statistics : Solution Essays

1. Let Ei be the mutually exclusive and collectively exhaustive events with i=1,2,3,…,100; OD be the odds; P be the probabilities.
Assume that OD = (i+1)^2.
Calculate the probabilities P and plot the histogram of i with class boundaries at 0.5,1.5,2.5,…,100.5

2. For question 1, plot the histogram of X where X=cos(pi*i/20)  

3. Let Ei be the mutually exclusive and collectively exhaustive events with i=1,2,3,4,…,Infinity; OD be the odds; P be the probabilities.
Assume that OD = 1/i^2.
Determine whether the event A = (E1 or E3) is independent of event B = (E3 or E4). 

4. Let Ei be the mutually exclusive and collectively exhaustive events with i=1,2,3,…,Infinity; OD be the odds; P be the probabilities.
Assume that OD = 1/i^2.
Let the event outcomes X be given by formula X = i+5.
Calculate the expected value and variance of X.

5. Let Ei be the mutually exclusive and collectively exhaustive events with i=1,2,…,Infinity; OD be the odds; P be the probabilities.
Assume that OD = 1/i^2.
Let the event outcomes X be given by formula X = 1/i.
Plot the sampling distribution of the average of 10 independent observations of X using N=100000 iid samples.

6. Let Ei be the mutually exclusive and collectively exhaustive events with i=1,2,…,10; OD be the odds; P be the probabilities.
Assume that OD = 1.
Let the event outcomes X be given by formula X = tanh(i).
Find the median X.

7. Let Ei be the mutually exclusive and collectively exhaustive events with i=1,2,…,10; OD be the odds; P be the probabilities.
Assume that OD = 1.
Let the event outcomes Y be given by formula Y = exp(-i).
Obtain an N=10000 iid sample of Y and calculate the quantile-quantile plot of the sample vs the normal distribution. Does the distribution appear to be normal? 

8. Calculate and plot the experimental cumulative density function for index X using the latest 500 daily prices of COS.TO stock, where
X=(High-Low)/Close  

9. For question 8, test whether the distribution of X is normal:
A) visually, with the quantile-quantile plot against the normal distribution;
B) using the Anderson-Darling normality test, with a significance level of 0.05;
C) using the Shapiro-Wilk normality test, with a significance level of 0.05.

10. Find the Spearman, Pearson and Kendall correlation between the latest 500 daily closing prices of SU.TO and COS.TO stocks. What do the correlations indicate?

11. Determine whether the daily volatility of SU.TO is higher than, lower than, or equal (three null hypotheses) to volatility of COS.TO by comparing index X=log(High/Close) for latest 50 daily prices for both stocks. Treat X as a sample from a much larger population, for both stocks. Use the significance level of 0.01 in each test.
A) Assume normality of both distributions.
B) Do not assume normality; instead, use the Wilcoxon non-parametric two-sample test. 
C) Do not assume normality; instead, use the two-sample Kolmogorov-Smirnov test.  

12. For the 50 latest daily opening prices of COS.TO stock, find whether the direction of daily changes is random by performing the runs test.

13. Assuming normality, test the null hypotheses for latest 50 closing prices of Suncor (SU) on New York Stock Exchange
Ho:(mean(X)=0)
Ho:(mean(X)>0)
Ho:(mean(X)<0)
where Xi=log(Close(i)/Close(i-1)), i=2,3,..,50
Can either of hypotheses be rejected at significance level 0.01?

14. Assuming symmetry about the median, test the null hypotheses for latest 50 closing prices of Suncor (SU) on New York Stock Exchange
Ho:(median(X)=0)
Ho:(median(X)>0)
Ho:(median(X)<0)
where Xi=log(Close(i)/Close(i-1)), i=2,3,..,50
Can either of hypotheses be rejected at significance level 0.01?