Question

In a statistical survey of 2000 families on ownership of cars 30 families responded that they do not own a car, 470 families owned 1, 850 families owned 2, 490 fam- ilies owned 3, 150 owned 4, and there were 10 families which owned 10 cars each. (a)[5 pts] Plot the sample-based Probability Mass Function (PMF) and Cu- mulative Distribution Function (CDF) of the random variable corresponding to number of cars per family. Calculate the expected number of cars per family and its vari- ance. Show the steps of your calculation.

Answer 1

Let x = No. of cars owned by family. X CDF (F) 0.015 frea (Fi) 30 470 850 490 0.250 Probability (Pi) 30/2000 : 0.015 470 (2000 - 0.235 0.425 0.245 0.075 0.005 0.675 0.920 o 995 150 10 2000 PMF plot: probability (P) -0.05 0 4 10 1 2 3 No. of cars

Plot for CDF 0.8 10 1 2 3 4 No. of cars. b) Expected value: E(x) = 2x. P(x) = 0 x 0.015 + 1xo 235troon + loxo.005 = 2.17 variance ! E (X²) = {X?P(x) = 0% 0.015 + 2² x 0.235ter...-t 10²x0.005 = 5.84 V(x) = E(x²) - 5.84 - 1.1311 [E(X)]2 (2.1702 -