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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. 02:23반응형
Further statistics 1
Table of contents
- Probability mass function
- Probability generating function
- Moment generating function
- Moments
- Using moment generating function
- Central moments
- Standardised moments
- Edexcel FS1 Ch Exercise (under construction)
1. Probability mass function
The probability mass function for a geometric distribution, X∼Geo(p), is P(x)=pqx−1,x=1,2,3,⋯ where p+q=1.Sum of probabilities is 1
∞∑x=1P(x)=∞∑x=1pqx−1=p(11−q)=p⋅1p=12. Probability generating function
GX(t)=P(0)+P(1)t+P(2)t2+P(3)t3+⋯=∞∑x=1P(x)tx=⟨tX⟩=∞∑x=1pqx−1tx=pq∞∑x=1(qt)x=pq⋅qt1−qt=pt1−qt=pt1−(1−p)t3. Moment generating function
MX(t)=1+⟨X⟩t+12!⟨X2⟩t2+13!⟨X3⟩t3+⋯=⟨1+Xt+12!(Xt)2+13!(Xt)3+⋯⟩=⟨eXt⟩=∞∑x=1P(x)ext=∞∑x=1pqx−1(et)x=pq∞∑x=1(qet)x=pq⋅qet1−qet=pet1−qet=pet1−(1−p)et4. Moments
4.1. Mean = First moment
E(X)=∞∑x=1xP(X=x)=∞∑x=1xpqx−1=pq∞∑x=1xqx=pq(q∂∂q)∞∑x=1qx=pq(q∂∂q)q1−q=pq(q∂∂q)[q−1+11−q]=pq(q∂∂q)[−1+11−q]=pq⋅q(1−q)2=p(1−q)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+⋯⇒(1−q)S=q+q2+q3+q4+⋯=∞∑x=1qx=q1−q⇒S=q(1−q)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=1xnpqx−1=pq∞∑x=1xnqx=pq(q∂∂q)n∞∑x=1qx=pq(q∂∂q)nq1−q It is convenient to work out: (q∂∂q)2=q∂∂q+q2∂2∂q2(q∂∂q)3=q∂∂q(q∂∂q+q2∂2∂q2)=(q∂∂q+q2∂2∂q2)+(2q2∂2∂q2+q3∂3∂q3)=q∂∂q+3q2∂2∂q2+q3∂3∂q3(q∂∂q)4=q∂∂q(q∂∂q+3q2∂2∂q2+q3∂3∂q3)=(q∂∂q+q2∂2∂q2)+(6q2∂2∂q2+3q3∂3∂q3)+(3q3∂3∂q3+q4∂4∂q4)=q∂∂q+7q2∂2∂q2+6q3∂3∂q3+q4∂4∂q4(q∂∂q)5=q∂∂q(q∂∂q+7q2∂2∂q2+6q3∂3∂q3+q4∂4∂q4)=(q∂∂q+q2∂2∂q2)+(14q2∂2∂q2+7q3∂3∂q3)+(18q3∂3∂q3+6q4∂4∂q4)+(4q4∂4∂q4+q5∂5∂q5)=q∂∂q+15q2∂2∂q2+25q3∂3∂q3+10q4∂4∂q4+q5∂5∂q5(q∂∂q)6=q∂∂q(q∂∂q+15q2∂2∂q2+25q3∂3∂q3+10q4∂4∂q4+q5∂5∂q5)=(q∂∂q+q2∂2∂q2)+(30q2∂2∂q2+15q3∂3∂q3)+(75q3∂3∂q3+25q4∂4∂q4)+(40q4∂4∂q4+10q5∂5∂