340 CHAPTER 6 Inferential Statistics
Exercises
- Suppose that you are going to toss a fair coin 200 times. Therefore,
you know that the expected number of heads obtained is 100, and
the variance is 50. IfX is the actual number of heads, what does
Chebyshev’s Inequality say about the probability thatXdeviates
from the mean of 50 by more than 15? - Suppose that you are playing an arcade game in which the prob-
ability of winning isp=.2.
(a) If you play 100 times, how many games do you expect to win?
(b) If you play 100 times, what is the probability that you will
win more than 30 games?
(c) If you play until you win exactly 20 games, how many games
will you expect to play?
(d) If you stop after winning 20 games, what is the probability
that this happens no later than on the 90-th game?- Prove that the sum of two independent binomial random variables
with the same success probability pis also binomial with success
probabilityp. - Prove that the result of Exercise 3 is correct if “ binomial” is
replaced with “negative binomial.” - Prove that the sum of two independent Poisson random variables
is also Poisson. - Suppose that N men check their hats before dinner. However,
the clerk then randomly permutes these hats before returning the
hats to the N men. What is the expected number of men who
will receive their own hats? (This is actually easier than it looks:
let Bi be the Bernoulli random variable which is 1 if the man
receives his own hat and 0 otherwise. WhileB 1 , B 2 , ..., BN are
not independent (why not?), the expectation of the sum is still the
sum of the expectations.)