odds.tex

Paper VI: Transfer Semigroups: Stochastic, Bayesian, Fractional, and PDE Realizations

odds.tex

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1 \begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]
2 \path[use as bounding box] (0,-.6) rectangle (14,4.8);
3 \draw[black!13,line width=.25pt] (0.8,0.6) -- (0.8,3.9);
4 \draw[] (0.8,0.54) -- (0.8,0.66);
5 \node[font=\small,align=center,font=\scriptsize] at (0.8,0.34) {0};
6 \draw[black!13,line width=.25pt] (2.975,0.6) -- (2.975,3.9);
7 \draw[] (2.975,0.54) -- (2.975,0.66);
8 \node[font=\small,align=center,font=\scriptsize] at (2.975,0.34) {1};
9 \draw[black!13,line width=.25pt] (5.15,0.6) -- (5.15,3.9);
10 \draw[] (5.15,0.54) -- (5.15,0.66);
11 \node[font=\small,align=center,font=\scriptsize] at (5.15,0.34) {2};
12 \draw[black!13,line width=.25pt] (7.325,0.6) -- (7.325,3.9);
13 \draw[] (7.325,0.54) -- (7.325,0.66);
14 \node[font=\small,align=center,font=\scriptsize] at (7.325,0.34) {3};
15 \draw[black!13,line width=.25pt] (9.5,0.6) -- (9.5,3.9);
16 \draw[] (9.5,0.54) -- (9.5,0.66);
17 \node[font=\small,align=center,font=\scriptsize] at (9.5,0.34) {4};
18 \draw[black!13,line width=.25pt] (0.8,0.6) -- (9.5,0.6);
19 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.64,0.6) {0};
20 \draw[black!13,line width=.25pt] (0.8,2.25) -- (9.5,2.25);
21 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.64,2.25) {0.5};
22 \draw[black!13,line width=.25pt] (0.8,3.9) -- (9.5,3.9);
23 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.64,3.9) {1};
24 \draw[->,line width=.6pt] (0.8,0.6) -- (9.62,0.6);
25 \draw[->,line width=.6pt] (0.8,0.6) -- (0.8,4.02);
26 \node[font=\small,align=center,font=\scriptsize] at (5.15,0.05) {time $t$};
27 \node[font=\small,align=center,anchor=west,font=\scriptsize] at (0.8,4.3) {probability};
28 \draw[line width=1.1pt] plot coordinates {(0.8,2.25) (0.887,2.282996) (0.974,2.315965) (1.061,2.348881) (1.148,2.381719) (1.235,2.414452) (1.322,2.447055) (1.409,2.479503) (1.496,2.51177) (1.583,2.543833) (1.67,2.575669) (1.757,2.607255) (1.844,2.638568) (1.931,2.669588) (2.018,2.700293) (2.105,2.730666) (2.192,2.760686) (2.279,2.790338) (2.366,2.819603) (2.453,2.848467) (2.54,2.876916) (2.627,2.904935) (2.714,2.932513) (2.801,2.959639) (2.888,2.986302) (2.975,3.012493) (3.062,3.038205) (3.149,3.06343) (3.236,3.088163) (3.323,3.112398) (3.41,3.136132) (3.497,3.159361) (3.584,3.182084) (3.671,3.2043) (3.758,3.226007) (3.845,3.247207) (3.932,3.2679) (4.019,3.28809) (4.106,3.307777) (4.193,3.326966) (4.28,3.345661) (4.367,3.363865) (4.454,3.381585) (4.541,3.398825) (4.628,3.415592) (4.715,3.431891) (4.802,3.447731) (4.889,3.463117) (4.976,3.478057) (5.063,3.492559) (5.15,3.50663) (5.237,3.52028) (5.324,3.533515) (5.411,3.546345) (5.498,3.558779) (5.585,3.570823) (5.672,3.582489) (5.759,3.593783) (5.846,3.604716) (5.933,3.