•
While some households may be more predisposed to the particular service
e.g. richer households may have a different p from poorer ones; this
couldn’t be determined beforehand. With random sampling (so that one is
likely to ask both rich and poor households) then the constant p represents
an average over the whole population.
(b) Use the binomial distribution tables to calculate: P(
X
= 4), P(
X
< 4),
P(
X
≥ 1).
We have
n=10, p=.2
𝑃
(
𝑋
= 4) = 0.0881
𝑃
(
𝑋
< 4) =
𝑃
(
𝑋 ≤
3) = 0.8791
𝑃
(
𝑋 ≥
1) = 1
− 𝑃
(
𝑋
= 0) = 1
−
0.1074 = 0.8926
3.
A believer in the “random walk” theory of stock markets thinks that an
index of stock prices has a probability of 0.65 of increasing in any one year.
Let
X
be the number of years among the next 5 years in which the index
rises.
(a)
What do we need to assume for
X
to have a binomial distribution.
What are
n
and
p
? What are the possible values that
X
can take?
•
Each of the n=5 years represent identical and independent trials. For this
to be true the fact that the index went up last year does not change the
probability it will go up in the subsequent year.
•
Index going up is denoted a success and all other outcomes (going down or
staying unchanged)d are taken to be a failure.
•
The probability of a success p=0.65 is the same for each trial.
(b)
Assuming
X
has a binomial distribution, construct the probability
distribution of
X
and draw the associated probability histogram.

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