Taking the total derivative we get
lɪ+y, [σ^ φ ( χσ—c ) - 2σ2φ, ( χσ—c ) [Bkr+2xcr+r2+дст2∣] h(χ°^c}dc
{£ [r [ɪ - * ■ i)] + ⅛φ ■ c) [Bkr + . Xcr + 2 + ∆σ2∣] "«.} d = 0
and hence
√= ʃ ∞ [r [ɪ - Φ (—)] + ɪφ (—) [Bkr + 2Xcr + r2 +∆σ2∣] dh(xc) dXc
dc _ J-∞ LL ' σ,- ∕J 2σv ∖ σv ) I 1 c ' 1 "lJ dσχc c
dσχc - ɪ + - ʃɪ [rφ (χσ∏ - ⅛φ' (χσ-) [Bkr+2χr+r2+^2∣] ⅛(χc)dχc
= ʃɪ [r [ɪ - * (χσ-c)] + 2⅛φ (χσ-c) [Bkr+2χcr+r2+1] ⅜1 dχc
ɪ + - ʃɪ Φ ( xσ-c ) [r + x2σ-c [Bkr + 2χcr + r2 + ^21] ⅛(χc )dχc
where the second line uses the fact that φ'(x) =
—
ɪ— e
√2πσχc
[χc-μχc ]2
2σ2
χc
we get
[χc-μχc ]2
dh(Χc) ɪ - ~~2σζ
χ χc
dσχc √2π
—xφ(x). Assuming a normal density h(Xc) =
[χc-μχc ]2 ɪ
σ4 σ2
χc χc
and hence
- [Xc-μXc ]2
d-c σXc ʃɪ [r [ɪ- * (χσ-c)] + 2⅛φ (χσ-c) [Bkr + 2χcr+ r2 + ^2]][ɪ- χ -X x ^∏σχ-e 2σχc dχc
dσχc - [Xc-μXc]2
ɪ + ɪ ʃ ∞ φ ( ' ) [r + ¾-c [Bkr + 2Χcr + r2 + ∆σ21] ɔ-i---e 27XC dΧc
1 σv .1 ∞ σ∙. ; L 2σJ; l ' c 1 1 2jJ √2πσXc c
33
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