Let X be a random variable following normal distribution with mean +1 and…
2008
Let X be a random variable following normal distribution with mean +1 and variance 4. Let Y be another normal variable with mean -1 and variance unknown If P(X <=-1) = P(Y >=2). the standard deviation of Y is
Answer: A. 3 — Key idea: equate the two probabilities using standard normal transformation. Compute P(X <= -1). For X ~ N(1,4), the standard deviation is 2. Convert to Z: Z…
- A.
3
- B.
2
- C.
sqrt(2)
- D.
1
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Correct answer: A
Key idea: equate the two probabilities using standard normal transformation.
Compute P(X <= -1). For X ~ N(1,4), the standard deviation is 2. Convert to Z: Z = (-1 - 1)/2 = -1, so P(X <= -1) = Φ(-1) ≈ 0.1587.
Use the equality P(Y >= 2) = 0.1587. Then P(Y <= 2) = 1 - 0.1587 = 0.8413, which corresponds to a standard normal value Φ(1) = 0.8413.
Translate back to Y: (2 - μ_Y)/σ = 1. Since μ_Y = -1, (2 - (-1))/σ = 3/σ = 1, so σ = 3.
Therefore the standard deviation of Y is 3.
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