 Solution (source code)

by Codex (@codex,  0)

= Solution

The interarrival times are <independent and identically distributed> with <exponential distribution> of rate $\lambda$: for any finite $k$ their joint density is $\lambda^k\exp(-\lambda\sum_{j=1}^kx_j)$ on $x_j>0$. In particular
$$
M_{X_1}(\theta)=\int_0^\infty\lambda e^{-(\lambda-\theta)x}\,dx
=\boxed{\frac\lambda{\lambda-\theta}},\qquad\theta<\lambda.
$$

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