Let $X$ be a random variable and I now assume for simplicity that it is uniformly distributed on $[0,1]$.
Now fix $t \in [0,1]$ and let $Y_t$ be the random variable $Y_t:=\min \{t,X\}$.
I have problems computing for example the density
Computing the cdf is easy: Of course $P(Y_t \leq y)= 1- P(t>y,X>y)$. Now I tried to split this apart as if $t$ was just random variable independet of $X$ and this gives $P(Y_t \leq y)=y$ if $y
However, I feel that taking the derivative is not the appropriate her since $Y_t$ attains the value $t$ with positive probability. So $Y_t$ seems to be a random variable that is neither a discrete nor a continuous random variable. So maybe a density does not make sense.
However, in order to be able to compute expectation and so on, I would need a density. In fact I guess everything becomes easy if I knew on which space $Y_t$ actually lived. However, I have problems formalizing that. Can anybody help.
Maybe for the uniform distribution everything is easy by direct arguments. But I would like to see a general approach which is valid for $X$ having an continuous distribution.