verifying whether a conditional density function is valid

Question

I want to verify whether a given conditional probability function is valid or not.

$\mathsf P(y\mid x)=\begin{cases}c\, e^{(-y/x)} & : y\geqslant 0, x>0;\\ 0 & :\text{otherwise}\end{cases}$

How should I perform my integration, there are no marginals and no joint probability function given. Can I perform with respect to $d(y/x)$? or should I proceed using a double integral?

thanks in advance, phani tej

Answer 1

Assuming $\mathsf P$ is meant to be the conditional probability density function (pdf), you should integrate $\int_0^\infty \mathsf P(y\mid x) \operatorname d y$, which should equal $1$ iff valid.

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Assuming $\mathsf P$ is meant to be the conditional cumulative probability function (CDF), then validity requires: $$\forall x\in(0,\infty):[\mathsf P(0\mid x)=0\bigwedge \lim\limits_{y\to\infty}\mathsf P(y\mid x)=1\bigwedge \forall y\in[0,\infty):\frac{\mathrm d \mathsf P(y\mid x)}{\mathrm d y}\geqslant 0]$$