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Unformatted text preview: 2 2σ 2 x = σ ⋅ z + µ Let the Jacobian be J: J= dx d
= (σ ⋅ z + µ ) = σ dz dz −
1
f z ( z ) = J  ⋅ f x ( x) = σ ⋅
e
2πσ ∴Z = X −µ σ ( x − µ )2
2σ 2 1−
=
e
2π i .i .d . Proof: M X (t ) = E (e tX ) = E (e 2 1 − z2
=
e 2π ~ N (0,1) e.g. If X1 , X 2 ,K , X n ~ N ( µ , σ 2 ) , then X ~ N ( µ , (σ ⋅ z + µ − µ )2
2σ 2 = E (et * t X 1 + X 2 +!+ X n
n ( X1 +L + X n ) σ2
n ) . ) ), where t * = t / n, = M X1 +!X n (t * ) = M X1 (t * )! M X n (t * ) = (e = e ∴ X ~ N (µ , 1
2 µt * + σ 2t *2 ) n t1
t
nµ + nσ 2 ( ) 2
n2
n σ2
n =e µt + 1σ 2 2
t
2n ) 6 e.g. Sgt. Jones wishes to select one army recruit into his unit. It is known that the IQ distribution for all recruits is normal with mean 180 and standard deviation 10. What is the chance that Sgt. Jones would select a recruit with an IQ of at least 200? Sol) Let X represents the IQ of a randomly selected recruit. X ~ N ( µ = 180, σ = 10) P( X ≥ 200) = P( X − 180 200 − 180
≥
) = P( Z ≥ 2) ≈ 2.28% 10
10 e.g. Sgt. Jones wishes to select three army recruits into his unit. It is known that the IQ distribution for all recruits is normal with mean 180 and standard deviation 10. What is the chance that Sgt. Jones would select three recruits with an average IQ of at least 200? Sol) Let...
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This document was uploaded on 03/15/2014 for the course AMS 412 at SUNY Stony Brook.
 Spring '14

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