Answer: 10.3

To find the conditional standard deviation of Y given X = 7, we need to: **Step 1: Find the conditional probabilities P(Y|X=7)** First, calculate the marginal probability P(X=7): P(X=7) = 0.05 + 0.03 + 0.13 + 0.11 = 0.32 Now, conditional probabilities: P(Y=10|X=7) = 0.05 / 0.32 = 0.15625 P(Y=20|X=7) = 0.03 / 0.32 = 0.09375 P(Y=30|X=7) = 0.13 / 0.32 = 0.40625 P(Y=40|X=7) = 0.11 / 0.32 = 0.34375 **Step 2: Calculate conditional mean E[Y|X=7]** E[Y|X=7] = (10 × 0.15625) + (20 × 0.09375) + (30 × 0.40625) + (40 × 0.34375) = 1.5625 + 1.875 + 12.1875 + 13.75 = 29.375 **Step 3: Calculate conditional variance Var[Y|X=7]** E[Y²|X=7] = (100 × 0.15625) + (400 × 0.09375) + (900 × 0.40625) + (1600 × 0.34375) = 15.625 + 37.5 + 365.625 + 550 = 968.75 Var[Y|X=7] = E[Y²|X=7] - (E[Y|X=7])² = 968.75 - (29.375)² = 968.75 - 862.890625 = 105.859375 **Step 4: Calculate conditional standard deviation** σ[Y|X=7] = √105.859375 ≈ 10.29 Therefore, the conditional standard deviation is approximately 10.3, which corresponds to option A.

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