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Cambridge Maths Academy
FS1 Mean and variance of a normal distribution 본문
FS1 Mean and variance of a normal distribution
Cambridge Maths Academy 2022. 9. 23. 21:45Further statistics 1
Table of contents
- Probability density functions
- Moment generating functions
- Moments
- Edexcel FS1 Ch Exercise
1. Probability density functions
The probability dentity function for a normal distribution, X∼N(μ,σ2), is f(x)=1√2πσe−12(x−μσ)2 The probability dentity function for the standard normal distribution, Z=X−μσ∼N(0,12), is f(z)=1√2πe−12z2
2. Moment generating functions
2.1 A normal distribution
For a normal distribution, X∼N(μ,σ2), MX(t)=1+⟨X⟩t+12!⟨X2⟩t2+13!⟨X3⟩t3+⋯=⟨1+tX+12!(tX)2+13!(tX)3+⋯⟩=⟨etX⟩=1√2πσ∫∞−∞etxe−12(x−μσ)2dx=1√2πσ∫∞−∞e−12σ2[(x2−2(μ+σ2t)x+μ2]dx=1√2πσ∫∞−∞e−12σ2[[x−(μ+σ2t)]2−(μ+σ2t)2+μ2]dx=1√2πσ∫∞−∞e−12(x−μ−σ2tσ)2e−12σ2[−(μ2+2μσ2t+σ4t2)+μ2]dx=e12σ2t2+μt1√2πσ∫∞−∞e−12(x−μ−σ2tσ)2dx=eμt+12σ2t2
2.2 The standard normal distribution
For the standard normal distribution, Z=X−μσ∼N(0,12), MZ(t)=1+⟨Z⟩t+12!⟨Z2⟩t2+13!⟨Z3⟩t3+⋯=⟨1+tZ+12!(tZ)2+13!(tZ)3+⋯⟩=⟨etZ⟩=1√2π∫∞−∞etze−12z2dz=1√2π∫∞−∞e−12(z2−2zt)dz=1√2π∫∞−∞e−12[(z−t)2−t2]dz=e12t21√2π∫∞−∞e−12(z−t)2dz=e12t2 which agrees with MX(t) when μ=0 and σ=1.
3. Moments
3.1 The standard normal distribution
The Maclaurin/Taylor expansion of the moment generating function: MZ(t)=e12t2=∞∑n=01n!(t22)n=∞∑n=0(2n)!2nn!t2n(2n)! which gives ⇒E(Zodd)=0E(Z2n)=(2n)!2nn!E(Z)=0E(Z2)=2!211!=1E(Z3)=0E(Z4)=4!222!=3E(Z5)=0E(Z6)=6!233!=15E(Z7)=0E(Z8)=8!244!=105E(Z9)=0E(Z10)=10!255!=945
Aside. Alternatively, we can evaluate: E(Zm)=1√2π∫∞−∞⏟evenzme−12z2⏟evendz and the integral vanishes when m is odd. Thus, we may restrict to even m, i.e. m=2n. E(Z2n)=1√2π∫∞−∞z2ne−12z2dz=2√2π∫∞0z2ne−12z2dz By the substitution u=12z2, we find z2=2u and du=zdz. This gives ⇒E(Z2n)=2√2π∫∞0(2u)ne−udu√2u=2n+12√π∫∞0un−12e−udu=2n√πΓ(n+12)=2n√π⋅(2n)!22nn!√π=(2n)!2nn!✓ since Γ(n+12)=(n−12)(n−32)(n−52)⋯12Γ(12)⏟√π=(2n−1)(2n−3)(2n−5)⋯12n√π=(2n)!22nn!√π In summary, the standardised moments for a normal distribution, X∼N(μ,σ2): E(Zn)=E[(X−μσ)n]=1√2πσ∫∞−∞(x−μσ)ne−12(x−μσ)2dxE(Zodd)=0E(Z2n)=(2n)!2nn!E(Z2)=1E(Z4)=3E(Z6)=15E(Z8)=105E(Z10)=945
3.2 A normal distribution
We can Maclaurin/Taylor expand MX(t): MX(t)=eμt+12σ2t2=eμte12σ2t2=(1+μt+(μt)22!+(μt)33!+(μt)44!+⋯)[1+(12σ2t2)+12!(12σ2t2)2+⋯]=(1+μt+μ2t22!+μ3t33!+μ4t44!+⋯)[1+σ2t22!+4!2!(σ22)2t44!+⋯]=1+μt+(μ2+σ2)+(μ3+3μσ2)t33!+[μ4+4!2!2!μ2σ2+4!2!(σ22)2]t44!+⋯=1+μt+(μ2+σ2)+(μ3+3μσ2)t33!+(μ4+6μ2σ2+3σ4)t44!+⋯ Aside. Alternatively, we can evaluate: \begin{align} \mathbb E \left( X^n \right) &= \frac{1}{ \sqrt{2\pi} \sigma} \int_{-\infty}^{\infty} x^n \textrm e^{ - \frac12 \left( \frac{ x - \mu }{ \sigma } \right)^2 } \, \textrm dx \\ &= \frac{1}{ \sqrt{2\pi}} \int_{-\infty}^{\infty} ( \mu + \sigma z )^n \textrm e^{ - \frac12 z^2 } \, \textrm dz \\ &= \sum_{k=0}^n \binom{n}{k} \mu^{n-k} \sigma^{k} \left\langle Z^k \right\rangle \end{align} By the binomial expansion and using \left\langle Z^{ \textrm{odd} } \right\rangle = 0 and \left\langle Z^{2n} \right\rangle = \frac{(2n)!