Paper 2, Section II, F

Probability | Part IA, 2013

Let ZZ be an exponential random variable with parameter λ=1\lambda=1. Show that

P(Z>s+tZ>s)=P(Z>t)\mathbb{P}(Z>s+t \mid Z>s)=\mathbb{P}(Z>t)

for any s,t0s, t \geqslant 0.

Let Zint =ZZ_{\text {int }}=\lfloor Z\rfloor be the greatest integer less than or equal to ZZ. What is the probability mass function of Zint Z_{\text {int }} ? Show that E(Zint )=1e1\mathbb{E}\left(Z_{\text {int }}\right)=\frac{1}{e-1}.

Let Zfrac=ZZintZ_{\mathrm{frac}}=Z-Z_{\mathrm{int}} be the fractional part of ZZ. What is the density of ZfracZ_{\mathrm{frac}} ?

Show that Zint Z_{\text {int }} and Zfrac Z_{\text {frac }} are independent.

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