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Expected Value

If you play roulette, are you going to break-even? Maybe sometimes, but not if you play forever, because the expected value is negative. The expected value finds the average outcome of random events.

Expected Values


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

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

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

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

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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