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The problem is to find a joint distribution for $X$, $Y$ given $X=Y$ where $X$ has some pdf $f_x(x)$ . In my notation X denotes a random variable and x a realization of $X$.

Here is what I got so far:

The cdf should be $F_{XY}(x,y) = P( X \leq x \land Y \leq y ) = P ( X \leq x \land X \leq y)$

Then I defined my modified "min" function $\hat{min}$ $:= x\;$if $x \leq y$ and $y \;$otherwise. Using this I can simplify the cdf: $$ F_{XY}(x,y) = P( X \leq \hat{min}(x,y) \land X \leq \hat{min}(x,y) ) = P( X \leq \hat{min}(x,y) ) $$ $$ = \int_{-\infty}^{\hat{min}(x,y)} f_x(x)\,dx = F_x(\hat{min}(x,y))-F_x(-\infty) = F_x(\hat{min}(x,y)) $$ According to our lecture notes I should be able to obtain the pdf like this:

$$ f_{XY}(x,y) = \frac{d^2}{dxdy} F_{XY}(x,y) = \frac{d^2}{dxdy} F_x(\hat{min}(x,y)) $$

Unfortunately this differentiation is not very nice and would involve some Dirac-"functions". So I tried to guess the joint pdf. Obviously the support of the joint pdf $f_{XY}$ should by the $x=y$ line. Therefore I would guess something like

$$ f_{XY}(x,y) = \delta(x-y) \cdot C(x,y,f_x) \quad C \in C^{0}. $$

Does anyone know what to choose for $C$ or has any hint for me, how I could approach this problem?

user1462040
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1 Answers1

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If you take the joint probability as $p(y|x)p(x)$ instead you already know $p(x)=f_X(x)$. $p(y|x)$ on the other hand is what you had $\delta(x=y)$

$$f_X(x)\delta(x-y)$$

sachinruk
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