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Conditional mean from a joint table

ProbabilityEasy

Flow state $X$ and order size $Y$ have this joint distribution: $P (X = 0, Y = 1) = 0.1$ , $P (X = 0, Y = 2) = 0.3$ , $P (X = 1, Y = 1) = 0.4$ and $P (X = 1, Y = 2) = 0.2$ . What is $E [Y ∣ X = 1]$ ? Give the answer as a fraction in lowest terms.

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Hint

Keep only the entries with $X = 1$ , then rescale them by their total.

Solution

The slice with $X = 1$ has total $P (X = 1) = 0.4 + 0.2 = 0.6$ .
Rescale it: $P (Y = 1 ∣ X = 1) = \frac{0.4}{0.6} = \frac{2}{3}$ and $P (Y = 2 ∣ X = 1) = \frac{0.2}{0.6} = \frac{1}{3}$ .
Average over the slice: $E [Y ∣ X = 1] = 1 \times \frac{2}{3} + 2 \times \frac{1}{3} = \frac{4}{3}$ .

Answer: $\frac{4}{3}$

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