Derivative for cdf in normal distribution
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Just apply the chain rule for differentiation. The CDF FX(x;μ,σ2) of a N(μ,σ2) random variable X is Φ(
x−μ
σ
) and so
∂
∂μ
FX(x;μ,σ2)=
∂
∂μ
Φ(
x−μ
σ
)=ϕ(
x−μ
σ
)
−1
σ
=−[
1
σ
ϕ(
x−μσ)]
where ϕ(x) is the standard normal density and the quantity in square brackets on the rightmost expression above can be recognized as the density of X∼N(μ,σ2).
I will leave the calculation of the derivative with respect to σ or σ2 for you to work out for yourself.
x−μ
σ
) and so
∂
∂μ
FX(x;μ,σ2)=
∂
∂μ
Φ(
x−μ
σ
)=ϕ(
x−μ
σ
)
−1
σ
=−[
1
σ
ϕ(
x−μσ)]
where ϕ(x) is the standard normal density and the quantity in square brackets on the rightmost expression above can be recognized as the density of X∼N(μ,σ2).
I will leave the calculation of the derivative with respect to σ or σ2 for you to work out for yourself.
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