615295) (6.02,3.62553) (6.107,3.635429) (6.194,3.645002) (6.281,3.654256) (6.368,3.6632) (6.455,3.671843) (6.542,3.680193) (6.629,3.688259) (6.716,3.696049) (6.803,3.70357) (6.89,3.71083) (6.977,3.717838) (7.064,3.724601) (7.151,3.731127) (7.238,3.737422) (7.325,3.743495) (7.412,3.749351) (7.499,3.754999) (7.586,3.760444) (7.673,3.765693) (7.76,3.770753) (7.847,3.77563) (7.934,3.78033) (8.021,3.784858) (8.108,3.789222) (8.195,3.793425) (8.282,3.797474) (8.369,3.801374) (8.456,3.80513) (8.543,3.808747) (8.63,3.81223) (8.717,3.815583) (8.804,3.818812) (8.891,3.82192) (8.978,3.824912) (9.065,3.827792) (9.152,3.830564) (9.239,3.833231) (9.326,3.835798) (9.413,3.838269) (9.5,3.840646)};
29 \draw[dashed,line width=.8pt] plot coordinates {(0.8,2.25) (0.887,2.217004) (0.974,2.184035) (1.061,2.151119) (1.148,2.118281) (1.235,2.085548) (1.322,2.052945) (1.409,2.020497) (1.496,1.98823) (1.583,1.956167) (1.67,1.924331) (1.757,1.892745) (1.844,1.861432) (1.931,1.830412) (2.018,1.799707) (2.105,1.769334) (2.192,1.739314) (2.279,1.709662) (2.366,1.680397) (2.453,1.651533) (2.54,1.623084) (2.627,1.595065) (2.714,1.567487) (2.801,1.540361) (2.888,1.513698) (2.975,1.487507) (3.062,1.461795) (3.149,1.43657) (3.236,1.411837) (3.323,1.387602) (3.41,1.363868) (3.497,1.340639) (3.584,1.317916) (3.671,1.2957) (3.758,1.273993) (3.845,1.252793) (3.932,1.2321) (4.019,1.21191) (4.106,1.192223) (4.193,1.173034) (4.28,1.154339) (4.367,1.136135) (4.454,1.118415) (4.541,1.101175) (4.628,1.084408) (4.715,1.068109) (4.802,1.052269) (4.889,1.036883) (4.976,1.021943) (5.063,1.007441) (5.15,0.99337) (5.237,0.97972) (5.324,0.966485) (5.411,0.953655) (5.498,0.941221) (5.585,0.929177) (5.672,0.917511) (5.759,0.906217) (5.846,0.895284) (5.933,0.884705) (6.02,0.87447) (6.107,0.864571) (6.194,0.854998) (6.281,0.845744) (6.368,0.8368) (6.455,0.828157) (6.542,0.819807) (6.629,0.811741) (6.716,0.803951) (6.803,0.79643) (6.89,0.78917) (6.977,0.782162) (7.064,0.775399) (7.151,0.768873) (7.238,0.762578) (7.325,0.756505) (7.412,0.750649) (7.499,0.745001) (7.586,0.739556) (7.673,0.734307) (7.76,0.729247) (7.847,0.72437) (7.934,0.71967) (8.021,0.715142) (8.108,0.710778) (8.195,0.706575) (8.282,0.702526) (8.369,0.698626) (8.456,0.69487) (8.543,0.691253) (8.63,0.68777) (8.717,0.684417) (8.804,0.681188) (8.891,0.67808) (8.978,0.675088) (9.065,0.672208) (9.152,0.669436) (9.239,0.666769) (9.326,0.664202) (9.413,0.661731) (9.5,0.659354)};
30 \node[font=\small,align=center] at (11.6,3.7) {$p_0(t)=\dfrac{e^t}{1+e^t}$};
31 \node[font=\small,align=center] at (11.6,2.7) {$p_1(t)=\dfrac{1}{1+e^t}$};
32 \node[font=\small,align=center] at (11.6,1.5) {$\dot p_0=p_0(1-p_0)$};
33 \node[font=\small,align=center,font=\scriptsize] at (11.6,0.65) {Mass stays one.\\Mixtures are not preserved.};
34 \end{tikzpicture}

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