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### Stocks and the Binomial Distribution

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Date: 12/09/98 at 03:06:52
From: Tom
Subject: Probability: binomial distribution

Hi there. I have a question I would like to ask you.

A believer in the 'random walk' theory of the behaviour of stock prices
thinks that an index of stock prices has probability 0.65 of increasing
in any year. Moreover, the change in the index in any given year is not
influenced by whether it rose or fell in earlier years. Let X be the
number of years among the next 6 years in which the index rises.

a) What n are p and in the binomial distribution of X?
b) Give the possible values that X can take and the probability of
each value. Draw a probability histogram for the distribution of
X.
c) Find the mean of the number X of years in which the stock price
index rises according to the random walk stock price model and
mark the mean on your probability histogram for this
distribution.
d) Compute the standard deviation of X. What is the probability
that X takes a value within one standard deviation of its mean?

Actually, I'm not understanding the last part. What is the meaning of
"probability that X takes a value within one standard deviation of its
mean "?

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Date: 12/09/98 at 03:30:48
From: Doctor Pat
Subject: Re: Probability : binomial distribution

Tom,

From the top,

a) Each year is a new "independent trial" so n = 6, and the probability
of a success (which is when the index increases) is p = .65.

b) If you track six years, what is the smallest number of years the
stock could go up (zero?) and the most would be six, so the domain is
0, 1, 2, ..., 6.

Now find the probability of having six failures, (1-.65)^6, then of 1
success and five failures, and so on, to make a probability
distribution, then graph.

c) The mean of a binomial distribution is n*p  = 6*.65 = 3.9 (memorize
this formula and the next, if you are taking the AP exam because it is
on your list of formulae). The standard deviation is sqrt(n*p*q) which
I think we will need for the next question.

d) Find the standard deviation from the formula above and then find
the values for one standard deviation below the mean (call it L, so
L = mean - standard deviation) and one above the mean (call it H, so
H = mean + standard deviation) and now find all the possible values
between 0-6 that are in this interval and add up their probabilities.

Remember that a normal curve has approximately 68% of the curve within
one standard deviation. Binomial curves for large enough n will tend
to be close to this value. (I don't think this is a large enough value
though, 30 is the usual rule of thumb).

Good luck,

- Doctor Pat, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
College Probability
High School Probability

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