Random Stock Prices. A believer in the random walk theory of stock market thinks that an index of stock?

prices has teh 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 teh number of years among the next 5 years in which the index rises.








(a) What are the possible values X can take?





(b) Find te probability of each value of X.





(c) What are the mean and standard deviation of this distribution?








Please help me; I am very confused by binomial Distributions!

Random Stock Prices. A believer in the random walk theory of stock market thinks that an index of stock?
a) what you are describing is a memoryless distribution. That is, the prior year's result has no influence on the current year's result.





In that case, X can increase either 0, 1, 2, 3, 4 or 5 times over the next 5 years.





b) This is a binomial distribution. So





Pr[X=k] = 5Ck p^k (1-p)^(5-k) where nCm = n!/[m!(n-m)!]





Pr[k=0] = 5C0 (0.65)^0 (.35)^5 = 0.005252


Pr[k=1] = 5C1 (0.65)^1 (.35)^4 = 0.048770


Pr[k=2] = 5C2 (0.65)^2 (.35)^3 = 0.181147


Pr[k=3] = 5C3 (0.65)^3 (.35)^2 = 0.336416


Pr[k=4] = 5C4 (0.65)^4 (.35)^1 = 0.312386


Pr[k=5] = 5C5 (0.65)^5 (.35)^0 = 0.116029





c) Here I'm not certain which distribution you are referring to. Assuming it is the number of years that X can increase over the next 5 years, the mean and variance are given by





mean = np = 5(0.65) = 3.25


variance = np(1-p) = 1.1375





The std deviation is the square root of the variance so





std deviation = 1.066536