4. (a) Show that E(X | X > 0) ≤ E(X²)/E(X) for any random variable X taking non-negative values. (b) Let Zn be the size of the nth generation of a branching process with Zo = 1 and P(Z₁ = k) =qpk for k≥ 0, where p > . Use part (a) to show that E(Zn/µ" | Zn > 0) ≤ 2p/(p-q), where μ = p/q. (c) Show that, in the notation of part (b), E(Zn/" | Zn > 0)→ p/(p-q) as n → ∞o.

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