}{\sigma_y} \le \frac{Y}{\sigma_y} \le \frac{b}{\sigma_y} \right].$$

The left hand side is the conditional expectation of a standard Normal variate $X/\sigma_x$ while the right hand side is just $\rho$ times the expectation of a truncated standard Normal variate $Y/\sigma_y.$

If you get used to standardizing variables automatically--which amounts to choosing a particularly convenient unit of measurement for them--then simply by visualizing the regression of $X$ against $Y$ you will say to yourself

oh yes, since after standardization $X$ is just a multiple $\rho$ of $Y$, then the conditional expectation of $X$ must be $\rho$ times the corresponding expectation of $Y.$

That is the heart of the matter.

Answer 3

I am posting the complete answer for reference: \begin{eqnarray*} E(X|a<Y<b) &= &\int_a^bE(X|Y=y)f_{Y|a<Y<b}(y)dy \\ & = &\rho_{xy}\frac{\sigma_x}{\sigma_y}\int_a^bf_{Y|a<Y<b}(y)dy \\ & = &\rho_{xy}\frac{\sigma_x}{\sigma_y}\sigma_y\frac{\phi(a)-\phi(b)}{\Phi(b)-\Phi(a)} \\ & = &\frac{\sigma_{xy}}{\sigma_y}\frac{\phi(a)-\phi(b)}{\Phi(b)-\Phi(a)} ,\end{eqnarray*} where $\rho_{xy} = E(xy) / (\sigma_x\sigma_y) =\sigma_{xy} / (\sigma_x \sigma_y)$.

The second-to-last step is derived from the properties of the univariate truncated normal distribution, easily found on its wikipedia page. A reference for the first step is here

With $a=0,b=\infty$ we have $$ E(X,Y>0) = \rho_{xy}\sigma_x\frac{\phi(0)}{\Phi(0)}= \rho_{xy}\sigma_x\cdot \frac{2}{\sqrt{2\pi}}= \frac{\sigma_{xy}}{\sigma_y}\cdot \frac{2}{\sqrt{2\pi}} $$ Thanks for pointing in the right direction.