The exponential is the analog of the geometric in continuous time. this problem explores
the connection between exponential and geometric in more detail, asking what
happens to a geometric in a limit where the bernoulli trials are performed faster and
faster but with smaller and smaller success probabilities.
suppose that bernoulli trials are being performed in continuous time; rather than only
thinking about first trial, second trial, etc., imagine that the trials take place at points
on a timeline. assume that the trials are at regularly spaced times 0,δt,2δ where
δt is a small positive number. let the probability of success of each trial be λδt, where
λ is a positive constant. let g be the number of failures before the first success (in
discrete time), and t be the time of the first success (in continuous time).
(a) find a simple equation relating g to t.
hint: draw a timeline and try out a simple example.
(b) find the cdf of t.
hint: first find p(t> t).
(c) show that as δt→0, the cdf of t converges to the expo(λ) cdf, evaluating all
the cdfs at a fixed t≥0.