A viral linear DNA molecule of length, say, 1 is frequently known to contain certain "marked position," with exact location of this mark being unknown. One approach to locating markedposition is to cut molecule by agents that break it at points chosen according to Poisson process with rate λ. It is then possible to find out fragments which contains marked position. For example, letting mdenote location on line of the marked position, then if L1 denotes last Poisson event time beforem (or 0 if there are no Poisson events in[0,m]), and R1 signifies the first Poisson event time after m (or 1 if there are no Poisson events in [m,1]), then it would be learned that marked position lies betweenL1 and R1.
Determine:

(a) P{L1 = 0}
(b) P{L1 < x}, 0 < x (c) P{R1 = 1}
(d) P{R1 > x}, m < x< 1

By repeating preceding process on identical copies of DNAmolecule, we are able to zero in on location of  marked position. If cutting procedure is used on identical copies of molecule, yielding data Li, Ri, i = 1,...,n, then it follows that marked position lies between L and R where

(e) Determine E[R - L], and in doing so, illustrate that E[R - L] ~ 2/nλ