Stocks and the Binomial Distribution

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 "?

Please help me. Thank you.

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/