# PDF of sum of truncated exponential distribution

Let $x_i$ represent samples from a Truncated Exponential distribution between $0$ and $1$, with rate parameter $\lambda$.

Defining

$\tilde x = \dfrac{\sum_{i=1}^{n}x_i}{n}$

What is the PDF of $\tilde x$?

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 $X \sim \text{TruncatedExponential}(\lambda)$ (truncated above at 1), with pdf:

where:

Then if the random variables ${X_1, X_2, ...}$ are iid, by the Central Limit Theorem:

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 $n = 6$: Even with this tiny sample size, the simple Normal approximation already performs well in the $\lambda = 1$ case (LHS diagram). If $\lambda$ 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: 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 $f(x,y)$: Then, the cdf of $S=X+Y$, i.e. $P(X+Y is: where I am using the Prob function from the mathStatica package for Mathematica to automate the calculation.

The pdf of $S=X+Y$is just the derivative of the cdf wrt $s$: Here is a plot of the exact pdf just derived in the $n= 2$ case (here for the sample sum) when $\lambda = 1$: 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.