Subject | Mathematics |

Due By (Pacific Time) | 12/15/2016 12:00 pm |

**Conceptual Problems: Probability Density Function and Binomial Convergence **

**Problem 1 **

a. Use Excel/Megastat to find the discrete probability and cumulative probability of the Binomial distribution with probability of success p = 0.4 and n = 30. Find its mean and variance.

b. Based upon the chart on Excel, what can you conclude about the binomial convergence?

**Problem 2 **

Use Excel/Megastat to create the density probability plot of normal distribution. Take Î¼ = 100 and Ïƒ = 7. Set up the range values of x from 80 to 120.

**Problem 3**

Demonstrate the Central Limit Theorem (CLT) of sample mean by sampling a 100 uniform distribution data with 50 variables. Verify the result by computing the sample mean, sample variance and sketch the histogram on Excel/Megastat.

**Practical Problems: Historical Trends and Patterns in Stock Prices **

Choose two publicly traded companies with a sample size of 50 and a time range (long term/short term) for its stock prices so that you can download the data from quote.yahoo.com or google.com/finance

**Problem 3 **

Use a statistical package (Excel/Megastat) to find the summary measurements: min, max, mean, median, standard deviation of the historical prices of 2 stocks

a.Â Â Â Â Draw a Box Plot for both stocks that you picked. Identify the highest and lowest stock prices.

b.Â Â Â Â Repeat this exercise with histogram/bar-charts and scatter plots for each stock.

c.Â Â Â Â Write a short paragraph describing how the two stock prices are different.

Hint: Base your answer on any notable differences you observe in the two Box Plots.

Â **Problem 4 **

Use Megastat/Excel to find the value of the linear correlation coefficient between these 2 stock prices? Is the correlation significant? Explain the reason for your answer.

a. Â If the correlation is significant, what does it imply about the trend in the predicted stock?

b. Â Find the equation for the least squares regression (LSR) line.

c. Â Interpret the meaning of the slope of the LSR line.

d. Based on the equation of LSR line, what is the â€œbest predictedâ€ value for the stock that you treated as a dependent variable?

**Problem 5 **

Run the hypothesis testing for difference in mean prices of 2 stocks that you picked from the sample size of 50. Compare the result with equal/unequal variances.

**Problem 6 **

Test the hypothesis that the periodic returns follow a normal distribution, as required by certain models of mathematical finance. Predict the periodic returns that you want to be tested for both population stock prices and run the hypothesis testing based upon your prediction.

Hint: Use T.DIST.RT to test the hypothesis and periodic return r = __Price _{2} â€“ Price_{1}__Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Â Â Â Â Â Price_{1}

**Problem 7 **

Investigate models which predict the stock price at time â€˜tâ€™ as a function of the stock price at previous times (S_{t-1 }and S_{t-2} for example).

Hint: Run a multiple linear regression model.Â Â Â

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