Question

(b) Consider the experiment on pp. 149-156 of the online notes tossing a coin three times). Consider the following discrete random variable: Y = 2[number of H-3[number of T). (For example, Y (HHT) = 2.2-3.1=1, while Y (TTH) = 2.1-3.2 = -4.) Repeat the analysis found on pp. 149-156. That is, (i) find the range of values of Y: (ii) find the value of Y(s) for each s ES: (iii) find the outcomes in the events A -Y = for each possible value i of Y: (iv) find the probability mae function f(y) =PY = y) for each value y in the Range(Y) (v) find the cumulative distribution function F(a) - P(Y S a) of Y as it was done on p. 156 in the notes (vi) graph the cumulative distribution function Fla) = P(Y S a) as a function of a € (-0, 0) (it should be a step function). This is not in the notes, but it was done in class (see also Fig. 3.3, p. 87, in the textbook for another example) (c) Consider the experiment of rolling two dice, and let X be the sum of the two numbers. (For example, X((2, 3)) = 5 = X((3,2)).) Repeat the above analysis from pp. 149-156 in the notes. That is, repeat parts (1)-(vi) above.
part C

Answer 1

Experiment sum of WUUUU of the rolling two dice and x two neembers that show represents up. the (1,1) (6,6) outcome outcome Minimum value for X = 2 Maximum value for X = 12 Range of x = {2,3,.., 12} integers. S = 11,00 (1,2), ---,0,6), (2,1), (2,2), . . . (2,6), . {2,12] but all & by for each summing e.g. VVVV X=2 X -3 x =4 x =5 Nm I noot as (6, 1), (6,2), ..., 66,6) } ses, we can very easily find X up the two numbers s= (11) X(8) = 2 s = (3,5) X(6) = 8 { 16,1)} {(1, 2), (2,3)} {(1,3), (2, 2), (3,0} $04), (2,3), (3,2), (4,1); f05),(2,4), (3,3),(4,2);(5,)} f (1,6), (2,5),(3,4), 14,3), (5,2), (6,107 { (2,6), (3,5), (4,4), (5,3), (6,2)} x=7 x=8. { (6,6)} X = 12 favourable outcomes in faut cing, help of X 5 4 36 civ Cleanly with the we have pmf 1 2 3 4 6(x) t 2 3 36 36 37 of as 6 5 37 7 12 8 5 9 4 10 3 11 2 6 1 5 4 3 2 36 36 36 36

(V) Fla) = Plica) a 2 3 4 5 7 8 9 10 11 12 Fla) Bolso (vi) The above be photted and Fla) probabilities (cumulative as a step function on y-ans. probabilities can with ' l on x-anis