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5982概率中等derivationmedium

Variance of Fills Over a Binomial Number of Quotes

题目

Out of n=10n=10 resting quotes, each fills independently with probability 0.30.3, so the number of fills NN is Binomial(10,0.3)(10,0.3). Each fill produces an i.i.d. PnL XiX_i with E[Xi]=2E[X_i]=2 and Var(Xi)=9\mathrm{Var}(X_i)=9, independent of which quotes fill. For the stopped sum SN=i=1NXiS_N=\sum_{i=1}^N X_i, compute Var(SN)\mathrm{Var}(S_N).

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