Cambridge Maths Academy

FS1 §3.2 Mean and variance of a geometric distribution 본문

A-level Further Maths/Further Statistics 1

FS1 §3.2 Mean and variance of a geometric distribution

Cambridge Maths Academy 2022. 9. 16. 02:23
반응형

Further statistics 1

Table of contents

  1. Probability mass function
  2. Probability generating function
  3. Moment generating function
  4. Moments
  5. Using moment generating function
  6. Central moments
  7. Standardised moments
  8. Edexcel FS1 Ch Exercise (under construction)

1. Probability mass function

The probability mass function for a geometric distribution, XGeo(p), is P(x)=pqx1,x=1,2,3, where p+q=1.

Sum of probabilities is 1

x=1P(x)=x=1pqx1=p(11q)=p1p=1

2. Probability generating function

GX(t)=P(0)+P(1)t+P(2)t2+P(3)t3+=x=1P(x)tx=tX=x=1pqx1tx=pqx=1(qt)x=pqqt1qt=pt1qt=pt1(1p)t

3. Moment generating function

MX(t)=1+Xt+12!X2t2+13!X3t3+=1+Xt+12!(Xt)2+13!(Xt)3+=eXt=x=1P(x)ext=x=1pqx1(et)x=pqx=1(qet)x=pqqet1qet=pet1qet=pet1(1p)et

4. Moments

4.1. Mean = First moment

E(X)=x=1xP(X=x)=x=1xpqx1=pqx=1xqx=pq(qq)x=1qx=pq(qq)q1q=pq(qq)[q1+11q]=pq(qq)[1+11q]=pqq(1q)2=p(1q)2=1p since p+q=1.

 

Aside. An alternative way to evaluate xqx: S=x=1xqx=q+2q2+3q3+4q4+qS=x=1xqx+1=q2+2q3+3q4+4q5+(1q)S=q+q2+q3+q4+=x=1qx=q1qS=q(1q)2 When dealing with X2, we shall also look at a similar way to evaluate x2qx.

4.2. A preparation for higher moments

E(Xn)=x=1xnP(X=x)=x=1xnpqx1=pqx=1xnqx=pq(qq)nx=1qx=pq(qq)nq1q It is convenient to work out: (qq)2=qq+q22q2(qq)3=qq(qq+q22q2)=(qq+q22q2)+(2q22q2+q33q3)=qq+3q22q2+q33q3(qq)4=qq(qq+3q22q2+q33q3)=(qq+q22q2)+(6q22q2+3q33q3)+(3q33q3+q44q4)=qq+7q22q2+6q33q3+q44q4(qq)5=qq(qq+7q22q2+6q33q3+q44q4)=(qq+q22q2)+(14q22q2+7q33q3)+(18q33q3+6q44q4)+(4q44q4+q55q5)=qq+15q22q2+25q33q3+10q44q4+q55q5(qq)6=qq(qq+15q22q2+25q33q3+10q44q4+q55q5)=(qq+q22q2)+(30q22q2+15q33q3)+(75q33q3+25q44q4)+(40q44q4+10q55q5)+(5q55q5+q66q6)=qq+31q22q2+90q33q3+65q44q4+15q55q5+q66q6(qq)7=qq(qq+31q22q2+90q33q3+65q44q4+15q55q5+q66q6)=(qq+q22q2)+(62q22q2+31q33q3)+(270q33q3+90q44q4)+(260q44q4+65q55q5)+(75q55q5+15q66q6)+(6q66q6+q77q7)=qq+63q22q2+301q33q3+350q44q4+140q55q5+21q66q6+q77q7(qq)8=qq(qq+63q22q2+301q33q3+350q44q4+140q55q5+21q66q6+q77q7)=(qq+q22q2)+(126q22q2+63q33q3)+(903q33q3+301q44q4)+(1400q44q4+350q55q5)+(700q55q5+140q66q6)+(126q66q6+21q77q7)+(7q77q7+q88q8)=qq+127q22q2+966q33q3+1701q44q4+1050q55q5+266q66q6+28q77q7+q88q8

 

