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Let a~$\mathcal{N}(\mu_a,{\sigma_a}^2)$,b~$\mathcal{N}(\mu_b,{\sigma_b}^2)$ and c~$\mathcal{N}(\mu_c,{\sigma_c}^2)$.

We construct two normal variables x~$a-b$ and y~$a-c$.

Can we find the covariance between these two random variables i.e. $\textrm{cov}(x,y)$ by hand? Sorry if my question is trivial.

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  • $\begingroup$ Are $a$,$b$ and $c$ independent or at least uncorrelated? if not, what are the covariances between them? $\endgroup$
    – Glen_b
    Commented Aug 26, 2014 at 11:10
  • $\begingroup$ Be aware that $a-b$ (respectively $a-c$) is not necessarily normal unless $a$ and $b$ (respectively $a$ and $c$) are jointly normal. The covariance can be worked out as indicated in Glen_b's answer. However, while that calculation is valid regardless of the distribution of the variables, it cannot be used to assert the normality of $a-b$ or $a-c$. $\endgroup$ Commented Aug 26, 2014 at 11:37

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Let $a\sim N(μ_a,σ_a^2)$,$b\sim N(μ_b,σ_b^2)$ and $c\sim N(μ_c,σ_c^2)$.

Let $x=a−b$ and $y=a−c$.

Substitute in the definition of $x$ and $y$:

$\text{cov}(x,y)=\text{cov}(a−b,a−c)$

From there, just use basic properties of covariance (or even just definition of covariance plus linearity of expectation):

$\text{cov}(aX+bY, cW+dV) \\ \quad\quad= ac\,\text{cov}(X,W)+ad\,\text{cov}(X,V)+bc\,\text{cov}(Y,W)+bd\,\text{cov}(Y,V)$

where in your problem the constants are all $1$ or $-1$, and then use the fact that $\text{cov}(X,X) = \text{Var}(X)$.

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