Discussion
Let X be a normal random variable with mean 1 and variance 1. Define the events :
\(A = \{-2 \lt X \lt 1\}\),
\(B = \{-1 \lt X \lt 1\}\),
\(C = \{0 \lt X \lt 2\}\).
Which of the following statements is correct ?
Answer: D
The correct ordering is \(P(B) \lt P(A) \lt P(C)\).
Standardise with \(Z = X - 1\), since the mean is 1 and the standard deviation is 1:
- \(P(A) = P(-3 \lt Z \lt 0) \approx 0.4987\)
- \(P(B) = P(-2 \lt Z \lt 0) \approx 0.4772\)
- \(P(C) = P(-1 \lt Z \lt 1) \approx 0.6827\)
A and B both end at the mean, and A reaches further into the tail, so \(P(B) \lt P(A)\). C is an interval of the same length as B but centred on the mean, where the density is highest, so it is the largest.
The orderings putting C first or making A and B equal ignore that the intervals sit at different distances from the mean of 1, not 0.
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