}{2^n n!} , we find: \begin{align} \mathbb E \left( X \right) &= \frac{1}{ \sqrt{2\pi}} \int_{-\infty}^{\infty} ( \mu + \sigma z ) \textrm e^{ - \frac12 z^2 } \, \textrm dz \\ &= \mu \\ \\ \mathbb E \left( X^2 \right) &= \frac{1}{ \sqrt{2\pi}} \int_{-\infty}^{\infty} ( \mu + \sigma z )^2 \textrm e^{ - \frac12 z^2 } \, \textrm dz \\ &= \left\langle ( \mu + \sigma Z )^2 \right\rangle \\ &= \left\langle \mu^2 + 2 \mu \sigma Z + \sigma^2 Z^2 \right\rangle \\ &= \mu^2 + \sigma^2 \\ \\ \mathbb E \left( X^3 \right) &= \frac{1}{ \sqrt{2\pi}} \int_{-\infty}^{\infty} ( \mu + \sigma z )^3 \textrm e^{ - \frac12 z^2 } \, \textrm dz \\ &= \left\langle ( \mu + \sigma Z )^3 \right\rangle \\ &= \left\langle \mu^3 + 3 \mu^2 \sigma Z + 3 \mu \sigma^2 Z^2 + \sigma^3 Z^3 \right\rangle \\ &= \mu^3 + 3 \mu \sigma^2 \\ \\ \mathbb E \left( X^4 \right) &= \frac{1}{ \sqrt{2\pi}} \int_{-\infty}^{\infty} ( \mu + \sigma z )^4 \textrm e^{ - \frac12 z^2 } \, \textrm dz \\ &= \left\langle ( \mu + \sigma Z )^4 \right\rangle \\ &= \left\langle \mu^4 + 4 \mu^3 \sigma Z + 6 \mu^2 \sigma^2 Z^2 + 4 \mu \sigma^3 Z^3 + \sigma^4 Z^4 \right\rangle \\ &= \mu^4 + 6 \mu^2 \sigma^2 + 3 \sigma^4 \\ \\ \mathbb E \left( X^5 \right) &= \frac{1}{ \sqrt{2\pi}} \int_{-\infty}^{\infty} ( \mu + \sigma z )^5 \textrm e^{ - \frac12 z^2 } \, \textrm dz \\ &= \left\langle ( \mu + \sigma Z )^5 \right\rangle \\ &= \left\langle \mu^5 + 5 \mu^4 \sigma Z + 10 \mu^3 \sigma^2 Z^2 + 10 \mu^2 \sigma^3 Z^3 + 5 \mu \sigma^4 Z^4 + \sigma^5 Z^5 \right\rangle \\ &= \mu^5 + 10 \mu^3 \sigma^2 + 15 \mu \sigma^4 \end{align} In summary, the moments for a normal distribution, X \sim \textrm N( \mu , \sigma^2 ) : \begin{align} \mathbb E \left( X^n \right) &= \frac{1}{ \sqrt{2\pi} \sigma} \int_{-\infty}^{\infty} x^n \textrm e^{ - \frac12 \left( \frac{ x - \mu }{ \sigma } \right)^2 } \, \textrm dx \\ \\ \mathbb E \left( X \right) &= \mu \\ \mathbb E \left( X^2 \right) &= \mu^2 + \sigma^2 \\ \mathbb E \left( X^3 \right) &= \mu^3 + 3 \mu \sigma^2 \\ \mathbb E \left( X^4 \right) &= \mu^4 + 6 \mu^2 \sigma^2 + 3 \sigma^4 \\ \mathbb E \left( X^5 \right) &= \mu^5 + 10 \mu^3 \sigma^2 + 15 \mu \sigma^4 \\ \mathbb E \left( X^6 \right) &= \mu^6 + 15 \mu^4 \sigma^2 + 45 \mu^2 \sigma^4 + 15 \sigma^6 \\ \mathbb E \left( X^7 \right) &= \mu^7 + 21 \mu^5 \sigma^2 + 105 \mu^3 \sigma^4 + 105 \mu \sigma^6 \\ \mathbb E \left( X^8 \right) &= \mu^8 + 28 \mu^6 \sigma^2 + 210 \mu^4 \sigma^4 + 420 \mu^2 \sigma^6 + 105 \sigma^8 \\ \mathbb E \left( X^9 \right) &= \mu^9 + 36 \mu^7 \sigma^2 + 378 \mu^5 \sigma^4 + 1260 \mu^3 \sigma^6 + 945 \mu \sigma^8 \\ \end{align}
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