Probability & Statistics

25 methods · 72 worked questions using these methods

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To find the expected count of something (matches, rings, runs, distinct items collected), write the count as a sum $X = i \sum I_{i}$ of $0/1$ indicators and use $E (X) = i \sum P (I_{i} = 1)$ , which needs no independence. For the variance, expand $E (X^{2}) = i \sum E (I_{i}) + i \neq = j \sum E (I_{i} I_{j})$ using only pairwise probabilities.

Trigger: "expected number of" a count that is awkward to handle directly but is a sum of simple yes/no events.

Instances:
(i) $I_{i} = 1$ if object $i$ has property (correct shirt, fixed card), so $E = \sum P (I_{i} = 1)$ even when the $I_{i}$ are dependent;
(ii) collector problems where the wait splits as $N = N_{1} + \dots + N_{r}$ of independent geometric stages;
(iii) variance via the pair probability $P (I_{i} = 1 and I_{j} = 1)$ .

Linked questions for this method (3)

STEP 2 2008 · Q12

STEP 3 2008 · Q13

STEP 2 2011 · Q13