(qq)2=qq+q22q(qq)3=qq(qq+q22q)=(qq+q22q)+(2q22q+q33q)=qq+3q22q+q33q(qq)4=qq(qq+3q22q+q33q)=(qq+q22q)+(6q22q+3q33q)+(3q33q+q44q)=qq+7q22q+6q33q+q44q(qq)5=qq(qq+7q22q+6q33q+q44q)=(qq+q22q)+(14q22q+7q33q)+(18q33q+6q44q)+(4q44q+q55q)=qq+15q22q+25q33q+10q44q+q55q(qq)6=qq(qq+15q22q+25q33q+10q44q+q55q)=(qq+q22q)+(30q22q+15q33q)+(75q33q+25q44q)+(40q44q+10q55q)+(5q55q+q66q)=qq+31q22q+90q33q+65q44q+15q55q+q66q(qq)7=qq(qq+31q22q+90q33q+65q44q+15q55q+q66q)=(qq+q22q)+(62q22q+31q33q)+(270q33q+90q44q)+(260q44q+65q55q)+(75q55q+15q66q)+(6q66q+q77q)=qq+63q22q+301q33q+350q44q+140q55q+21q66q+q77q(qq)8=qq(qq+63q22q+301q33q+350q44q+140q55q+21q66q+q77q)=(qq+q22q)+(126q22q+63q33q)+(903q33q+301q44q)+(1400q44q+350q55q)+(700q55q+140q66q)+(126q66q+21q77q)+(7q77q+q88q)=qq+127q22q+966q33q+1701q44q+1050q55q+266q66q+28q77q+q88q

 

In summary, (qq)2=qq+q22q(qq)3=qq+3q22q+q33q(qq)4=qq+7q22q+6q33q+q44q(qq)5=qq+15q22q+25q33q+10q44q+q55q(qq)6=qq+31q22q+90q33q+65q44q+15q55q+q66q(qq)7=qq+63q22q+301q33q+350q44q+140q55q+21q66q+q77q(qq)8=qq+127q22q+966q33q+1701q44q+1050q55q+266q66q+28q77q+q88q

 

It is useful to Taylor expand qn in powers of (1q). Note that we observe the binomial coefficients: q=1(1q)q2=12(1q)+(1q)2q3=13(1q)+3(1q)2(1q)3q4=14(1q)+6(1q)24(1q)3+(1q)4q5=15(1q)+10(1q)210(1q)3+5(1q)4(1q)5q6=16(1q)+15(1q)220(1q)3+15(1q)46(1q)5+(1q)6q7=17(1q)+21(1q)2+35(1q)335(1q)4+21(1q)57(1q)6+(1q)7q8=18(1q)+28(1q)2+56(1q)370(1q)4+56(1q)528(1q)6+8(1q)7(1q)8 The Taylor expansions above are used to express the following derivatives in partial fractions: (qq)q1q=(qq)[q1+11q]=(qq)[1+11q]=q(1q)2(qq)2q1q=(qq+q22q)[q1+11q]=(qq+q22q)[1+11q]=q(1q)2+2!q2(1q)3=q[1(1q)2+2!q(1q)3]=q[1(1q)2+2[1(1q)](1q)3]=q[1(1q)2+2(1q)32(1q)2]=q[1(1q)2+2(1q)3](qq)3q1q=(qq+3q22q+q33q)[1+11q]=q(1q)2+(32!)q2(1q)3+3!q3(1q)4=q[1(1q)2+6q(1q)3+6q2(1q)4]=q[1(1q)2+6[1(1q)](1q)3+6[12(1q)+(1q)2](1q)4]=q[1(1q)2+6(1q)36(1q)2+6(1q)412(1q)3+6(1q)2]=q[1(1q)26(1q)3+6(1q)4](qq)4q1q=(qq+7q22q+6q33q+q44q)[1+11q]=q(1q)2+(72!)q2(1q)3+(63!)q3(1q)4+4!q4(1q)5=q[1(1q)2+14q(1q)3+36q2(1q)4+24q3(1q)5]=q[1(1q)2+14[1(1q)](1q)3+36[12(1q)+(1q)2](1q)4+24[13(1q)+3(1q)2(1q)3](1q)5]=q[1(1q)2+14(1q)314(1q)2+36(1q)472(1q)3+36(1q)2+24(1q)572(1q)4+72(1q)324(1q)2]=q[1(1q)2+14(1q)336(1q)4+24(1q)5](qq)5q1q=(qq+15q22q+25q33q+10q44q+q55q)[1+11q]=q[1(1q)230(1q)3+150(1q)4240(1q)5+120(1q)6](qq)6q1q=(qq+31q22q+90q33q+65q44q+15q55q+q66q)[1+11q]=q[1(1q)2+62(1q)3540(1q)4+1560(1q)51800(1q)6+720(1q)7](qq)7q1q=(qq+63q22q+301q33q+350q44q+140q55q+21q66q+q77q)[1+11q]=q[1(1q)2126(1q)3+1860(1q)48400(1q)5+16800(1q)615120(1q)7+5040(1q)8](qq)8q1q=(qq+127q22q+966q33q+1701q44q+1050q55q+266q66q+28q77q+q88q)[1+11q]=q[1(1q)2+254(1q)35796(1q)4+40824(1q)5126000(1q)6+191520(1q)7141120(1q)8+40320(1q)9]

