Suppose A and B are two random variables and given by $A = s\delta_1$ and $B = s\delta_2$, where $\delta_1$ and $\delta_2$ are fixed and known, however $s \sim N(0,1)$.
What does the joint distribution of $p(A,B)$ look like ?
In general, what is $p(A,A)$ ?
It seems to be that the joint is only dictated by $s$ which is the only random number in this case, and therefore the joint $p(A,B)$ can be equivalently written as $p(s)$, however I'm not a mathematician hence I'm not sure about the exact math. Thanks!
Best Answer
The joint probability density function of $A \sim N(0,\delta_1^2)$ and $B \sim N(0,\delta_2^2)$ is a degenerate joint density since all the mass lies along a straight line through the origin instead of being spread all over the plane. Problems involving $A$ and $B$ are best solved in terms of $s$ alone. For example, $$\begin{align}P\{A\leq a, B\leq b\}=F_{A,B}(a,b)&=P\left\{s\leq\frac{a}{\delta_1},s\leq\frac{b}{\delta_2}\right\}\\&=P\left\{s\leq\min\left(\frac{a}{\delta_1},\frac{b}{\delta_2}\right)\right\}\\&=\Phi\left(\min\left(\frac{a}{\delta_1},\frac{b}{\delta_2}\right)\right),\end{align}$$