q5)+(5q5∂5∂q5+q6∂6∂q6)=q∂∂q+31q2∂2∂q2+90q3∂3∂q3+65q4∂4∂q4+15q5∂5∂q5+q6∂6∂q6(q∂∂q)7=q∂∂q(q∂∂q+31q2∂2∂q2+90q3∂3∂q3+65q4∂4∂q4+15q5∂5∂q5+q6∂6∂q6)=(q∂∂q+q2∂2∂q2)+(62q2∂2∂q2+31q3∂3∂q3)+(270q3∂3∂q3+90q4∂4∂q4)+(260q4∂4∂q4+65q5∂5∂q5)+(75q5∂5∂q5+15q6∂6∂q6)+(6q6∂6∂q6+q7∂7∂q7)=q∂∂q+63q2∂2∂q2+301q3∂3∂q3+350q4∂4∂q4+140q5∂5∂q5+21q6∂6∂q6+q7∂7∂q7(q∂∂q)8=q∂∂q(q∂∂q+63q2∂2∂q2+301q3∂3∂q3+350q4∂4∂q4+140q5∂5∂q5+21q6∂6∂q6+q7∂7∂q7)=(q∂∂q+q2∂2∂q2)+(126q2∂2∂q2+63q3∂3∂q3)+(903q3∂3∂q3+301q4∂4∂q4)+(1400q4∂4∂q4+350q5∂5∂q5)+(700q5∂5∂q5+140q6∂6∂q6)+(126q6∂6∂q6+21q7∂7∂q7)+(7q7∂7∂q7+q8∂8∂q8)=q∂∂q+127q2∂2∂q2+966q3∂3∂q3+1701q4∂4∂q4+1050q5∂5∂q5+266q6∂6∂q6+28q7∂7∂q7+q8∂8∂q8(q∂q)2=q∂q+q2∂2q(q∂q)3=q∂q(q∂q+q2∂2q)=(q∂q+q2∂2q)+(2q2∂2q+q3∂3q)=q∂q+3q2∂2q+q3∂3q(q∂q)4=q∂q(q∂q+3q2∂2q+q3∂3q)=(q∂q+q2∂2q)+(6q2∂2q+3q3∂3q)+(3q3∂3q+q4∂4q)=q∂q+7q2∂2q+6q3∂3q+q4∂4q(q∂q)5=q∂q(q∂q+7q2∂2q+6q3∂3q+q4∂4q)=(q∂q+q2∂2q)+(14q2∂2q+7q3∂3q)+(18q3∂3q+6q4∂4q)+(4q4∂4q+q5∂5q)=q∂q+15q2∂2q+25q3∂3q+10q4∂4q+q5∂5q(q∂q)6=q∂q(q∂q+15q2∂2q+25q3∂3q+10q4∂4q+q5∂5q)=(q∂q+q2∂2q)+(30q2∂2q+15q3∂3q)+(75q3∂3q+25q4∂4q)+(40q4∂4q+10q5∂5q)+(5q5∂5q+q6∂6q)=q∂q+31q2∂2q+90q3∂3q+65q4∂4q+15q5∂5q+q6∂6q(q∂q)7=q∂q(q∂q+31q2∂2q+90q3∂3q+65q4∂4q+15q5∂5q+q6∂6q)=(q∂q+q2∂2q)+(62q2∂2q+31q3∂3q)+(270q3∂3q+90q4∂4q)+(260q4∂4q+65q5∂5q)+(75q5∂5q+15q6∂6q)+(6q6∂6q+q7∂7q)=q∂q+63q2∂2q+301q3∂3q+350q4∂4q+140q5∂5q+21q6∂6q+q7∂7q(q∂q)8=q∂q(q∂q+63q2∂2q+301q3∂3q+350q4∂4q+140q5∂5q+21q6∂6q+q7∂7q)=(q∂q+q2∂2q)+(126q2∂2q+63q3∂3q)+(903q3∂3q+301q4∂4q)+(1400q4∂4q+350q5∂5q)+(700q5∂5q+140q6∂6q)+(126q6∂6q+21q7∂7q)+(7q7∂7q+q8∂8q)=q∂q+127q2∂2q+966q3∂3q+1701q4∂4q+1050q5∂5q+266q6∂6q+28q7∂7q+q8∂8q
In summary, (q∂q)2=q∂q+q2∂2q(q∂q)3=q∂q+3q2∂2q+q3∂3q(q∂q)4=q∂q+7q2∂2q+6q3∂3q+q4∂4q(q∂q)5=q∂q+15q2∂2q+25q3∂3q+10q4∂4q+q5∂5q(q∂q)6=q∂q+31q2∂2q+90q3∂3q+65q4∂4q+15q5∂5q+q6∂6q(q∂q)7=q∂q+63q2∂2q+301q3∂3q+350q4∂4q+140q5∂5q+21q6∂6q+q7∂7q(q∂q)8=q∂q+127q2∂2q+966q3∂3q+1701q4∂4q+1050q5∂5q+266q6∂6q+28q7∂7q+q8∂8q
