SECTION 6.1 Discrete Random Variables 341
- My motorcycle has a really lousy starter; under normal conditions
my motorcycle will start with probability 1/3 when I try to start it.
Given that I need to recharge my battery after every 200 attempts
at starting my motorcycle, compute the probability that I will have
to recharge my battery after one month. (Assume that I need to
start my motorcycle twice each day.) - On page 334 we saw that if we toss a fair coin in succession, the
expected waiting time before seeing two heads in a row is 6. Now
play the same game, stopping after the sequenceHT occurs. Show
that expected length of this game is 4. Does this seem intuitive? - Do the same as in the above problem, comparing the waiting times
before seeing the sequences THH versusTHT. Are the waiting
times the same? - (A bit harder) Show that on tossing a coin whose probability of
heads isp the expected waiting time before seeing k heads in a
row is
1 −pk
(1−p)pk.
- As we have seen the binomial distribution is the result of witness-
ing one of two results, often referred to as success and failure. The
multinomial distribution is where there is a finite number of
outcomes,O 1 , O 2 , ...,Ok. For example we may consider the out-
comes to be your final grade in this class: A, B, C, D, or F. Suppose
that on any given trial the probability that outcomeOi results is
pi, i= 1, 2 ,...,kNaturally, we must have thatp 1 +p 2 +···+pk= 1.
Again, to continue my example, we might assume that my grades
are assigned according to a more-or-less traditional distribution:
A: 10%
B: 20%
C: 40%
D: 20%
F: 10%
If we performntrials, and we denotebyXithe number of times we witness outcomeOi, then the proba-
bilities in question are of the formP(X 1 =x 1 , X 2 =x 2 , ...,Xk=