Let Ei Be The Mutually Exclusive And Collectively Exhaustive

Let Ei Be The Mutually Exclusive And Collectively Exhaustive Events

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)

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)

Paper For Above instruction

The set of problems presented involves advanced statistical analysis of probabilistic events, stock market data, and hypothesis testing. They cover a wide spectrum of topics including probability calculations, histogram plotting, independence testing, expectation and variance computation, distribution sampling, and several hypothesis tests tailored to financial data. This comprehensive discussion aims to elucidate each problem through theoretical explanations, methodological approaches, and relevant statistical techniques, all contextualized within real-world stock market analysis.

Introduction

Understanding probabilities and their implications plays a critical role in quantitative finance and statistical modeling. When modeling mutually exclusive and collectively exhaustive events, accurate calculation of probabilities and related measures like odds is essential. Such models underpin the analysis of stock price movements, risk assessment, and the testing of underlying assumptions about data distributions. This paper addresses these concepts systematically, applying probability theory, hypothesis testing, and statistical sampling to analyze given scenarios involving stock data and theoretical events.

Problem 1 & 2: Probabilities and Distribution of Events

Consider the first problem where Ei are mutually exclusive and collectively exhaustive events from 1 to 100, with odds OD expressed as (i+1)². Probabilities P are derived from odds using the relationship P = OD / (1 + OD). Substituting OD = (i+1)², yields:

P(i) = (i+1)^2 / [1 + (i+1)^2].

This formula ensures that all probabilities sum to unity across the 100 events, satisfying the axioms of probability. Plotting the probability histogram with class boundaries at 0.5, 1.5, ..., 100.5 provides a visual insight into the distribution of these event probabilities, highlighting how the likelihood varies with increasing i.

For the second problem, plotting the histogram of X=cos(pi*i/20) for i=1 to 100 gives a distribution reflecting the oscillatory nature of cosine over discrete intervals, illustrating the periodic behavior and distribution spread of the function over the specified domain.

Problem 3: Independence of Events

With odds OD = 1/i^2 for i=1 to infinity, probabilities are derived as P(i) = OD / (1 + OD) = 1/i^2 / (1 + 1/i^2). The events A = (E1 or E3) and B = (E3 or E4) are tested for independence via P(A ∩ B) = P(A) * P(B). Calculating these probabilities involves summing over the relevant event combinations, and comparing the joint probability with the product of individual probabilities confirms their independence or dependence.

Problem 4 & 5: Expectations, Variance, and Sampling Distribution

Using the derived probabilities, the expected value of X= i+5 is obtained as E[X] = Σ i P(i), summing over all i. Variance is computed as Var[X] = E[X^2] - (E[X])^2, where E[X^2] = Σ (i+5)^2 P(i).

For the sampling distribution, X=1/i with OD = 1/i^2 is considered. Generating 100,000 IID samples, the distribution of the mean of 10 observations is approximated. The Central Limit Theorem indicates that the sampling distribution tends toward normality as the sample size increases, which can be visualized through histograms or density plots.

Problems 6 & 7: Statistical Quantiles and Distribution Normality

In problem 6, with outcomes X = tanh(i), the median can be approximated by evaluating the 50th percentile of the sample for i=1 to 10, considering the symmetry of tanh about zero. The median can also be analytically deduced recognizing the monotonicity of tanh.

In problem 7, Y=exp(-i) for i=1 to 10, and an iid sample of N=10,000 is used to construct a QQ plot against the standard normal distribution. Deviations from the straight line indicate non-normality, often attributable to skewness or kurtosis inherent in the exponential decay function.

Problems 8 & 9: Stock Data Analysis and Normality Testing

Using actual stock data, the empirical cumulative distribution function (ECDF) of X=(High-Low)/Close over 500 days quantifies the distribution of intraday volatility. Testing the normality of X involves visual tools like the QQ plot and formal tests including Anderson-Darling and Shapiro-Wilk. These tests assess whether the data conforms to a normal distribution, essential for parametric inference.

Problem 10: Correlation Analysis

The correlation between stock indices SU.TO and COS.TO is estimated using Pearson's r, Spearman's rho, and Kendall's tau, each capturing different dependence structures. High positive correlations suggest synchronized price movements, affecting portfolio diversification and risk management strategies.

Problems 11-14: Volatility, Randomness, and Hypotheses Testing

Volatility comparison employs indices X=log(High/Close) over 50 days, tested via normal-based t-tests, Wilcoxon, and Kolmogorov-Smirnov tests to determine statistical significance. Non-parametric methods are preferred when data normality cannot be assumed.

Runs tests evaluate the randomness of daily price changes, testing the hypothesis of independent, identically distributed sequences. Similarly, tests concerning the mean and median of the log returns assess whether the underlying distribution centers around zero, indicative of market efficiency and lack of bias in price changes.

Conclusion

This comprehensive analysis integrates probability theory, statistical inference, hypothesis testing, and real stock data application to underpin decision-making and risk assessment in financial markets. The multifaceted approach ensures robust insights into event probabilities, distribution characteristics, dependence structures, and market behavior, demonstrating the importance of rigorous statistical methods in financial analytics.

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