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

First let's find the GCF of 27, 36, and 72:
-->Find the factors of each number:
27: 1, 3, 9, 27
36: 1, 2, 3, 4, 9, 12, 18, 36
72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
-->Now look for the factors they have in common.
1, 3, and 9
-->Now see which one is the biggest number in value
9
--> So therefore, 9 is the GCF

Now let's find the LCM of 7, 4, 10, and 12
-->Start listing all the multiples of of each number
4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40... 420
7: 7, 14, 21, 28, 35, 42, 49, 56, 64, 70... 420
10: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 120... 420
12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120... 420
(sorry got bored of typing so I fast forwarded at the '...'s)
--> The least number in value that they have in common is the LCM of those numbers
The LCM is 420

*Just saying that took forever to do that LCM part but not way too long thank goodness :p