X~N(3, 1) and Y~N(3, 2), then

Question

X~N(3, 1) and Y~N(3, 2),  then

BlackTom AI · Accepted Answer

We start by clearly identifying the given distributions: X ~ N(3, 1) and Y ~ N(3, 2). Here the second parameter is the variance, so X has variance 1 and Y has variance 2, both centered at the same mean 3. Option 1: P(X < -36) = P(Y < -36) - Compute standardized values: for X, z = (-36 - 3)/sqrt(1) = -39; for Y, z = (-36 - 3)/sqrt(2) = -39/√2 ≈ -27.6. - The left tails of a normal distribution decay very rapidly as the z-score becomes large in magnitude. A z of -39 is far further into the tail than -27.6, so P(X < -36) is much smaller than P(Y < -36). - Therefore, P(X < -36) ≠ P(Y < -36); the two probabilities are not equal, with the X-tail being much tinier. Option 2: None of the other three - This statement would be true if each of the other three options is false. - We will evaluate the remaining options to see whether they are all false or not. Option 3: P(X > 3) > P(Y > 3) - Since both X and Y are centered at 3, P(X > 3) and P(Y > 3) are both equal to 0.5 due to symmetry of the normal distribution about its mean. - The comparison P(X > 3) > P(Y > 3) asserts a strict inequality that does not hold; the two probabilities are equal, not one strictly larger. Option 4: P(X > 36) > P(Y > 36) - Compute standardized values: for X, z = (36 - 3)/sqrt(1) = 33; for Y, z = (36 - 3)/sqrt(2) = 33/√2 ≈ 23.3. - The tail probability P(X > 36) = P(Z > 33) for Z ~ N(0,1) is astronomically small. P(Y > 36) = P(Z > 23.3) is also astronomically small, but tail probabilities decrease with larger z, and 33 is much larger than 23.3, making P(X > 36) far smaller than P(Y > 36). - Therefore, P(X > 36) > P(Y > 36) is false; in fact the inequality goes the other way around (P(X > 36) < P(Y > 36)). Summary of option analyses - Option 1 is false because the left-tail probabilities are not equal; the X-tail is far lighter than the Y-tail. - Option 3 is false because both P(X > 3) and P(Y > 3) equal 0.5, so there is no strict greater-than relationship. - Option 4 is false because the tail probability for X at 36 is much smaller than that for Y at 36, so the inequality direction stated does not hold. - Since each of the three non- "None of the other three" options is false, Option 2 (None of the other three) would be the only one that remains consistent with the evaluations above.

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