Question

For each of the random quantities X,Y, and Z, defined below

(a) Plot the probability mass function PMS (in the discrete case) , or the probability density function PDF (in the continuous case)

(b) Calculate and plot the cumulative distribution function CDF

(c) Calculate the mean and variance, and the moment function m(n), and plot the latter.

The random quantities are as follows:
X is a discrete r.q. taking values k=0,1,2,3,... with probabilities p(1-p)^k, where p is a parameter with values in the interval (0,1). Do the plots for a selected value of p (e.g., p=0.3).
Y is a continuous r.q. with the PDF f(x) = C[1/(1+ |x|^(10/3)) ], for all x in R. (find C first) . What can you say about the existence of the moments of different order?
Z is a continuous r.q. with the PDF f(x) = Cx^2, for -1<x<1, and =0 elsewhere. (find C first)

Answer 1