# Compute the mean and std of a random variable from its correlation with known random variables?

Context

I'm running statistical simulations on IQ distribution among people and I would like to sample the IQ of a child given :

1. The general distribution of IQ following : $$\mathcal{N}(100, 15.5) \text{ for male}$$ $$\mathcal{N}(100, 14.5) \text{ for female}$$
2. The correlation between the average of parents' IQ and their child IQ (0.5).
3. Both parents' IQs
4. The sex of the child

Question

Is it possible to do that and if it is, how ?

• Do you know the average of the parents' IQ, and the sex of the child? May 31, 2019 at 9:25
• Yes I know that ! May 31, 2019 at 9:27
• I edited the question since it was not clear enough. May 31, 2019 at 9:34

Let $$X_1$$ and $$X_2$$ be random variables corresponding to the IQ of the mother and father. The average of their IQs also has a normal distribution with

$$Y_1 \equiv \frac{X_1 + X_2}{2} \sim \mathcal{N}(100, \frac{1}{4}(15.5 + 14.5 + 2\mathrm{Cov}(X_1, X_2)).$$

It's probably the case that $$\mathrm{Cov}(X_1, X_2) \neq 0$$ for parents of the child - you could include this in your simulations if you wished. Anyway, continue by letting $$\sigma^2 = \frac{1}{4}(15.5 + 14.5 + 2\mathrm{Cov}(X_1, X_2))$$.

Now if $$Y_2$$ is another random variable for the IQ of the child (say it's a girl) then the joint distribution of $$Y_1$$ and $$Y_2$$ is multivariate normal with

$$\begin{pmatrix} Y_1 \\ Y_2 \\ \end{pmatrix} \sim \mathcal{N}\left( \begin{pmatrix} 100 \\ 100 \\ \end{pmatrix}, \begin{pmatrix} \sigma^2 & \mathrm{Cov}(Y_1, Y_2) \\ \mathrm{Cov}(Y_1, Y_2) & 14.5\\ \end{pmatrix} \right).$$

The off-diagonal covariance terms can be found by the definition of correlation

$$\frac{\mathrm{Cov}(Y_1, Y_2)}{\sqrt{14.5}\sqrt{\sigma^2}} = 0.5.$$

Now you can use the formula for the conditional distribution of the multivariate normal to find the distribution of $$Y_2 | Y_1 = y_1$$ where $$y_1$$ is the realisation of the parent's average IQ, which you have. This should give you a univariate distribution which can be simulated from easily (say using rnorm if you are using R.)

• Thank you very much, this is exactly what I was looking for ! May 31, 2019 at 10:56
• How does this answer the question as phrased? The question says the parents' IQs are given; their constant, not random variables. The random variable was the IQ of the child. May 31, 2019 at 19:23
• I interpreted the parent's IQs to be given as realisations from the general distribution of IQ. If it were instead a constant, how would one interpret $\rho = 0.5$? The correlation would be undefined I think May 31, 2019 at 21:27