 

The moments: (qq)q1q=qp2E(X)=pq(qq)q1q=1p(qq)2q1q=q[1p2+2p3]E(X2)=pq(qq)2q1q=1p+2p2(qq)3q1q=q[1p26p3+6p4]E(X3)=pq(qq)3q1q=1p6p2+6p3(qq)4q1q=q[1p2+14p336p4+24p5]E(X4)=pq(qq)4q1q=1p+14p236p3+24p4(qq)5q1q=q[1p230p3+150p4240p5+120p6]E(X5)=pq(qq)5q1q=1p30p2+150p3240p4+120p5(qq)6q1q=q[1p2+62p3540p4+1560p51800p6+720p7]E(X6)=pq(qq)6q1q=1p+62p2540p3+1560p41800p5+720p6(qq)7q1q=q[1p2126p3+1860p48400p5+16800p615120p7+5040p8]E(X7)=pq(qq)7q1q=1p126p2+1860p38400p4+16800p515120p6+5040p7(qq)8q1q=q[1p2+254p35796p4+40824p5126000p6+191520p7141120p8+40320p9]E(X8)=pq(qq)8q1q=1p+254p25796p3+40824p4126000p5+191520p6141120p7+40320p8

 

For a geometric distribution, XGeo(p), E(Xn)=x=1xnpqx1 E(X)=1pE(X2)=1p+2p2E(X3)=1p6p2+6p3E(X4)=1p+14p236p3+24p4E(X5)=1p30p2+150p3240p4+120p5E(X6)=1p+62p2540p3+1560p41800p5+720p6E(X7)=1p126p2+1860p38400p4+16800p515120p6+5040p7E(X8)=1p+254p25796p3+40824p4126000p5+191520p6141120p7+40320p8

 

E(X)=1pE(X2)=1p+2p2=2pp2E(X3)=1p6p2+6p3=p26p+6p3E(X4)=1p+14p236p3+24p4=(2p)(1212p+p2)p4E(X5)=1p30p2+150p3240p4+120p5=p430p3+150p2240p+120p5E(X6)=1p+62p2540p3+1560p41800p5+720p6=(2p)(360720p+420p260p3+p4)p6E(X7)=1p126p2+1860p38400p4+16800p515120p6+5040p7=p6126p5+1860p48400p3+16800p215120p+5040p7E(X8)=1p+254p25796p3+40824p4126000p5+191520p6141120p7+40320p8=(2p)(2016060480p+65520p230240p3+5292p4252p5+p6)p8

4.3. E(X2) and variance

For variance,  Var(X)=E[(Xμ)2]=X2X2=(1p+2p2)(1p)2=1p+1p2=1pp2=qp2

 

Aside. An alternative way to evaluate x2qx: S=x=1x2qx=q+4q2+9q3+16q4+qS=x=1x2qx+1=q2+4q3+9q4+16q5+(1q)S=q+3q2+5q3+7q4+=x=1(2x1)qxq(1q)S=q2+3q3+5q4+7q5+=x=1(2x1)qx+1(1q)2S=q+2q2+2q3+2q4+=2(q+q2+q3+)q=2q1qq=2qq(1q)1qS=q(1+q)(1q)3=q(2p)p3

4.4. E(X3) and skewness

Recall: E(X)=1pE(X2)=1p+2p2E(X3)=1p6p2+6p3E(X4)=1p+14p236p3+24p4 The skewness is defined by E[(Xμσ)3]=E[X33μX2+3μ2Xμ3σ3]=X33μX2+2μ3σ3 and thus E[(Xμσ)3]=X33μX2+2μ3σ3=(1p6p2+6p3)3p(1p+2p2)+2p3(qp2)32=23p+p2p3(qp2)32=(1p)(2p)q32=2p1p

4.5. E(X4) and skewness

Recall: E(X)=1pE(X2)=1p+2p2E(X3)=1p6p2+6p3E(X4)=1p+14p236p3+24p4 The kurtosis is defined by E[(Xμσ)4]=E[X44μX3+6μ2X24μ3X+μ4σ4]=X44μX3+6μ2X23μ4σ4 and thus E[(Xμσ)4]=X44μX3+6μ2X23μ4σ4=(1p+14p236p3+24p4)4p(1p6p2+6p3)+6p2(1p+2p2)3p4q2p4=918p+10p2p3p4q2p4=(1p)(99p+p2)(1p)2=99p+p21p=9+p21p

