A believer in the \"random walk\" theory of stock markets thinks the value of an

Question

A believer in the "random walk" theory of stock markets thinks the value of an index of stock prices has a 0.8 probability of rising in any year. The change in the value of the index in any given year is not affected by whether it rose or fell in earlier years. You plan to record the value for the index in each of the next eight years. Let the random variable, X, represent the number of years (out of the next eight years) in which the value of the index rises. If X is distributed according the binomial probability distribution, what are the values for n and p? What are the possible values X can take, or equal? Calculate the probability X = 5. Calculate the expected value for X. Calculate the standard deviation for X. Use two or fewer words to describe the shape of the distribution of X.

Explanation / Answer

( a )

Binomial Distribution:

n = 8

p = 0.8

( b )

X can the following values:

x = 0 , 1 , 2, 3, 4, 5, 6, 7 and 8

( c )

P ( x = 5 )

P ( x) = C (n,x) *p^x * (1-p)^(n-x)

P (x=5) = C(8,5) *0.8^5*(1-0.8)^(8-5)

              = 56 * 0.32768 * 0.008

             = 0.1468

Answer: 0.1468

( d )

Mean = np = 8 *0.8 = 6.4

( e )

Standard deviation = Önpq = Ö(8*0.8*(1-0.8)) = Ö1.28 = 1.13

( f )

Here the sample size is small (<30) the shape of the distribution will not be approximately normal. It will be right skewed.

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