Arrange the probabilities for a standard normal variate Z over the following…
2024
Arrange the probabilities for a standard normal variate Z over the following intervals in increasing order.
A. −∞ ≤ Z ≤ 1
B. 0 ≤ Z ≤ 3
C. −1 ≤ Z ≤ 1
D. Z ≥ 1
Choose the correct answer from the options given below.
Answer: A. D, B, C, A — ConceptFor a standard normal variate Z, interval probabilities are obtained from the cumulative distribution function Φ(z) = P(Z ≤ z). Symmetry gives Φ(−z) =…
- A.
D, B, C, A
- B.
C, B, D, A
- C.
A, C, B, D
- D.
B, D, A, C
Attempted by 2 students.
Show answer & explanation
Correct answer: A
Concept
For a standard normal variate Z, interval probabilities are obtained from the cumulative distribution function Φ(z) = P(Z ≤ z). Symmetry gives Φ(−z) = 1 − Φ(z), P(Z ≥ z) = 1 − Φ(z), and Φ(0) = 0.5.
To compare several intervals, express every probability in terms of Φ and then order the resulting numerical values.
Application
For Z ≥ 1: P(Z ≥ 1) = 1 − Φ(1) ≈ 1 − 0.8413 = 0.1587.
For 0 ≤ Z ≤ 3: P(0 ≤ Z ≤ 3) = Φ(3) − Φ(0) ≈ 0.9987 − 0.5000 = 0.4987.
For −1 ≤ Z ≤ 1: P(−1 ≤ Z ≤ 1) = Φ(1) − Φ(−1) ≈ 0.8413 − 0.1587 = 0.6826.
For −∞ ≤ Z ≤ 1: P(−∞ ≤ Z ≤ 1) = Φ(1) ≈ 0.8413.
Therefore, 0.1587 < 0.4987 < 0.6826 < 0.8413, corresponding to D, B, C, A.
Cross-check
By symmetry, P(−1 ≤ Z ≤ 1) = 1 − 2P(Z ≥ 1) ≈ 1 − 2(0.1587) = 0.6826. This independently confirms the central-interval probability used in the ordering.
Result
The increasing order is D, B, C, A.