5. Using moment generating function

5.1 Series expansion

As we Taylor-expand MX(t), MX(t)=pet1qet=pet(1qet)1=p(n=0tnn!)[1+qet+q2e2t+q3e3t+]=p(n=0tnn!)[k=0(qet)k]=p(1+t+t22!+t33!+t44!+)×[1+q(1+t+t22!+t33!+t44!+)+q2(1+2t+22t22!+23t33!+24t44!+)+q3(1+3t+32t22!+33t33!+34t44!+)+q4(1+4t+42t22!+43t33!+44t44!+)+]=p(1+t+t22!+t33!+t44!+)×[(k=0qk)+(k=0kqk)t+(k=0k2qk)t22!+(k=0k3qk)t33!+(k=0k4qk)t44!+] Note that: k=0qk=11qk=0kqk=(qq)k=0qk=q(1q)2=1(1q)+1(1q)2k=0k2qk=(qq)2k=0qk=q(1+q)(1q)3=1(1q)3(1q)2+2(1q)3k=0k3qk=(qq)3k=0qk=q(1+4q+q2)(1q)4=1(1q)+7(1q)212(1q)3+6(1q)4k=0k4qk=(qq)4k=0qk=q(1+11q+11q2+q3)(1q)5=1(1q)15(1q)2+50(1q)360(1q)4+24(1q)5 We find: MX(t)=p(1+t+t22!+t33!+t44!+)×[11q+(q(1q)2)t+(q(1+q)(1q)3)t22!+(q(1+4q+q2)(1q)4)t33!+(q(1+11q+11q2+q3)(1q)5)t44!+]=p1q+p(11q+q(1q)2)t+p(1(1q)+2!q(1q)2+q(1+q)(1q)3)t22!+p(1(1q)+3!q2!(1q)2+3!q(1+q)2!(1q)3+q(1+4q+q2)(1q)4)t33!+p(1(1q)+4!q3!(1q)2+4!q(1+q)2!2!(1q)3+4!q(1+4q+q2)3!(1q)4+q(1+11q+11q2+q3)(1q)5)t44!+=p1q+p(1q)2t+p(1+q)(1q)3t22!+p(1+4q+q2)(1q)4t33!+p(1+q)(1+10q+q2)(1q)5t44!+=1+1pt+(2p)p2t22!+(66p+p2)p3t33!+(2p)(1212p+p2)p4t44!+=1+1pt+(1p+2p2)t22!+(1p6p2+6p3)t33!+(1p+14p236p3+24p4)t44!+ which yields: X=1pX2=(2p)p2=1p+2p2X3=66p+p2p3=1p6p2+6p3X4=(2p)(1212p+p2)p4=1p+14p236p3+24p4X5=120240p+150p230p3+p4p5=1p30p2+150p3240p4+120p5X6=(2p)(360720p+420p260p3+p4)p6=1p+62p2540p3+1560p41800p5+720p6X7=504015120p+16800p28400p3+1806p4126p5+p6p7=1p126p2+1860p38400p4+16800p515120p6+5040p7X8=(2p)(2016060480p+65520p230240p3+5292p4252p5+p6)p8=1p+254p25796p3+40824p4126000p5+191520p6141120p7+40320p8

5.2 Derivatives of MGF

Xn=dnMX(t)dtn|t=0=M(n)X(0) We find the derivatives: MX(t)=pet1qetddtMX(t)=ddt(pet1qet)=pet(1qet)+petqet(1qet)2=pet(1qet)2d2dt2MX(t)=ddt(pet(1qet)2)=pet(1qet)2+2petqet(1qet)(1qet)4=pet(1qet)+2petqet(1qet)3=pet(1+qet)(1qet)3d3dt3MX(t)=ddt(pet(1+qet)(1qet)3)=[pet(1+qet)+petqet](1qet)3+3petqet(1+qet)(1qet)2(1qet)6=pet(1+2qet)(1qet)+3petqet(1+qet)(1qet)4=pet[(1+2qet)(1qet)+3qet(1+qet)](1qet)4=pet[(1+qet2q2e2t)+(3qet+3q2e2t)](1qet)4=pet(1+4qet+q2e2t)(1qet)4d4dt4MX(t)=ddt(pet(1+4qet+q2e2t)(1qet)4)=[pet(1+4qet+q2e2t)+pet(4qet+2q2e2t)](1qet)4(1qet)8+pet4qet(1+4qet+q2e2t)(1qet)3(1qet)8=pet(1+8qet+3q2e2t)(1qet)+pet4qet(1+4qet+q2e2t)(1qet)5=pet[(1+7qet5q2e2t3q3e3t)+(4qet+16q2e2t+4q3e3t)](1qet)5=pet(1+11qet+11q2e2t+q3e3t)(1qet)5=pet(1+qet)(1+10qet+11q2e2t)(1qet)5d5dt5MX(t)=pet(1+26qet+66q2e2t+26q3e3t+q4e4t)(1qet)6d6dt6MX(t)=pet(1+57qet+302q2e2t+302q3e3t+57q4e4t+q5e5t)(1qet)7=pet(1+qet)(1+56qet+246q2e2t+56q3e3t+q4e4t)(1qet)7d7dt7MX(t)=pet(1+120qet+1191q2e2t+2416q3e3t+1191q4e4t+120q5e5t+q6e6t)(1qet)8d8dt8MX(t)=pet(1+247qet+4293q2e2t+15619q3e3t+15619q4e4t+4293q5e5t+247q6e6t+q7e7t)(1qet)9=pet(1+qet)(1+246qet+4047q2e2t+11572q3e3t+4047q4e4t+246q5e5t+q6e6t)(1qet)9

