Let $X$ be a (discrete) random variable with mass point $x_i$ and probabilities $p_i$, i.e., $Pr(X=x_i)=p_i$. Let $F_X(x)=Pr(X \leq x)$ denote the CDF of $X$. Suppose $F_X(0)=0$ and $F_X(1)=1$, that is: $X\in[0,1]$.
I want to defined a smoothed version of $X$ where the CDF $F_{\tilde{X}}$ of $\tilde{X}$ is equal to that of $X$ at the points $x_i$ and 0 and 1, but the function $F_{\tilde{X}}$ is piecewise linear, that is, the value of $F_{\tilde{X}}$ at any point other than $x_i$ is linear interpolation between the given points.
The quastion is, do you see a nice way to describe the $F_{\tilde{X}}$ in terms of $F_X$.
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