Can someone help me solve this.

E[X ] and OfA ] In terms of how many years are expected before in S.. Part 3 [Written] (Theoretical Analysis - Exact Computations - (Optional)): Compute the expected value, variance, and standard deviation of the random variable X. See the additional notes for discussion on these computations, in brief: (Expected Value) E[X] = _ _ ji . P(X = i) . (Variance) var[X] = E[(X - E[X])2] = E[X2] - (E[X])2 (Standard Deviation) o[X] = Vvar[X] Hint: For computing these without directly manipulating the infinite summations that appear in the definitions of E[X] and var[X], let A be the event that a 1 is rolled on the first throw and A' be the complementary event, namely, that a 1 is not rolled on the first throw. It is clear that E[X A'] = E[1 + X] = 1 + E[X] since what is rolled after the first roll is just like starting over. It is also clear that E[X | A] = 1. The Law of Total Expectation (see notes) gives: E[X] = E[X | A] . P(A) + E[X A'] . P(A') = P(A) + (1 + E[X]) . (1 - P(A)) This makes it quite simple to find E[X]. A similar "trick" can be used to find E[X2], here you will use E[X2 | A'] = E[(1 + X)?] = E [1 + 2X + X2] = 1 + 2E[X] +E[X2]. Again, the Law of Total Expectation gives: E[X2] = E[X2 | A] . P(A) + E[X2 | A'] . P(A')