Suppose the random variables X, Y, Z are obtained by spinning the adjoining roulette wheel, with X given by the outermost circle, Y given by the intermediate circle and Z given by the inner circle. (a) Calculate E[Z] (the expectation of Z). (b) Calculate E[X + YJ. (c) Calculate E[X • Z]. Is any of the three random variables X, Y, Z dependent on another? Justify your answer. (e) Are any of the three pairs (X, Y), (X, Z), (Y, Z) independent? If so, which? Justify your answer. (1) Compute P[Y=4 | Z=1]. (g) Compute E[Y|Z=1]. ( 2 .5 -3 1/6 1/12 | 1/12

Statement reason 1. δabc is similar to δced. given 2. 3. definition of slope 4. slope of slope of definition of slope 5. slope of × slope of multiplying the slopes 6. slope of × slope of substitution property of equality 7. slope of × slope of simplifying the right side the table contains the proof of the relationship between the slopes of two perpendicular lines. what is the reason for statement 2? a. parallel line segments that meet a common perpendicular line are proportional in length. b. the lengths of vertical and horizontal sides in congruent triangles are in a common ratio. c. trigonometric identities determine the lengths of the legs in a right triangle. d. corresponding side lengths in similar triangles are proportional in length.