I'll bug you guys until I get comfortable enough with all these techniques.

So, I have a cdf $F_{K}$ for a random variable K, which is strictly increasing and continuous function over the entire space of reals.

In order to find the distribution of $F(K)$ I define another random variable as a one-to-one correspondence $M = g(K)$, where $g^{-1}$ is also continuous, increasing over all the real space.

Then, by def of cdf, I have:

$F_{M}(m) = P[M \leq m] = P[g(K) \leq m] = P[K \leq g^{-1}(m)] = F_{K}(g^{-1}(m)).$

Thus, the distribution of the $F_{K}$ is defined as follows:

if $g^{-1}(m) \leq 0 , F_{K} = 0$

if $0 < g^{-1}(m) < 1 , F_{K} = k$

if $1 \leq g^{-1}(m) , F_{K} = 1$.

Kind of makes sense, although if there is a simpler way to define the distribution let me know.

Now, the actual question comes after - assuming that the RV K has a density function $f$, then I need to find the pmf of the $\lceil K \rceil$ (in the sense that for $k = 0.1$, $\lceil k \rceil = 1$, as is the case for the ceiling function).

I know that a pmf of a discrete random variable Y is defined as $P(Y = y)$, but how to transform this continues (or real-valued) K into a discrete and make up a pmf for it, confuses me a bit.

Could you please help me here

Thanks a lot