It is useful to Taylor expand qn in powers of (1−q). Note that we observe the binomial coefficients: q=1−(1−q)q2=1−2(1−q)+(1−q)2q3=1−3(1−q)+3(1−q)2−(1−q)3q4=1−4(1−q)+6(1−q)2−4(1−q)3+(1−q)4q5=1−5(1−q)+10(1−q)2−10(1−q)3+5(1−q)4−(1−q)5q6=1−6(1−q)+15(1−q)2−20(1−q)3+15(1−q)4−6(1−q)5+(1−q)6q7=1−7(1−q)+21(1−q)2+35(1−q)3−35(1−q)4+21(1−q)5−7(1−q)6+(1−q)7q8=1−8(1−q)+28(1−q)2+56(1−q)3−70(1−q)4+56(1−q)5−28(1−q)6+8(1−q)7−(1−q)8 The Taylor expansions above are used to express the following derivatives in partial fractions: (q∂∂q)q1−q=(q∂∂q)[q−1+11−q]=(q∂∂q)[−1+11−q]=q(1−q)2(q∂∂q)2q1−q=(q∂q+q2∂2q)[q−1+11−q]=(q∂q+q2∂2q)[−1+11−q]=q(1−q)2+2!q2(1−q)3=q[1(1−q)2+2!q(1−q)3]=q[1(1−q)2+2[1−(1−q)](1−q)3]=q[1(1−q)2+2(1−q)3−2(1−q)2]=q[−1(1−q)2+2(1−q)3](q∂∂q)3q1−q=(q∂q+3q2∂2q+q3∂3q)[−1+11−q]=q(1−q)2+(3⋅2!)q2(1−q)3+3!q3(1−q)4=q[1(1−q)2+6q(1−q)3+6q2(1−q)4]=q[1(1−q)2+6[1−(1−q)](1−q)3+6[1−2(1−q)+(1−q)2](1−q)4]=q[1(1−q)2+6(1−q)3−6(1−q)2+6(1−q)4−12(1−q)3+6(1−q)2]=q[1(1−q)2−6(1−q)3+6(1−q)4](q∂∂q)4q1−q=(q∂q+7q2∂2q+6q3∂3q+q4∂4q)[−1+11−q]=q(1−q)2+(7⋅2!)q2(1−q)3+(6⋅3!)q3(1−q)4+4!q4(1−q)5=q[1(1−q)2+14q(1−q)3+36q2(1−q)4+24q3(1−q)5]=q[1(1−q)2+14[1−(1−q)](1−q)3+36[1−2(1−q)+(1−q)2](1−q)4+24[1−3(1−q)+3(1−q)2−(1−q)3](1−q)5]=q[1(1−q)2+14(1−q)3−14(1−q)2+36(1−q)4−72(1−q)3+36(1−q)2+24(1−q)5−72(1−q)4+72(1−q)3−24(1−q)2]=q[−1(1−q)2+14(1−q)3−36(1−q)4+24(1−q)5](q∂∂q)5q1−q=(q∂q+15q2∂2q+25q3∂3q+10q4∂4q+q5∂5q)[−1+11−q]=q[1(1−q)2−30(1−q)3+150(1−q)4−240(1−q)5+120(1−q)6](q∂∂q)6q1−q=(q∂q+31q2∂2q+90q3∂3q+65q4∂4q+15q5∂5q+q6∂6q)[−1+11−q]=q[−1(1−q)2+62(1−q)3−540(1−q)4+1560(1−q)5−1800(1−q)6+720(1−q)7](q∂∂q)7q1−q=(q∂q+63q2∂2q+301q3∂3q+350q4∂4q+140q5∂5q+21q6∂6q+q7∂7q)[−1+11−q]=q[1(1−q)2−126(1−q)3+1860(1−q)4−8400(1−q)5+16800(1−q)6−15120(1−q)7+5040(1−q)8](q∂∂q)8q1−q=(q∂q+127q2∂2q+966q3∂3q+1701q4∂4q+1050q5∂5q+266q6∂6q+28q7∂7q+q8∂8q)[−1+11−q]=q[−1(1−q)2+254(1−q)3−5796(1−q)4+40824(1−q)5−126000(1−q)6+191520(1−q)7−141120(1−q)8+40320(1−q)9]
