assignment 3 1. Let X be the maximum of the number of heads obtained when Carlos and Michael each flip a fair coin twice. (a) Describe the underlying space S of this random experiment and specify the probabilities of its elementary events. (b) Show the mapping form S to SX , the range of X. (c) Find the probabilities for the various values of X. 2. A dart is equally likely to land at any point inside a circular target of radius 2. Let R be the distance of the landing point from the origin. (a) Find the sample space S and the sample space of R, SR . (b) Show the mapping form S to SR . (c) The \"bull's eye\" is the central disk in the target of radius 0.25. Find the event A in SR corresponding to \"dart hits the bull's eye.\" Find the equivalent event in S and P [A]. (d) Find and plot cdf of R. 3. The cdf of the random variable X is given by: 0 0.5 FX (x) = (1 + x)/2 1 x 10]. 4. The random variable X has cdf shown in the figure below. (a) What type of random variable is X ? (b) Find the following probabilities:P [X 0, 0 if t = 0, and 1 if t