PDF of sum of truncated exponential distribution

Let xi represent samples from a Truncated Exponential distribution between 0 and 1, with rate parameter λ.

Defining

˜x=ni=1xin

What is the PDF of ˜x?

Answer

Updated answer

The solution is going to be an n-part piecewise pdf on (0,1). Given that the OP has noted he is interested in large n, expressing the exact pdf of the sample mean is likely to get messy. For large n (as given), one should obtain an excellent neat simple approximation via the Central Limit Theorem.

Structure

Let XTruncatedExponential(λ) (truncated above at 1), with pdf:

f(x)=λeλx1eλ for 0<x<1

where:

E[X]=1λ+11eλandVar(X)=1λ2eλ(eλ1)2

Then if the random variables X1,X2,... are iid, by the Central Limit Theorem:

ˉXnaN(E[X],Var(X)n)

All done. The following diagram compares:

  • the EXACT distribution of the sample mean (blue curve) with
  • the asymptotic Normal distribution (dashed red curve)

when the sample size is just n=6:

enter image description here

Even with this tiny sample size, the simple Normal approximation already performs well in the λ=1 case (LHS diagram). If λ becomes larger, the distribution becomes more peaked and shifts to the left, and larger sample sizes will be needed ... but will still perform extremely well for large n.

For comparison, the exact pdf when n=6 is:

enter image description here

Derivation of Exact PDF

To illustrate the calculation of the exact pdf, consider first two independent Truncated Exponential variables, say X and Y which will have joint pdf f(x,y):

enter image description here

Then, the cdf of S=X+Y, i.e. P(X+Y<s) is:

enter image description here

where I am using the Prob function from the mathStatica package for Mathematica to automate the calculation.

The pdf of S=X+Yis just the derivative of the cdf wrt s:

enter image description here

Here is a plot of the exact pdf just derived in the n=2 case (here for the sample sum) when λ=1:

enter image description here

One can derive the exact pdf of the sample sum (or sample mean) for larger n in this same manner ... though for large n, the Central Limit Theorem will rapidly become your friend.

Attribution
Source : Link , Question Author : Diogo Santos , Answer Author : wolfies

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