Let X be a (discrete) random variable with mass point xi and probabilities pi, i.e., Pr(X=xi)=pi. Let FX(x)=Pr(X≤x) denote the CDF of X. Suppose FX(0)=0 and FX(1)=1, that is: X∈[0,1].
I want to defined a smoothed version of X where the CDF F˜X of ˜X is equal to that of X at the points xi and 0 and 1, but the function F˜X is piecewise linear, that is, the value of F˜X at any point other than xi is linear interpolation between the given points.
The quastion is, do you see a nice way to describe the F˜X in terms of FX.
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