HW3

Problem 3.1 (Probability Distributions):

Complete the following tasks for each of the probability distributions below:

• Generate plots of the density, CDF, and the quantile (inverse-CDF) functions Links to an external site.
• Report the first 4 moments (mean, variance, skewness, kurtosis) Links to an external site.
• Complete the discrete probability distributions table below. The cell values in the table represent the values of the quantile function for the corresponding p-value (column) and distribution (row).

Problem 3.2 (Matrix equation solution):

Solve the following system of linear equations and validate your solution. Validate your solution.

6x + 3y - 3z + w = 2
7x + y + 2z + 2w = 5
5x + 3y - 3z + w = 3
-6x - 2y + 3z = 6

 Problem 3.3 (Dimensionality reduction)

Use PCA and t-SNE to analyze and interpret the monthly US Federal Reserve Monetary-Base Data (1959-2009) Links to an external site..

Problem 3.4 (Least Squares Estimation)

Use the SOCR Knee Pain dataset Links to an external site., extract the RB = Right-Back locations (x,y), and fit in a linear model for vertical location (y) in terms of the horizontal location (x). Display the linear model on top of the scatter plot of the paired data.

Problem 3.5 (Extra Challenge)

Generate a 3D plot (static or dynamic, and include snapshots for your report) of the following 3 features of the ALS dataset: Age_mean; ALSFRS_slope and trunk_median. The plot should have a 3D scatter of all subjects and two planes representing the 2D PCA & ICA (or PCA & t-SNE) dimensionality reductions.

Remember that 2D planes in R³ are defined by aX + bY +cZ +d = 0, where the normal vector to the plane is (a,b,c) and the free-parameter "d" is defined so that the plane passes through the mean - a special point in the 3D scatter plot. That is once you compute the first 2 principal vectors v1 and v2, their cross-product (v1 x v2) is the plane normal vector (a,b,c). Then force the plane to go through the scatter mean and this will restrict the final "d" (free) parameter.

Please try to come up with elegant solutions and pretty visualizations with detailed documentation.