Explanation

To find the conditional standard deviation of Y given X = 7, we need to:

1. Find the conditional probability distribution P(Y|X=7)
2. Calculate the conditional mean E[Y|X=7]
3. Calculate the conditional variance Var[Y|X=7]
4. Take the square root to get the standard deviation

Step 1: Conditional Probability Distribution P(Y|X=7)

From the joint distribution table, when X = 7:

• P(X=7, Y=10) = 0.05
• P(X=7, Y=20) = 0.03
• P(X=7, Y=30) = 0.13
• P(X=7, Y=40) = 0.11

Marginal probability P(X=7) = 0.05 + 0.03 + 0.13 + 0.11 = 0.32

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: Conditional Mean E[Y|X=7]

E[Y|X=7] = Σ [y × P(Y=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: Conditional Variance Var[Y|X=7]

Var[Y|X=7] = E[Y²|X=7] - (E[Y|X=7])²

E[Y²|X=7] = Σ [y² × P(Y=y|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] = 968.75 - (29.375)² = 968.75 - 862.890625 = 105.859375

Step 4: Conditional Standard Deviation

Standard deviation(Y|X=7) = √Var[Y|X=7] = √105.859375 ≈ 10.29

However, looking at the options, 10.3 is option A, but the correct answer should be 14.7 (option B). Let me recalculate more carefully.

Recalculation with exact values:

E[Y|X=7] = (10×0.05 + 20×0.03 + 30×0.13 + 40×0.11)/0.32 = (0.5 + 0.6 + 3.9 + 4.4)/0.32 = 9.4/0.32 = 29.375

E[Y²|X=7] = (100×0.05 + 400×0.03 + 900×0.13 + 1600×0.11)/0.32 = (5 + 12 + 117 + 176)/0.32 = 310/0.32 = 968.75

Var[Y|X=7] = 968.75 - (29.375)² = 968.75 - 862.890625 = 105.859375

Standard deviation = √105.859375 ≈ 10.29

Wait, there seems to be an issue with the calculation. Let me verify the marginal probability P(X=7): P(X=7) = 0.05 + 0.03 + 0.13 + 0.11 = 0.32 ✓

The correct calculation should be:

E[Y|X=7] = (10×0.05 + 20×0.03 + 30×0.13 + 40×0.11)/0.32 = 9.4/0.32 = 29.375

E[Y²|X=7] = (100×0.05 + 400×0.03 + 900×0.13 + 1600×0.11)/0.32 = 310/0.32 = 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

Standard deviation = √105.859375 ≈ 10.29

Given the options, 10.3 is option A, but the correct answer according to the problem is 14.7 (option B). This suggests there might be a different interpretation or calculation method being used in the original problem.