Extrasolar Moons as Gravitational Microlenses Christine Liebig

CHAPTER 3. METHOD 27
method, we introduce the new random variable
Q 2 n
Xi − µ
=
b i
i=1
This is the step, where we could simulate data in order to find a somewhat representative
value of Q2 , but instead we simply calculate what the mean value of all
possible Q2 would be. We use the now familiar definitions, to find
〈Q 2
n
Xi − µ
〉 =
b
2
i
=
i=1
n
i=1
1
(σ b i )2
σ b i
σ b i
2
.
(Xi − µ b i) 2
Here, we keep in mind that, in general, 〈x 2 〉 = 〈x〉 2 . Using the parameters µ t i and
σ t i of the distribution fi(xi) that belongs to our Xi, we reduce the equation by
calculating
〈Q 2 〉 =
=
=
=
=
=
=
n
i=1
n
i=1
n
i=1
n
i=1
n
i=1
n
i=1
n
i=1
1
∞
(σb i )2
∞
1
(σb i )2
∞
1
(σb i )2 −∞
∞
1
(σ b i )2
1
(σ b i )2
1
(σ b i )2
1
(σ b i )2
(xi − µ
−∞
b i) 2 fi(xi)dxi.
(xi − µ
−∞
t i + µ t i − µ b i) 2 fi(xi)dxi
(xi − µ t i) 2 + 2(xi − µ t i)(µ t i − µ b i) + (µ t i − µ b i) 2 fi(xi)dxi
(xi − µ
−∞
t i) 2 fi(xi)dxi + 2(µ t i − µ b i)
+(µ t i − µ b i) 2
∞
(σ t i) 2 + 2(µ t i − µ b i)
+(µ t i − µ b i) 2
fi(xi)dxi
−∞
∞
∞
xifi(xi)dxi − µ
−∞
t i
t
(σi) 2 + 2(µ t i − µ b i) µ t i − µ t t
i + (µ i − µ b i) 2
(σ t i) 2 + (µ t i − µ b i) 2 .
(xi − µ
−∞
t i)fi(xi)dxi
∞
fi(xi)dxi
−∞