The moments: (q∂∂q)q1−q=qp2⇒E(X)=pq(q∂∂q)q1−q=1p(q∂∂q)2q1−q=q[−1p2+2p3]⇒E(X2)=pq(q∂∂q)2q1−q=−1p+2p2(q∂∂q)3q1−q=q[1p2−6p3+6p4]⇒E(X3)=pq(q∂∂q)3q1−q=1p−6p2+6p3(q∂∂q)4q1−q=q[−1p2+14p3−36p4+24p5]⇒E(X4)=pq(q∂∂q)4q1−q=−1p+14p2−36p3+24p4(q∂∂q)5q1−q=q[1p2−30p3+150p4−240p5+120p6]⇒E(X5)=pq(q∂∂q)5q1−q=1p−30p2+150p3−240p4+120p5(q∂∂q)6q1−q=q[−1p2+62p3−540p4+1560p5−1800p6+720p7]⇒E(X6)=pq(q∂∂q)6q1−q=−1p+62p2−540p3+1560p4−1800p5+720p6(q∂∂q)7q1−q=q[1p2−126p3+1860p4−8400p5+16800p6−15120p7+5040p8]⇒E(X7)=pq(q∂∂q)7q1−q=1p−126p2+1860p3−8400p4+16800p5−15120p6+5040p7(q∂∂q)8q1−q=q[−1p2+254p3−5796p4+40824p5−126000p6+191520p7−141120p8+40320p9]⇒E(X8)=pq(q∂∂q)8q1−q=−1p+254p2−5796p3+40824p4−126000p5+191520p6−141120p7+40320p8
For a geometric distribution, X∼Geo(p), E(Xn)=∞∑x=1xnpqx−1 E(X)=1pE(X2)=−1p+2p2E(X3)=1p−6p2+6p3E(X4)=−1p+14p2−36p3+24p4E(X5)=1p−30p2+150p3−240p4+120p5E(X6)=−1p+62p2−540p3+1560p4−1800p5+720p6E(X7)=1p−126p2+1860p3−8400p4+16800p5−15120p6+5040p7E(X8)=−1p+254p2−5796p3+40824p4−126000p5+191520p6−141120p7+40320p8
E(X)=1pE(X2)=−1p+2p2=2−pp2E(X3)=1p−6p2+6p3=p2−6p+6p3E(X4)=−1p+14p2−36p3+24p4=(2−p)(12−12p+p2)p4E(X5)=1p−30p2+150p3−240p4+120p5=p4−30p3+150p2−240p+120p5E(X6)=−1p+62p2−540p3+1560p4−1800p5+720p6=(2−p)(360−720p+420p2−60p3+p4)p6E(X7)=1p−126p2+1860p3−8400p4+16800p5−15120p6+5040p7=p6−126p5+1860p4−8400p3+16800p2−15120p+5040p7E(X8)=−1p+254p2−5796p3+40824p4−126000p5+191520p6−141120p7+40320p8=(2−p)(20160−60480p+65520p2−30240p3+5292p4−252p5+p6)p8
4.3. E(X2) and variance
