Joint distribution of two dependent random variables 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!
 A: Let me rephrase the question in a somewhat more orthodox language. One considers two random variables $X=uZ$ and $Y=vZ$, where $Z$ is standard normal and $u$ and $v$ are real numbers such that $(u,v)\ne(0,0)$.
Then $(X,Y)\in D$ almost surely, where $D$ is the straight line $D=\{(x,y)\in\mathbb R^2\mid vx=uy\}$. Since $D$ has Lebesgue measure zero, the distribution of $(X,Y)$ has no density. 
Note that all that matters is that one can still evaluate expectations, using the density $f_Z$ of the random variable $Z$. To wit, for every measurable function $g$,
$$
\mathbb E(g(X,Y))=\int_{-\infty}^{+\infty}g(uz,vz)\,f_Z(z)\,\mathrm dz.
$$
Edit: The fact that $D$ has Lebesgue measure zero is not a probability result since the Lebesgue measure on the plane has infinite mass but here is a proof. Consider without loss of generality the line $\Delta=\mathbb R\times\{0\}$, then $\Delta=\bigcup\limits_{n\geqslant1}\Delta_n$ with $\Delta_n=[-n,n]\times\{0\}$. Note that, for every $\varepsilon\gt0$, $\Delta_n\subset[-n,n]\times[-\varepsilon,\varepsilon]$. The rectangle $[-n,n]\times[-\varepsilon,\varepsilon]$ has Lebesue measure $4n\varepsilon$, hence the Lebesgue measure of $\Delta_n$ is at most $4n\varepsilon$, for every $\varepsilon\gt0$, hence the Lebesgue measure of $\Delta_n$ is zero, hence the Lebesgue measure of their union $\Delta$ is zero.
Edit: See Elementary Probability for Applications, by Rick Durrett.
A: 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, $$\begin{align}P\{A\leq a, B\leq b\}=F_{A,B}(a,b)&=P\left\{s\leq\frac{a}{\delta_1},s\leq\frac{b}{\delta_2}\right\}\\&=P\left\{s\leq\min\left(\frac{a}{\delta_1},\frac{b}{\delta_2}\right)\right\}\\&=\Phi\left(\min\left(\frac{a}{\delta_1},\frac{b}{\delta_2}\right)\right),\end{align}$$
