Let Y follow N(-3,4). Compute $ P(-10\leq Y \leq 1) $

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BlackTom AI · Accepted Answer

We need to compute P(-10 ≤ Y ≤ 1) where Y ~ N(-3, 4).
First, confirm the distribution parameters: N(mu, sigma^2) is typically written as N(-3, 4), so the mean mu = -3 and the variance sigma^2 = 4. This means the standard deviation sigma = sqrt(4) = 2.
Next, standardize the endpoints to Z ~ N(0,1):
Z = (Y - mu) / sigma = (Y + 3) / 2.
Compute the Z-values corresponding to Y = -10 and Y = 1:
For Y = -10: Z = (-10 + 3) / 2 = -7 / 2 = -3.5.
For Y = 1: Z = (1 + 3) / 2 = 4 / 2 = 2.
Thus P(-10 ≤ Y ≤ 1) = P(-3.5 ≤ Z ≤ 2) = Phi(2) - Phi(-3.5).
Using standard normal values (or a Z-table): Phi(2) ≈ 0.97725 and Phi(-3.5) ≈ 0.00023.
Subtracting gives approximately 0.97725 - 0.00023 ≈ 0.97702.
This matches the provided answer 0.977 within typical rounding.
A quick note on the parameter interpretation: if instead the second parameter 4 were meant to be the standard deviation rather than the variance, i.e., sigma = 4, the Z-values would be (-10 + 3)/4 = -1.75 and (1 + 3)/4 = 1, yielding a different probability. However, by convention N(-3, 4) denotes variance 4, so sigma = 2 is the correct reading here.

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Let Y follow N(-3,4). Compute \( P(-10\leq Y \leq 1) \)

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