 

In summary, \begin{align} M_X(t) &= \frac{ p \textrm{e}^{t} }{ 1 - q \textrm{e}^{t} } \\ \\ \frac{ \textrm d }{ \textrm dt } M_X(t) &= \frac{ p \textrm{e}^{t} }{ \left( 1 - q \textrm{e}^{t} \right)^2 } \\ \\ \frac{ \textrm d^2 }{ \textrm dt^2 } M_X(t) &= \frac{ p \textrm{e}^{t} \left( 1 + q \textrm{e}^{t} \right) }{ \left( 1 - q \textrm{e}^{t} \right)^3 } \\ \\ \frac{ \textrm d^3 }{ \textrm dt^3 } M_X(t) &= \frac{ p \textrm{e}^{t} \left( 1 + 4 q \textrm{e}^{t} + q^2 \textrm{e}^{ 2t } \right) }{ \left( 1 - q \textrm{e}^{t} \right)^4 } \\ \\ \frac{ \textrm d^4 }{ \textrm dt^4 } M_X(t) &= \frac{ p \textrm{e}^{t} \left( 1 + 11 q \textrm{e}^{t} + 11 q^2 \textrm{e}^{ 2t } + q^3 \textrm{e}^{ 3t } \right) }{ \left( 1 - q \textrm{e}^{t} \right)^5 } \\ &= \frac{ p \textrm{e}^{t} \left( 1 + q \textrm{e}^{t} \right) \left( 1 + 10 q \textrm{e}^{t} +11 q^2 \textrm{e}^{ 2t } \right) }{ \left( 1 - q \textrm{e}^{t} \right)^5 } \\ \\ \frac{ \textrm d^5 }{ \textrm dt^5 } M_X(t) &= \frac{ p \textrm{e}^{t} \left( 1 + 26 q \textrm{e}^{t} + 66 q^2 \textrm{e}^{ 2t } + 26 q^3 \textrm{e}^{ 3t } + q^4 \textrm{e}^{ 4t } \right) }{ \left( 1 - q \textrm{e}^{t} \right)^6 } \\ \\ \frac{ \textrm d^6 }{ \textrm dt^6 } M_X(t) &= \frac{ p \textrm{e}^{t} \left( 1 + 57 q \textrm{e}^{t} + 302 q^2 \textrm{e}^{ 2t } + 302 q^3 \textrm{e}^{ 3t } + 57 q^4 \textrm{e}^{ 4t } + q^5 \textrm{e}^{ 5t } \right) }{ \left( 1 - q \textrm{e}^{t} \right)^7 } \\ &= \frac{ p \textrm{e}^{t} \left( 1 + q \textrm{e}^{t} \right) \left( 1 + 56 q \textrm{e}^{t} + 246 q^2 \textrm{e}^{ 2t } + 56 q^3 \textrm{e}^{ 3t } + q^4 \textrm{e}^{ 4t } \right) }{ \left( 1 - q \textrm{e}^{t} \right)^7 } \\ \\ \frac{ \textrm d^7 }{ \textrm dt^7 } M_X(t) &= \frac{ p \textrm{e}^{t} \left( 1 + 120 q \textrm{e}^{t} + 1191 q^2 \textrm{e}^{ 2t } + 2416 q^3 \textrm{e}^{ 3t } + 1191 q^4 \textrm{e}^{ 4t } + 120 q^5 \textrm{e}^{ 5t } + q^6 \textrm{e}^{ 6t } \right) }{ \left( 1 - q \textrm{e}^{t} \right)^8 } \\ \\ \frac{ \textrm d^8 }{ \textrm dt^8 } M_X(t) &= \frac{ p \textrm{e}^{t} \left( 1 + 247 q \textrm{e}^{t} + 4293 q^2 \textrm{e}^{ 2t } + 15619 q^3 \textrm{e}^{ 3t } + 15619 q^4 \textrm{e}^{ 4t } + 4293 q^5 \textrm{e}^{ 5t } + 247 q^6 \textrm{e}^{ 6t } + q^7 \textrm{e}^{ 7t } \right) }{ \left( 1 - q \textrm{e}^{t} \right)^9 } \\ &= \frac{ p \textrm{e}^{t} \left( 1 + q \textrm{e}^{t} \right) \left( 1 + 246 q \textrm{e}^{t} + 4047 q^2 \textrm{e}^{ 2t } + 11572 q^3 \textrm{e}^{ 3t } + 4047 q^4 \textrm{e}^{ 4t } + 246 q^5 \textrm{e}^{ 5t } + q^6 \textrm{e}^{ 6t } \right) }{ \left( 1 - q \textrm{e}^{t} \right)^9 } \end{align}

