PDF of double truncated exponential function [closed]

Question

I am trying to find the pdf of a double truncated exponential distribution using the following code:

x = TruncatedDistribution[{t1, t2}, ExponentialDistribution] // PiecewiseExpand
fx = PDF[x]
fx

but an errror occurs.

Answer 1

You can obtain it as follows:

truncated = TruncatedDistribution[{t1, t2}, ExponentialDistribution[lambda]];
Assuming[{t1 > 0, t2 > t1}, PDF[truncated, x]]

(* Out:
Piecewise[
 {
  {lambda/(E^(lambda*x)*(E^((-lambda)*t1) - E^((-lambda)*t2))), t1 - x < 0 && t2 - x >= 0 && x >= 0}, 
  {0, x < 0 || t1 - x >= 0 || x < 0 || t2 - x < 0}
 },
 Indeterminate
]
*)

For a graphical comparison, invluding the full non-truncated distribution from which the truncated one derives (choosing arbitrary values of $\lambda,t_1,t_2$):

values = {lambda -> 1, t1 -> 1/2, t2 -> 2};
full = ExponentialDistribution[lambda];

Plot[
 Evaluate[{
    1/(CDF[full, t2] - CDF[full, t1]) PDF[full, x],
    PDF[truncated, x]
    } /. values
 ],
 {x, 0, 6},
 Exclusions -> None, PlotRange -> All, AspectRatio -> 1,
 PlotLegends -> {"full", "truncated"},
 AxesStyle -> Directive[Black, 14], Filling -> Axis
]