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FS1 Mean and variance of a normal distribution 본문

A-level Further Maths/Further Statistics 1

FS1 Mean and variance of a normal distribution

Cambridge Maths Academy 2022. 9. 23. 21:45
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Further statistics 1

Table of contents

  1. Probability density functions
  2. Moment generating functions
  3. Moments
  4. Edexcel FS1 Ch Exercise

1. Probability density functions

The probability dentity function for a normal distribution, XN(μ,σ2), is f(x)=12πσe12(xμσ)2 The probability dentity function for the standard normal distribution, Z=XμσN(0,12), is f(z)=12πe12z2

2. Moment generating functions

2.1 A normal distribution

For a normal distribution, XN(μ,σ2), MX(t)=1+Xt+12!X2t2+13!X3t3+=1+tX+12!(tX)2+13!(tX)3+=etX=12πσetxe12(xμσ)2dx=12πσe12σ2[(x22(μ+σ2t)x+μ2]dx=12πσe12σ2[[x(μ+σ2t)]2(μ+σ2t)2+μ2]dx=12πσe12(xμσ2tσ)2e12σ2[(μ2+2μσ2t+σ4t2)+μ2]dx=e12σ2t2+μt12πσe12(xμσ2tσ)2dx=eμt+12σ2t2

2.2 The standard normal distribution

For the standard normal distribution, Z=XμσN(0,12), MZ(t)=1+Zt+12!Z2t2+13!Z3t3+=1+tZ+12!(tZ)2+13!(tZ)3+=etZ=12πetze12z2dz=12πe12(z22zt)dz=12πe12[(zt)2t2]dz=e12t212πe12(zt)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)=12πevenzme12z2evendz and the integral vanishes when m is odd. Thus, we may restrict to even m, i.e. m=2n. E(Z2n)=12πz2ne12z2dz=22π0z2ne12z2dz By the substitution u=12z2, we find z2=2u and du=zdz. This gives E(Z2n)=22π0(2u)neudu2u=2n+12π0un12eudu=2nπΓ(n+12)=2nπ(2n)!22nn!π=(2n)!2nn! since Γ(n+12)=(n12)(n32)(n52)12Γ(12)π=(2n1)(2n3)(2n5)12nπ=(2n)!22nn!π In summary, the standardised moments for a normal distribution, XN(μ,σ2): E(Zn)=E[(Xμσ)n]=12πσ(xμσ)ne12(xμσ)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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