STAT 400 CD3, CD4
L. Shand
Spring 2017
Discussion 5
An event with only 2 outcomes, say ”sucess” and ”failure” is a Bernoulli trial with p =
probability of a success.
# successes in n Bernoulli trials ∼ Binomial(n, p)
P (X = x) =
n
x
px (1 − p)n−x , E[X] = np, V ar[X] = np(1 − p)
# trials until 1st success ∼ Geometric(p)
P (X = x) = p(1 − p)x−1 , E[X] = p1 , V ar[X] =
CDF : P (X ≤ x) = 1 − (1 − p)x
1−p
,
p2
P (X > x) = (1 − p)x
# successes in n draws w/o replacement from population of size N with K successes.
∼ Hypergeometric(n, p)
P (X = x) =
−K
(Kx )(Nn−x
)
N −K N −n
, E[X] = n K
, V ar[X] = n K
N
N
N N N −1
(n)
Exercises:
1. Jelly Belly makes 16oz bags of jelly beans which contain approximately 400 pieces with
49 different flavors. Although 15% of the time, the machines either under or overfill
the bag. Say you buy 10 of these 1lb. bags.
(a) How many of these bags do you expect to have exactly 400 pieces?
(b) What is the probability that at least 8 of the bags have exactly 400 pieces?
2. For now, assume that their machines fill every bag with exactly 400 jelly beans. Since
it is a popular flavor, they attempt have approximately 80 juicy pear flavor beans in
each bag, i.e. the probability of a randomly selected jelly bean from a bag is juicy pear
flavor is 20%.
(a) What is the expected number of juicy pear jelly beans in a given bag?
(b) What is the expected number of juicy pear jelly beans in 3 bags? Standard
Deviation?
(c) Say your friend will give you $0.25 for every juicy pear jelly bean you have. What
is the expected amount you could earn for a single bag? Standard Deviation?
(d) Consider a handful of 15 jelly beans from a perfect bag. What is the probability
more than 20% of them are juicy pear?Can this be approximated?
(e) Now consider a sample of 150 jelly beans from a perfect bag. What is the probability less than 2% of them are juicy pear?
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