WORKBOOK ASSIGNMENT 2: BASIC PROBABILITIES AND VISUALIZATIONS (2)

For each of the distributions below, please provide the requested graphics as well as the numeric results. In both cases, please provide how you realized these (calculations, code, steps...) and why it is the appropriate tools. Do not forget to include the scale of each graphics so a reader can read the numbers represented.

a) Consider the variables and . The realization of a sample of size 20 is given below (where is the first variable and is the second): (1.202, 563.024), (2.112, 291.072), (2.827, 893.619), (0.314, 1321.814), (1.477, 91.573), (6.516, 446.336), (0.920, 111.487), (3.477, 153.165), (7.273, 1076.221), (2.251, 477.931), (0.713, 909.696), (0.853, 226.865), (3.176, 389.413), (1.913, 47.169), (1.070, 178.695), (3.385, 744.486), (9.506, 362.670), (7.004, 364.578), (0.504, 324.975), (2.861, 360.571) Sketch an appropriate plot that displays the values of these points. Calculate the sample covariance as well as the sample's expectations and variances of and .

b) Consider that a ball is thrown with a random angle [0, 360) (in degrees) and a random radius [0, 1] (in meters) both independent and uniform. Calculate the density of the variables and (the cartesian coordinates of the point at angle and radius ) as well as their expectation and variance.