the data are only inaccurate due to rounding off when writing down
the table. Which are consistent with the production of a single
isotope, and which imply that more than one isotope was being
created?
7 Devise a method for testing experimentally the hypothesis that
a gambler’s chance of winning at craps is independent of her previous
record of wins and losses. If you don’t invoke the mathematical
definition of statistical independence, then you haven’t proposed a
test. This has nothing to do with the details of the rules of craps,
or with the fact that it’s a game played using dice.
8 A blindfolded person fires a gun at a circular target of radius
b, and is allowed to continue firing until a shot actually hits it. Any
part of the target is equally likely to get hit. We measure the random
distancerfrom the center of the circle to where the bullet went in.
(a) Show that the probability distribution ofrmust be of the form
D(r) =kr, wherekis some constant. (Of course we haveD(r) = 0
forr > b.)
(b) Determinekby requiringDto be properly normalized.√(c) Find the average value ofr.√(d) Interpreting your result from part c, how does it compare with
b/2? Does this make sense? Explain.
9 We are given some atoms of a certain radioactive isotope, with
half-lifet 1 / 2. We pick one atom at random, and observe it for one
half-life, starting at time zero. If it decays during that one-half-
life period, we record the timetat which the decay occurred. If it
doesn’t, we reset our clock to zero and keep trying until we get an
atom that cooperates. The final result is a time 0≤t≤t 1 / 2 , with a
distribution that looks like the usual exponential decay curve, but
with its tail chopped off.
(a) Find the distributionD(t), with the proper normalization.
√(b) Find the average value oft.√(c) Interpreting your result from part b, how does it compare with
t 1 / 2 /2? Does this make sense? Explain.
10 The speed,v, of an atom in an ideal gas has a probability
distribution of the formD(v) =bve−cv
2
, where 0≤v <∞,crelates
to the temperature, andbis determined by normalization.
(a) Sketch the distribution.
(b) Findbin terms ofc.√(c) Find the average speed in terms ofc, eliminatingb. (Don’t try
to do the indefinite integral, because it can’t be done in closed form.
The relevant definite integral can be found in tables or done with
computer software.)
√Problems 941