Problem Set 2

Question 1

Obtain daily returns data for Medical Marijuana Inc. (MJNA) between 2 Feb 2016 and 1 Feb 2017.

a. (5 points)

Compute the sample skewness of the data.

Solution:

b. (5 points)

Compute the sample kurtosis of the data.

Solution:

c. (10 points)

Create a normal probability plot of the data, including a line that depicts the normal quantiles. Do your estimates of skewness and kurtosis corroborate what you see in the QQ-plot? If so, how?

Solution:

Question 2

a. (15 points)

Simulate \(N = 100\) data observations from a distribution that is 87% \(\mathcal{N}(0, 1)\), 10% \(\mathcal{N}(0, 7.2)\) and 3% \(\mathcal{N}(0, 11)\)? What is the sample kurtosis of your data?

Solution:

b. (10 points)

Repeat part (a) for \(N = 1000\), \(N = 10,000\) and \(N = 100,000\). What is the sample kurtosis of each of your simulated datasets?

Solution:

c. (5 points)

Compute the true kurtosis of the distribution in part (a). How does this compare to the values computed in parts (a) and (b)?

Solution:

Question 3

(40 points)

Suppose that \(X_1, \ldots, X_n \stackrel{i.i.d.}{\sim} Poisson(\lambda)\). That is \[\begin{align} p_X(x) & = \frac{\lambda^x e^{-\lambda}}{x!}, \end{align}\]

for \(x \in \{0,1,2,\ldots\}\). Compute the MLE of \(\lambda\).

Solution: