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Unformatted text preview: NORMAL Distribution
Most important distribution in statistics 1
1
−2
f (x) = √ e
σ 2π x−µ 2
σ 1 E [X ] =
Var X =
P (X ≤ x) = F (x)
2
1
x
− 1 t−µ
= t=−∞ √ e 2 σ dt
σ 2π No formula for this integral. Tables exist C4 p5689 for “standard normal”
Standard Normal
µ = 0, σ = 1.
Notation: Z for random variable,
φ(z ) pdf, Φ(z ) cdf.
2 Using Standard Normal Table
P (Z ≤ 1.5) = P (Z > 1) = P (1 < Z ≤ 1.5) =
Finding percentiles of Z . Find 80th percentile = z0.80 of standard normal.
Look in body of normal table for value
nearest to 0.80. Read corresponding z .
May need to interpolate. 3 689599.7% Rule for Z
P (−1 < Z ≤ 1) =
P (−2 < Z ≤ 2) =
P (−3 < Z ≤ 3) =
Note. Doesn’t matter whether
use < or ≤ because normal is
continuous. 4 Standardization. Suppose want
probabilities for normal X ≈ N (µ =
10, σ = 2), say P (X ≤ 13). Write
X −µ
Z=
σ
Shown in text that Z is standard
normal. Convert P (X ≤ 13) into
statement about Z . This is called
standardization. Converts X into
dimensionless quantities and preserves probabilities.
5 P (X ≤ 13) =
=
= = 0.9332 fig_ 0 4 _ 1 5
6 General Standardization.
Suppose X ≈ N (µ, σ ).
a−µ
a−µ
<Z<
)
P (a < X ≤ b ) = P (
σ
σ
Special Case.
68 − 95 − 99.7% Rule.
Take a = µ ± σ or a = µ ± 2σ
or a ± µ + 3σ. Then µ cancels
in numerator and then σ cancels in numerator and denominator, reducing to the same probabilities as the standard normal
68 − 95 − 99.7% Rule. 7 689599.7% Rule for X
P (µ − σ < X ≤ µ + σ ) =
P (µ − 2σ < X ≤ µ + 2 σ ) =
P (µ − 3σ < X ≤ µ + 3 σ ) = fig_ 0 4 _ 1 2 8 Example 3.21, p 166.
X ≈ N (µ = 75, σ 2 = 100).
P (80 < X < 95)
=
=
=
= 0.9332 − 0.6915 = 0.2857. 9 Finding percentiles of X .
xp = µ + σzp
Find the 95th percentile of the
normal X in previous example.
x0.95 =
=
= 91.45 10 Time to assemble item is X ≈
N (µ = 15.8, σ = 2.4).
Find P (X > 17).
P (X > 17)
17 − 15.8
X − 15.8
>
)
= P(
2. 4
2. 4
= P (Z > 0.5)
= 1 − Φ(0.5)
= 1 − 0.6915 = 0.3085. 11 ...
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This note was uploaded on 01/03/2012 for the course EE 1244 taught by Professor Drera during the Fall '10 term at Conestoga.
 Fall '10
 drera

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