For variance, Var(X)=E[(X−μ)2]=⟨X2⟩−⟨X⟩2=(−1p+2p2)−(1p)2=−1p+1p2=1−pp2=qp2Aside. An alternative way to evaluate ∑x2qx: S=∞∑x=1x2qx=q+4q2+9q3+16q4+⋯qS=∞∑x=1x2qx+1=q2+4q3+9q4+16q5+⋯⇒(1−q)S=q+3q2+5q3+7q4+⋯=∞∑x=1(2x−1)qxq(1−q)S=q2+3q3+5q4+7q5+⋯=∞∑x=1(2x−1)qx+1⇒(1−q)2S=q+2q2+2q3+2q4+⋯=2(q+q2+q3+⋯)−q=2q1−q−q=2q−q(1−q)1−q⇒S=q(1+q)(1−q)3=q(2−p)p3✓
4.4. E(X3) and skewness
Recall: E(X)=1pE(X2)=−1p+2p2E(X3)=1p−6p2+6p3E(X4)=−1p+14p2−36p3+24p4 The skewness is defined by E[(X−μσ)3]=E[X3−3μX2+3μ2X−μ3σ3]=⟨X3⟩−3μ⟨X2⟩+2μ3σ3 and thus E[(X−μσ)3]=⟨X3⟩−3μ⟨X2⟩+2μ3σ3=(1p−6p2+6p3)−3p(−1p+2p2)+2p3(qp2)32=2−3p+p2p3(qp2)32=(1−p)(2−p)q32=2−p√1−p✓4.5. E(X4) and skewness
Recall: E(X)=1pE(X2)=−1p+2p2E(X3)=1p−6p2+6p3E(X4)=−1p+14p2−36p3+24p4 The kurtosis is defined by E[(X−μσ)4]=E[X4−4μX3+6μ2X2−4μ3X+μ4σ4]=⟨X4⟩−4μ⟨X3⟩+6μ2⟨X2⟩−3μ4σ4 and thus E[(X−μσ)4]=⟨X4⟩−4μ⟨X3⟩+6μ2⟨X2⟩−3μ4σ4=(−1p+14p2−36p3+24p4)−4p(1p−6p2+6p3)+6p2(−1p+2p2)−3p4q2p4=9−18p+10p2−p3p4q2p4=(1−p)(9−9p+p2)(1−p)2=9−9p+p21−p=9+p21−p✓5. Using moment generating function
5.1 Series expansion
As we Taylor-expand MX(t), MX(t)=pet1−qet=pet(1−qet)−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=11−q∞∑k=0kqk=(q∂∂q)∞∑k=0qk=q(1−q)2=−1(1−q)+1(1−q)2∞∑k=0k2qk=(q∂∂q)2∞∑k=0qk=q(1+q)(1−q)3=1(1−q)−3(1−q)2+2(1−q)3∞∑k=0k3qk=(q∂∂q)3∞∑k=0qk=q(1+4q+q2)(1−q)4=−1(1−q)+7(1−q)2−12(1−q)3+6(1−q)4∞∑k=0k4qk=(q∂∂q)4∞∑k=0qk=q(1+11q+11q2+q3)(1−q)5=1(1−q)−15(1−q)2+50(1−q)3−60(1−q)4+24(1−q)5 We find: ⇒MX(t)=p(1+t+t22!+t33!+t44!+⋯)×[11−q+(q(1−q)2)t+(q(1+q)(1−q)3)t22!+(q(1+4q+q2)(1−q)4)t33!+(q(1+11q+11q2+q3)(1−q)5)t44!+⋯]=p1−q+p(11−q+q(1−q)2)t+p(1(1−q)+2!q(1−q)2+q(1+q)(1−q)3)t22!+p(1(1−q)+3!q2!(1−q)2+3!q(1+q)2!(1−q)3+q(1+4q+q2)(1−q)4)t33!+p(1(1−q)+4!q3!(1−q)2+4!q(1+q)2!2!(1−q)3+4!q(1+4q+q2)3!(1−q)4+q(1+11q+11q2+q3)(1−q)5)t44!+⋯=p1−q+p(1−q)2t+p(1+q)(1−q)3t22!+p(1+4q+q2)(1−q)4t33!+p(1+q)(1+10q+q2)(1−q)5t44!+⋯=1+1pt+(2−p)p2t22!+(6−6p+p2)p3t33!+(2−p)(12−12p+p2)p4t44!+⋯=1+1pt+(−1p+2p2)t22!+(1p−6p2+6p3)t33!+(−1p+14p2−36p3+24p4)t44!