 

Evaluating the derivatives at t = 0 yield moments: \begin{align} \left\langle X \right\rangle = M'_X(0) &= \frac{ p }{ \left( 1 - q \right)^2 } = \frac1p \\ \\ \left\langle X^2 \right\rangle = M''_X(0) &= \frac{ 1 + q }{ p^2 } \\ \\ \left\langle X^3 \right\rangle = M^{ (3) }_X(0) &= \frac{ 1 + 4 q + q^2 }{ p^3 } \\ \\ \left\langle X^4 \right\rangle = M^{ (4) }_X(0) &= \frac{ \left( 1 + 11 q + 11 q^2 + q^3 \right) }{ p^4 } \\ &= \frac{ \left( 1 + q \right) \left( 1 + 10 q + 11 q^2 \right) }{ p^4 } \\ \\ \left\langle X^5 \right\rangle = M^{ (5) }_X(0) &= \frac{ 1 + 26 q + 66 q^2 + 26 q^3 + q^4 }{ p^5 } \\ \\ \left\langle X^6 \right\rangle = M^{ (6) }_X(0) &= \frac{  1 + 57 q + 302 q^2 + 302 q^3 + 57 q^4 + q^5 }{ p^6 } \\ &= \frac{ \left( 1 + q \right) \left( 1 + 56 q + 246 q^2 + 56 q^3 + q^4 \right) }{ p^6 } \\ \\ \left\langle X^7 \right\rangle = M^{ (7) }_X(0) &= \frac{ \left( 1 + 120 q + 1191 q^2 + 2416 q^3 + 1191 q^4 + 120 q^5 + q^6 \right) }{ p^7 } \\ \\ \left\langle X^8 \right\rangle = M^{ (8) }_X(0) &= \frac{ \left( 1 + 247 q + 4293 q^2 + 15619 q^3 + 15619 q^4 + 4293 q^5 + 247 q^6 + q^7 \right) }{ p^8 } \\ &= \frac{ \left( 1 + q \right) \left( 1 + 246 q + 4047 q^2 + 11572 q^3 + 4047 q^4 + 246 q^5 + q^6 \right) }{ p^8 } \end{align}

