Constructing a Correlated Normal Variable from Independent Normals
Summary
The document explains how to construct a standard normal variable Z with a chosen correlation to another standard normal variable X. Starting with independent standard normals X and Y, it defines Z as a weighted sum: the coefficient on X sets the target correlation, while the coefficient on Y supplies the remaining variance.
The proof uses the zero means, unit variances, and zero covariance of X and Y. These properties show that Z has unit variance and that its covariance with X equals the coefficient on X; since both variables have unit variance, that covariance is also their correlation. The result holds for the stated parameter range from negative one to positive one. This is a basic probability construction useful for simulating correlated normal quantities, but the document does not discuss extensions to multiple variables or non-normal distributions.
Key ideas
- A weighted sum of independent standard normals can create a standard normal variable with a specified correlation to one component.
- The independent component receives a coefficient that preserves unit variance.
- Zero covariance between the original normals makes the covariance calculation straightforward.
- For two unit-variance variables, covariance equals correlation.
Tags
Full text
# Prove ρ(X,Z) = ρ
# Prove ρ(X,Z) = ρ
The covariance of two random variables $X$ and $Y$ is defined by: $$\mathrm{Cov}(X,Y）= \operatorname{E}(X-\operatorname{E}(X))(Y-\operatorname{E}(Y))=\operatorname{E}(XY)-\operatorname{E}(X)\operatorname{E}(Y)$$
Another related definition is correlation coefficient $$\rho(X,Y) = \frac{\mathrm{Cov}(X,Y)}{\sqrt{\mathrm{Var}(X)\mathrm{Var}(Y)}}$$
It can be proved that the correlation coefficient $\rho(X,Y)$ always lies between −1 and +1. $X$ and $Y$ are two independent standard normal random variables. We now define another random variable $Z$ by $Z=\rho X+\sqrt{1-\rho^2}\cdot Y$ where $\rho \in [−1,1]$.
How can one prove hat $\rho(X,Z) = \rho$?
## Answer by ZRH (score 1, accepted)
https://quant.stackexchange.com/a/44508
as per definition $E[Y]=E[X]=0$, and also $\mathit{Var}(X)=\mathit{Var}(Y)=1$.
$\mathit{Var}(Z)=\mathit{Var}(\rho X+\sqrt{1-\rho^2}Y)=E\big[\big(\rho X+\sqrt{1-\rho^2}Y-\rho \langle X \rangle+\sqrt{1-\rho^2} \langle Y \rangle \big)^2\big]=E\big[\big(\rho X+\sqrt{1-\rho^2}Y \big)^2\big]=E\big[\rho^2 X^2+2\rho \sqrt{1-\rho^2}XY+(1-\rho^2)Y^2\big]=\rho^2E\big[X^2\big]+2\rho\sqrt{1-\rho^2}E\big[XY\big]+(1-\rho^2)E\big[Y^2\big]=1$
Therefore $\rho(X,Z)=\mathit{Cov}(X,Z)$. Further, also as per definition $E[XY]=0$. Applying the formula for the covariance:
$\mathit{Cov}(X,Z)=\mathit{Cov}(X,\rho X+\sqrt{1-\rho^2}Y)=E\big[(X-\langle X \rangle)(\rho X+\sqrt{1-\rho^2}Y-\rho \langle X \rangle-\sqrt{1-\rho^2}\langle Y \rangle)\big]=E\big[X(\rho X+\sqrt{1-\rho^2}Y)\big]=\rho E[X^2]+\sqrt{1-\rho^2}E[XY]=\rho E[X^2]=\rho$
## Answer by Hunaphu (score 1)
https://quant.stackexchange.com/a/44534
The covariance operator is linear and since $X$ and $Y$ are independent any covariance between them is zero. Therefore, the only part that matters is the X-part of the covariance:
$Cov[X, Z] = \rho Cov[X, X] + Cov[..., ...Y] = \rho Var[X] = \rho$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.