Assumptions of a probability distribution Let $X$ be a continuous real-valued random variable indicating the fragility of a firm. Suppose that the firm defaults if $X$ takes a value above a threshold $u>0$. Hence $$
Prob(X>u) 
$$
is the probability of default. 
A very basic intuition could be that the higher is the value taken by $X$, the higher should be the probability of default. Does this statement make sense? And, if it makes sense, which assumptions on the dstribution of $X$ can represent it?
 A: Maybe you are looking for a statement like this: The higher the probability of high values of $X$, the more likely it is that the firm defaults. That is, you are comparing probability distributions of $X$.
Mathematically, you could use the notion of first order stochastic dominance (FOSD). Suppose you have two different distributions for $X$. Denote the cumulative distribution functions by $F(z)=Pr(X_F\le z)$ and $G(z)=Pr(X_G\le z)$, respectively. 
$F$ first order stochastically dominates $G$ if $F(z)\le G(z)~\forall z\in\mathcal{X}$, where $\mathcal{X}$ is the set of possible $X$, and $F(z)< G(z)$ for at least one  $z\in\mathcal{X}$. Hence, if $X$ is distributed according to $F$, it tends to have higher values, and indeed has a higher expected value. In your problem, it means that default is more likely if $X$ is drawn from distribution $F$ than when it is drawn from $G$.
So one assumption on the distribution of $X$ where your intuition makes sense would be: "If higher values of $X$ are more likely in a FOSD sense, then probability of default is higher". But as the first comment says: as soon as you know the realization of $X$, there is no uncertainty any more.
In terms of economics, another approach you could take is to assume that the firm specific parameter $u_i$ is unknown, but the distribution of $u_i$ is known. Then for a higher $X$ the probability of default would be higher, because it is less likely that $u_i$ is above $X$ to prevent default.