6. Central moments

The central moments are the moments about the mean value, i.e. \begin{align} \mu_n = \mathbb E \left[ (X-\mu)^n \right] = \sum_{x} (x-\mu)^n \mathbb P(x) \end{align} For a geometric distribution, X \sim \textrm{Geo}(p) , \begin{align} \mu_n = \mathbb E \left[ (X-\mu)^n \right] &= \sum_{x=1}^\infty (x-\mu)^n pq^{x-1} \\ &= \sum_{x=1}^\infty (x-\mu)^n pq^{x-\mu+\mu-1} \\ &= pq^{\mu-1} \sum_{x=1}^\infty (x-\mu)^n q^{x-\mu} \\ &= pq^{\mu-1} \left( q \frac{ \partial }{ \partial q } \right)^n \sum_{x=1}^\infty q^{x-\mu} \\ &= pq^{\mu-1} \left( q \frac{ \partial }{ \partial q } \right)^n \frac{ q^{1 - \mu} }{ 1 - q } \end{align} We note that: \begin{align} \left( q \frac{ \partial }{ \partial q } \right) \frac{ q^{1 - \mu} }{ 1 - q } &= \frac{ (1-\mu) q^{1-\mu }}{ 1-q } + \frac{ q^{2-\mu} }{ (1-q)^2 } \\ &= \frac{ q^{1-\mu} [ - \mu (1-q) + 1 ] }{ (1-q)^2 } \\ \\ \left( q \frac{ \partial }{ \partial q } \right)^2 \frac{ q^{1 - \mu} }{ 1 - q } &= \frac{ (1 - \mu)^2 q^{1-\mu} }{ 1-q } + \frac{ (3 - 2\mu) q^{2-\mu} }{ (1-q)^2 } + \frac{ 2 q^{3-\mu} }{ (1-q)^3 } \\ &= \frac{ q^{1-\mu} \left[ \mu^2 (1-q)^2 - 2\mu (1-q) + (1+q) \right] }{ (1-q)^3 } \\ \\ \left( q \frac{ \partial }{ \partial q } \right)^3 \frac{ q^{1 - \mu} }{ 1 - q } &= \frac{ (1 - \mu)^3 q^{1-\mu} }{ 1-q } + \frac{ \left( 7 - 9 \mu + 3 \mu^2 \right) q^{2-\mu} }{ (1-q)^2 } + \frac{ 6 ( 2 - \mu ) q^{3-\mu} }{ (1-q)^3 } + \frac{ 6 q^{4-\mu} }{ (1-q)^4 } \\ &= \frac{ q^{1-\mu} \left[ - \mu^3 (1-q)^3 + 3 \mu^2 (1-q)^2 - 3\mu (1+q)(1-q) + \left( 1 + 4q + q^2 \right) \right] }{ (1-q)^4 } \\ \\ \left( q \frac{ \partial }{ \partial q } \right)^4 \frac{ q^{1 - \mu} }{ 1 - q } &= \frac{ (1 - \mu)^4 q^{1-\mu} }{ 1-q } + \frac{ \left( 15 - 28 \mu + 18 \mu^2 - 4 \mu^3 \right) q^{2-\mu} }{ (1-q)^2 } + \frac{ 2 \left( 25 - 24 \mu + 6 \mu^2 \right) q^{3-\mu} }{ (1-q)^3 } + \frac{ 12 ( 5 - 2 \mu ) q^{4-\mu} }{ (1-q)^4 } + \frac{ 24 q^{5-\mu} }{ (1-q)^4 } \\ &= \frac{ q^{1-\mu} \left[ \mu^4 (1-q)^4 - 4 \mu^3 (1-q)^3 + 6 \mu^2 (1+q)(1-q)^2 - 4 \mu \left(1 + 4q +q^2 \right)(1-q) + (1+q) \left( 1 + 10q + q^2 \right) \right] }{ (1-q)^5 } \end{align} and, noting that \mu (1-q) = \frac{1}{p} \times p = 1 , \begin{align} \mu_1 &= \left\langle (X-\mu) \right\rangle \\ &= \frac{ p [ - \mu (1-q) + 1 ] }{ (1-q)^2 } = 0 \\ \\ \mu_2 &= \left\langle (X-\mu)^2 \right\rangle \\ &= \frac{ p \left[ \mu^2 (1-q)^2 - 2 \mu (1-q) + (1+q) \right] }{ (1-q)^3 } \\ &= \frac{ p \left[ 1 - 2 + (1+q) \right] }{ p^3 } \\ &= \frac{ q }{ p^2 } \\ &= \frac{ 1-p }{ p^2 } \\ \\ \mu_3 &= \left\langle (X-\mu)^3 \right\rangle \\ &= \frac{ p \left[ - \mu^3 (1-q)^3 + 3 \mu^2 (1-q)^2 - 3\mu (1+q)(1-q) + \left( 1 + 4q + q^2 \right) \right] }{ (1-q)^4 } \\ &= \frac{ p \left[ - 1 + 3 - 3 (1+q) + \left( 1 + 4q + q^2 \right) \right] }{ p^4 } \\ &= \frac{ - 1 -3q + \left( 1 + 4q + q^2 \right) }{ p^3 } \\ &= \frac{ q + q^2 }{ p^3 } \\ &= \frac{ (1-p)(2-p) }{ p^3 } \\ \\ \mu_4 &= \left\langle (X-\mu)^4 \right\rangle \\ &= \frac{ p \left[ \mu^4 (1-q)^4 - 4 \mu^3 (1-q)^3 + 6 \mu^2 (1+q)(1-q)^2 - 4 \mu \left(1 + 4q +q^2 \right)(1-q) + (1+q) \left( 1 + 10q + q^2 \right) \right] }{ (1-q)^5 } \\ &= \frac{ p \left[ 1 - 4 + 6 (1+q) - 4 \left(1 + 4q +q^2 \right) + \left( 1 + 11q + 11q^2 + q^3 \right) \right] }{ p^5 } \\ &= \frac{ q + 7q^2 + q^3 }{ p^4 } \\ &= \frac{ 9 - 18 p + 10 p^2 - p^3 }{ p^4 } \\ &= \frac{ (1-p) \left( 9 - 9p + p^2 \right) }{ p^4 } \\ \end{align} In addition, \begin{align} \mu_5 &= \left\langle (X-\mu)^5 \right\rangle = \frac{ (1-p)(2-p) \left( p^2 - 22p + 22 \right) }{ p^5 } \\ \mu_6 &= \left\langle (X-\mu)^6 \right\rangle = \frac{ (1-p) \left( p^4 - 55 p^3 + 320 p^2 - 530 p + 265 \right) }{ p^6 } \\ \mu_7 &= \left\langle (X-\mu)^7 \right\rangle = \frac{ (1-p)(2-p) \left( p^4 - 116 p^3 + 1043 p^2 - 1854 p + 927 \right) }{ p^7 } \\ \mu_8 &= \left\langle (X-\mu)^8 \right\rangle = \frac{ (1-p) \left( p^6 - 245 p^5 + 4571 p^4 - 23485 p^3 + 48825 p^2 - 44499 p + 14833 \right) }{ p^8 } \\ \end{align}

