The behavior of a certain stock market index is considered over the next (n+1) consecutive trading days starting from tomorrow. For each day a ‘+’ will be recorded if the index rises above the previous day’s index and a ‘−’ will be recorded otherwise (you may assume that the probability that the index will be the same on two consecutive days is zero). Assume that rises and falls are independent and equally likely and that today’s index exceeds yesterday’s index. (a) Let N be the number of days of the next n for which a ‘+’ will be recorded. (i) Is N a binomial random variable? Justify your answer. (ii) What are the mean and standard deviation of N? (iii) Determine the probability that rises outnumber falls over the next 5 days. (b) The index is said to have a trend on day I if +,+,+ or −,−,− is recorded on days i−1, i, i+1 respectively. Given n 2, let Z be the total number of days of the next n for which a trend will be recorded. (i) Is Z a binomial random variable? Justify your answer. (ii) Show that E(Z) = n/4 [Hint: let Xi = 1 if there is a trend on day i, Xi = 0 otherwise and note that Z = X1 + X2 + · · · + Xn]
Contingency Table
A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.
Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
The behavior of a certain stock market index is considered over the next (n+1) consecutive trading days starting from tomorrow. For each day a ‘+’ will be recorded if the index rises above the previous day’s index and a ‘−’ will be recorded otherwise (you may assume that the
and falls are independent and equally likely and that today’s index exceeds yesterday’s index.
(a) Let N be the number of days of the next n for which a ‘+’ will be recorded.
(i) Is N a binomial random variable? Justify your answer.
(ii) What are the mean and standard deviation of N?
(iii) Determine the probability that rises outnumber falls over the next 5 days.
(b) The index is said to have a trend on day I if +,+,+ or −,−,− is recorded on days
i−1, i, i+1 respectively. Given n 2, let Z be the total number of days of the next n for
which a trend will be recorded.
(i) Is Z a binomial random variable? Justify your answer.
(ii) Show that E(Z) = n/4
[Hint: let Xi = 1 if there is a trend on day i, Xi = 0 otherwise and note that Z = X1 + X2 + · · · + Xn]
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