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Compute E(N^2), for N∼Geom(p), by conditioning on the outcome of the first trial. Use this to...

Compute E(N^2), for N∼Geom(p), by conditioning on the outcome of the first trial. Use this to calculate Var(N)

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Answer #1

The distribution of is

When a failure occurs on the first trial, (conditioning on the failure of first trial)

The conditional expectation is

Now the unconditional expectation is

Now the unconditional expectation is

The variance is

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