Question

Between 1896 when the Dow Jones Index was created and 2009, the index rose in 64% of the years. Sample size is 5 years. Based on this information and assuming a binomial distribution, answer the following questions

Build the probability distribution

Build the cumulative distribution

Graph the probability and cumulative distribution

Find the probability the stock market will rise one year in the next five years?

Find the probability the stock market will rise in three years of the next five years?

Find the probability the stock market will rise in at 3 or less years of the next five years?

Find the probability the stock market will rise in at 3 or more years of the next five years?

Find the mean number of stock market rises on the sample of 5 years?

Find the standard deviation of stock market rises on the sample of 5 years?

For this situation, what assumption of the binomial distribution might not be valid?

Answer 1

The probability distribution is,

X	f(X)
0	5C0 * 0.640 * (1 - 0.64)5-0 = 0.006046618
1	5C1 * 0.641 * (1 - 0.64)5-1 = 0.053747712
2	5C2 * 0.642 * (1 - 0.64)5-2 = 0.191102976
3	5C3 * 0.643 * (1 - 0.64)5-3 = 0.339738624
4	5C4 * 0.644 * (1 - 0.64)5-4 = 0.301989888
5	5C5 * 0.645 * (1 - 0.64)5-5 = 0.107374182

The cumulative distribution is,

X	F(X)
X < 0	0
0 X < 1	0.006046618
1 X < 2	0.006046618 + 0.053747712 = 0.05979433
2 X < 3	0.05979433 + 0.191102976 = 0.2508973
3 X < 4	0.2508973  + 0.339738624 = 0.5906359
4 X < 5	0.5906359 +  0.301989888 = 0.8926258
5 X	0.8926258 + 0.107374182 = 1

probability the stock market will rise one year in the next five years = P(X = 1) = 0.053747712

probability the stock market will rise in three years of the next five years = P(X = 3) = 0.339738624

probability the stock market will rise in at 3 or less years of the next five years = F(3) = 0.5906359

probability the stock market will rise in at 3 or more years of the next five years = 1 - F(3) = 1 - 0.5906359 = 0.4093641

mean number of stock market rises on the sample of 5 years = np = 5 * 0.64 = 3.2

standard deviation of stock market rises on the sample of 5 years =

= 1.073313

For this situation, the assumption of the binomial distribution might not be valid is that the index rise in each year is independent of previous years.