7. Standardised moments

\begin{align} \mathbb E \left[ \left( \frac{ X-\mu }{ \sigma } \right)^n \right] = \sum_{x=1}^\infty \left( \frac{ X-\mu }{ \sigma } \right)^n \mathbb P(x) \end{align}

 

\begin{align} \mathbb E \left[ \frac{ X-\mu }{ \sigma } \right] &= \frac{ \mu_1 }{ \sqrt{ \mu_2 } } = 0 \\ \mathbb E \left[ \left( \frac{ X-\mu }{ \sigma } \right)^2 \right] &= \frac{ \mu_2 }{ \mu_2 } = 1 \\ \mathbb E \left[ \left( \frac{ X-\mu }{ \sigma } \right)^3 \right] &= \frac{ \mu_3 }{ \mu_2^{\frac32} } = \frac{ (1-p)(2-p) }{ p^3 } \cdot \frac{ p^3 }{ (1-p)^{\frac32} } = \frac{ 2 - p }{ \sqrt{ 1 - p } } \\ \mathbb E \left[ \left( \frac{ X-\mu }{ \sigma } \right)^4 \right] &= \frac{ \mu_4 }{ \mu_2^2 } = \frac{ (1-p) \left( 9 - 9p + p^2 \right) }{ p^4 } \cdot \frac{ p^4 }{ (1-p)^2 } = \frac{ 9 - 9p + p^2 }{ 1 - p } = 9 + \frac{ p^2 }{ 1 - p } \\ \mathbb E \left[ \left( \frac{ X-\mu }{ \sigma } \right)^5 \right] &= \frac{ \mu_5 }{ \mu_2^{\frac52} } = \frac{ (1-p)(2-p) \left( p^2 - 22p + 22 \right) }{ p^5 } \cdot \frac{ p^5 }{ (1-p)^{\frac52} } \\ &= \frac{ (2-p) \left( p^2 - 22p + 22 \right) }{ (1-p)^{\frac32} } \\ \mathbb E \left[ \left( \frac{ X-\mu }{ \sigma } \right)^6 \right] &= \frac{ \mu_6 }{ \mu_2^3 } = \frac{ (1-p) \left( p^4 - 55 p^3 + 320 p^2 - 530 p + 265 \right) }{ p^6 } \cdot \frac{ p^6 }{ (1-p)^3 } \\ &= \frac{ p^4 - 55 p^3 + 320 p^2 - 530 p + 265 }{ (1-p)^2 } \\ \mathbb E \left[ \left( \frac{ X-\mu }{ \sigma } \right)^7 \right] &= \frac{ \mu_7 }{ \mu_2^{\frac72} } = \frac{ (1-p)(2-p) \left( p^4 - 116 p^3 + 1043 p^2 - 1854 p + 927 \right) }{ p^7 } \cdot \frac{ p^7 }{ (1-p)^{\frac72} } \\ &= \frac{ (2-p) \left( p^4 - 116 p^3 + 1043 p^2 - 1854 p + 927 \right) }{ (1-p)^{\frac52} } \\ \mathbb E \left[ \left( \frac{ X-\mu }{ \sigma } \right)^8 \right] &= \frac{ \mu_8 }{ \mu_2^4 } = \frac{ (1-p) \left( p^6 - 245 p^5 + 4571 p^4 - 23485 p^3 + 48825 p^2 - 44499 p + 14833 \right) }{ p^8 } \cdot \frac{ p^8 }{ (1-p)^4 } \\ &= \frac{ p^6 - 245 p^5 + 4571 p^4 - 23485 p^3 + 48825 p^2 - 44499 p + 14833 }{ (1-p)^3 } \\ \end{align}

 

반응형
Comments