+⋯ which yields: ⟨X⟩=1p⟨X2⟩=(2−p)p2=−1p+2p2⟨X3⟩=6−6p+p2p3=1p−6p2+6p3⟨X4⟩=(2−p)(12−12p+p2)p4=−1p+14p2−36p3+24p4⟨X5⟩=120−240p+150p2−30p3+p4p5=1p−30p2+150p3−240p4+120p5⟨X6⟩=(2−p)(360−720p+420p2−60p3+p4)p6=−1p+62p2−540p3+1560p4−1800p5+720p6⟨X7⟩=5040−15120p+16800p2−8400p3+1806p4−126p5+p6p7=1p−126p2+1860p3−8400p4+16800p5−15120p6+5040p7⟨X8⟩=(2−p)(20160−60480p+65520p2−30240p3+5292p4−252p5+p6)p8=−1p+254p2−5796p3+40824p4−126000p5+191520p6−141120p7+40320p85.2 Derivatives of MGF
⟨Xn⟩=dnMX(t)dtn|t=0=M(n)X(0) We find the derivatives: MX(t)=pet1−qetddtMX(t)=ddt(pet1−qet)=pet(1−qet)+pet⋅qet(1−qet)2=pet(1−qet)2d2dt2MX(t)=ddt(pet(1−qet)2)=pet(1−qet)2+2pet⋅qet(1−qet)(1−qet)4=pet(1−qet)+2pet⋅qet(1−qet)3=pet(1+qet)(1−qet)3d3dt3MX(t)=ddt(pet(1+qet)(1−qet)3)=[pet(1+qet)+pet⋅qet](1−qet)3+3pet⋅qet(1+qet)(1−qet)2(1−qet)6=pet(1+2qet)(1−qet)+3pet⋅qet(1+qet)(1−qet)4=pet[(1+2qet)(1−qet)+3qet(1+qet)](1−qet)4=pet[(1+qet−2q2e2t)+(3qet+3q2e2t)](1−qet)4=pet(1+4qet+q2e2t)(1−qet)4d4dt4MX(t)=ddt(pet(1+4qet+q2e2t)(1−qet)4)=[pet(1+4qet+q2e2t)+pet(4qet+2q2e2t)](1−qet)4(1−qet)8+pet⋅4qet(1+4qet+q2e2t)(1−qet)3(1−qet)8=pet(1+8qet+3q2e2t)(1−qet)+pet⋅4qet(1+4qet+q2e2t)(1−qet)5=pet[(1+7qet−5q2e2t−3q3e3t)+(4qet+16q2e2t+4q3e3t)](1−qet)5=pet(1+11qet+11q2e2t+q3e3t)(1−qet)5=pet(1+qet)(1+10qet+11q2e2t)(1−qet)5d5dt5MX(t)=pet(1+26qet+66q2e2t+26q3e3t+q4e4t)(1−qet)6d6dt6MX(t)=pet(1+57qet+302q2e2t+302q3e3t+57q4e4t+q5e5t)(1−qet)7=pet(1+qet)(1+56qet+246q2e2t+56q3e3t+q4e4t)(1−qet)7d7dt7MX(t)=pet(1+120qet+1191q2e2t+2416q3e3t+1191q4e4t+120q5e5t+q6e6t)(1−qet)8d8dt8MX(t)=pet(1+247qet+4293q2e2t+15619q3e3t+15619q4e4t+4293q5e5t+247q6e6t+q7e7t)(1−qet)9=pet(1+qet)(1+246qet+4047q2e2t+11572q3e3t+4047q4e4t+246q5e5t+q6e6t)(1−qet)9In 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}
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