Q function (gaussian random variable)

  • Thread starter magnifik
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  • #1
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For X ~ N(μ, σ), what is P[|X-μ] < σ] in terms of the Q function?

I know that P[|X-μ] < σ] can be decomposed into P[X > -σ + μ] + P[X < σ + μ] i'm not sure what to do next. i know P[X < σ + μ] can be expressed as 1 - phi(σ + μ - μ / σ) = Q(1), but i'm not sure how to approach P[X > -σ + μ]. I know the answer is 1 - 2Q(1)...not sure where this all comes from
 

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  • #2
I like Serena
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Hi magnifik! :smile:

I'm afraid you have the wrong decomposition.

Things should become clearer when you look at them graphically.
Draw a bell curve with μ in the middle and standard deviation σ.

P[|X-μ| < σ] corresponds to the surface under the bell curve between μ-σ and μ+σ

Q(1) corresponds to the surface under the bell curve to the right of μ+σ (the tail probability).
Since the bell curve is symmetric, Q(1) also corresponds to the surface to the left of μ-σ.

Since the total surface under the bell curve is 1, you can deduce your formula.
 

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