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Quantitative Finance

Probability Theory

Expected Value - Problem Solving


What is the expected number of coin flips we must make, before we see a Head followed immediately by a Tail?

Let X, Y, ZX, \ Y, \ Z be random variables over Z3, Z3, Z4,\mathbb{Z}_{3}, \ \mathbb{Z}_{3}, \ \mathbb{Z}_{4}, respectively, where Zn\mathbb{Z}_n denotes {1,2,,n}.\{1, 2, \dots, n\}. If PX,Y,Z(x,y,z)=xy144,P_{X, Y, Z}(x, y, z) = \frac{xy}{144}, what is the greatest integer less than or equal to E[X+YZ]?E[X + Y | Z]?

Suppose E[XY=0]=1E[X|Y=0] = 1 and E[XY=1]=9.E[X|Y=1] = 9. When P(Y=0)=0.2P(Y=0) = 0.2 and P(Y=1)=0.8,P(Y=1) = 0.8, what is E[X]?E[X]?

Given Z=9,Z=9, if XX and YY are uniform random variables over {0,1,,9},\{0, 1, \dots, 9\}, what is E[X+YZ=9]? E[ X + Y | Z=9 ] ?

When flipping a fair coin repeatedly, what is the expected value of the number of trials needed to get two tails in a row?


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