I have a truncated bivariate normal distribution

$$f(x,y)=\begin{cases}\frac{1}{2\pi \sigma_{1}\sigma_{2}\sqrt{1-\rho^{2}}}\exp\left(-\frac{z}{2(1-\rho^{2})}\right) &, |x| \leq a\\0 &, |x| > a\end{cases}$$

, where $$z=\frac{x^2}{\sigma_{1}^2}+\frac{y^2}{\sigma_{2}^2}-\frac{2\rho xy}{\sigma_{1}\sigma_{2}}$$

moreover, in case $f(x,y)$ would be not truncated, it would be normalized.

Is there a simple way to calculate variance of marginal distributions ?

I know that if the distribution is not truncated, variance of marginal distributions are $\sigma_{1}^2$ and $\sigma_{2}^2$, but what if the distribution is truncated ?

  • 1
    $\begingroup$ Is $f(x,y) a pdf? Or is the integral = 1? $\endgroup$
    – user158565
    Commented Jul 26, 2019 at 22:45
  • $\begingroup$ $f(x,y)$ is a normalized pdf, so integral=1 $\endgroup$ Commented Jul 26, 2019 at 22:58
  • $\begingroup$ $f(x,y)$ is plainly not normalised, as the normalising constant must depend on $a$ - and you do not even have a normalising constant. Also, you are confusing variance with standard deviation. $\endgroup$
    – wolfies
    Commented Jul 27, 2019 at 13:36
  • $\begingroup$ @wolfies You are right. $f(x,y)$ would be normalized only if not truncated. I corrected about viarance and standard deviation. $\endgroup$ Commented Jul 27, 2019 at 17:04

1 Answer 1


It depends what one means by simple ...

Given $\mu = (0,0)$, $\quad \Sigma =\left( \begin{array}{cc} \sigma _1^2 & \rho \sigma _1 \sigma _2 \\ \rho \sigma _1 \sigma _2 & \sigma _2^2 \\ \end{array} \right) \quad $ and $\quad a>0$.

Unconditional model

Let $(X,Y) \sim N( \mu, \Sigma)$ with pdf $f(x,y)$ and cdf $F(x,y)$:

enter image description here

Normalising constant

Let $c = P(-a<X<a) = F(a,\infty)-F(-a,\infty) =2 \big[ F(a,\infty) -\frac12 \big] = \text{Erf}\left(\frac{a}{\sqrt{2} \sigma _1}\right)$

where Erf denotes the error function. Or, automating it:

enter image description here

Doubly Truncated model

Let $\big(X \big| -a<X<a,Y\big)$ have pdf $g(x,y)$. Then:

enter image description here

Then, the variance-covariance matrix for $(X,Y)$ when $X$ is truncated above at $a$ and below at $-a$ is given by:

enter image description here

where I am using the Varcov function from the mathStatica package for Mathematica to automate the nitty gritties.

  • The top left element denotes $\text{Var}(X)$ for the conditional model $g$ and yields a closed-form solution. An interesting feature is that the conditional variance of $X$ does not depend on $\rho$ - this appears to be because the truncation is symmetrical around the mean 0.

  • The bottom right entry (with the integral sign) denotes $\text{Var}(Y)$ for conditional model $g$: since the software is unable to find a closed-form solution, the integration does not appear to be 'easy'.

  • The off-diagonal elements denote $\text{Cov}(X,Y)$ for conditional model $g$.

Hope this helps.

  • $\begingroup$ Thank you. Indeed, this helps a lot. I was mainly interested in $Var(Y)$, and now i understand calculation is not so easy as i wish. Thank you again! $\endgroup$ Commented Jul 28, 2019 at 9:20

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.