Questions / Quant research, Probability & inference
Improve an estimator by conditioning
For S = X₁ + X₂, what is E[X₁ | S = 1]?
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Answer: 0.5 probability (1/2)
- Given one success, it is equally likely to be X₁ or X₂.
- E[X₁ | S = 1] = 1/2.
- In general E[X₁ | S] = S/2.
Interview shortcut. Symmetric split: S/2.
Takeaway. Rao–Blackwellizing splits the sufficient statistic symmetrically.
In the app this has a follow-up where one thing about the setup changes. Try the follow-up.