A viral linear DNA molecule of length, say, 1 is often known to contain a certain "marked position," with the exact location of this mark being unknown. One approach to locating the marked position is to cut the molecule by agents that break it at points chosen according to a Poisson process with rate $\lambda$. It is then possible to determine the fragment that contains the marked position. For instance, letting $m$ denote the location on the line of the marked position, then if $L_{1}$ denotes the last Poisson event time before $m$ (or 0 if there are no Poisson events in $[0, m])$, and $R_{1}$ denotes the first Poisson event time after $m$ (or $I$ if there are no Poisson events in $[m, 1])$, then it would be learned that the marked position lies between $L_{1}$ and $R_{1} .$ Find
(a) $P\left\{L_{1}=0\right]$
(b) $P\left\{L_{1}<x\right\}, 0<x<m$
(c) $P\left[R_{1}=1\right\}$
(d) $P\left(R_{1}>x\right\}, m<x<1$
By repeating the preceding process on identical copies of the DNA molecule, we are able to zero in on the location of the marked position. If the cutting procedure is utilized on $n$ identical copies of the molecule, yielding the data $L_{i}, R_{i}, i=$ $1, \ldots, n$, then it follows that the marked position lies between $L$ and $R$, where
$$
L=\max _{i} L_{i}, \quad R=\min _{i} R_{i}
$$
(e) Find $E[R-L]$, and in doing so, show that $E[R-L] \sim \frac{2}{n \lambda}$.