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) A vote with outcome ππππππ or ππππππππππππππ follows a Bernoulli distribution where ππ(vote = "ππππππ") = 0.25. Represent the proportion of “for” and “against” in this single Bernoulli trial using a graphics and a percentage. Can an expectation be calculated? Justify your answer. b) The number of meteorites falling on an ocean in a given year can be modelled by a Poisson distribution with an expectation of ππ = 71. Explain why a Poisson distribution is a natural candidate for this phenomenon. Give a graphic showing the probability of one, two, three… meteorites falling (until the probability is less than 0.5%). Calculate the median and variance and show them graphically on this graphic. c) Let ππ be the random variable with the time to hear an owl from your room’s open window (in hours). Assume that the probability that you still need to wait to hear the owl after π¦π¦ hours is: 0.Μ Μ 6Μ 0e−0.5 π¦π¦ + 0.Μ Μ Μ 3Μ 9. 6e−0.25 π¦π¦ (attention: periodic numbers 2 Find the probability that you need to wait between 2 and 4 hours to hear the owl, compute and display the probability density function graph as well as a histogram by the minute. Compute and display in the graphics the mean, variance, and quartiles of the waiting times. 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 ππ and ππ. The realization of a sample of size 20 is given below (where ππ is the first variable and ππ is the second): (0.491, -193.12), (3.225, 493.81), (0.919, 319.33), (4.087, 125.92), (0.761, 221.01), (-1.965, -146.63), (-9.91, 1085.41), (3.882, -154.43), (3.166, - 344.76), (-11.081, 669.45), (6.252, -145.66), (0.863, 68.13), (2.26, 251.05), (0.329, 371.23), (0.231, 385.58), (4.713, -291.75), (0.246, -678.01), (2.775, 558.7), (-3.024, -122.3), (-9.453, -330.81) 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 ππ and ππ. b) Consider that a ball is thrown with a random angle ππ ∈ [0, 360] (in degrees) and a random radius ππ ∈ [0, 1] (in meters) both independent and uniform. Calculate the density of the variables ππ and ππ (the cartesian coordinates of the point at angle ππ and radius ππ) as well as their expectation and variance
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