# 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.