Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2003 / Paper 2 / 5 / b

An integrable nonnegative probability density of this form requires $k>0$. Normalization gives $1=A{\int }_0^{\infty }{e}^{-kt}dt=A/k$, so $\boxed{A=k}$. This is the exponential distribution with rate $k$.

The survival function of this exponential distribution is $\mathbb{P}(X\ge t)=e^{-kt}$ for $t\ge 0$. Since $\{X\ge 3\}\subseteq \{X\ge 1\}$, the conditional probability isThis is the memoryless property: after surviving the first unit, surviving two additional units has the same probability as it did initially.

The expectation is $\mathbb{E}[X]=k{\int }_0^{\infty }t{e}^{-kt}dt$. An integration by parts, with the boundary term $t{e}^{-kt}$ tending to zero, gives