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Derive the unshifted beta-geometric model and its key characteristics: P(t|alpha, beta), S(t|alpha, beta), and the formulas needed to compute P(t|alpha, beta) via forward recursion.

Derive the "unshifted" beta-geometric model and its key characteristics: P(t|alpha, beta), S(t|alpha, beta), and the formulas needed to compute P(t|alpha, beta) via forward recursion.

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