# When simulating a bivariate normal distribution, why is $\rho$ chosen instead of estimated from the data?

In a video lecture, MrProf shows the 3d-plot of a bivariate normal distribution $\mu_{x_1} = \mu_{x_2} = \sigma_1 = \sigma_2 = 1$ and chooses $\rho = 0.5$ .

If stick to Mathworld, $\rho$ simply is the Pearson correlation that is $\dfrac{\mbox{cov}(x_1, x_2) }{[ \mbox{var}(x_1) \mbox{var}(x_2)]^\frac{1}{2}}$.

I don't understand why one would have control over $\rho$ as in the example given by MrProf. I understood that $\rho$ is an output from the formula as I plug in $x_1$ and $x_2$, not an input.

Can anyone throw any light?

You need $\rho$ to account for how correlated the variables are.

If you had actual data, then $\rho$ would be output. But here MrProf is simulating data, so it is input. Just like, if you had actual data, $\mu$ and $\sigma$ would be output, but in order to simulate data, they are input.

• i had this distinction between real data and simulated ones on my mind, but i was not sure it actually is my answer thank you Peter – tagoma Sep 21 '12 at 12:04
• We're glad to help, @edouard, that's what CV is for. If Peter has provided the information you needed, you might consider accepting his answer by clicking on the check mark to the left underneath the vote total. – gung - Reinstate Monica Sep 21 '12 at 12:46