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Poisson binomial distribution
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<h2 id="definitions">Definitions</h2> <h3>Probability mass function</h3> <p>The probability of having <i>k</i> successful trials out of a total of <i>n</i> can be written as the sum <a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> Pr ( K = k ) = ∑ A ∈ F k ∏ i ∈ A p i ∏ j ∈ A c ( 1 − p j ) {\displaystyle \Pr(K=k)=\sum \limits _{A\in F_{k}}\prod \limits _{i\in A}p_{i}\prod \limits _{j\in A^{c}}(1-p_{j})} <p>where F k {\displaystyle F_{k}} is the set of all subsets of <i>k</i> integers that can be selected from { 1 , 2 , 3 , . . . , n } {\displaystyle \{1,2,3,...,n\}} . For example, if <i>n</i> = 3, then F 2 = { { 1 , 2 } , { 1 , 3 } , { 2 , 3 } } {\displaystyle F_{2}=\left\{\{1,2\},\{1,3\},\{2,3\}\right\}} . A c {\displaystyle A^{c}} is the <a href="/facts/Complement_(set_theory)/yWUePIYY">complement</a> of A {\displaystyle A} , i.e. A c = { 1 , 2 , 3 , … , n } ∖ A {\displaystyle A^{c}=\{1,2,3,\dots ,n\}\smallsetminus A} . </p><p> F k {\displaystyle F_{k}} will contain n ! / ( ( n − k ) ! k ! ) {\displaystyle n!/((n-k)!k!)} elements, the sum over which is infeasible to compute in practice unless the number of trials <i>n</i> is small (e.g. if <i>n</i> = 30, F 15 {\displaystyle F_{15}} contains over 1020 elements). However, there are other, more efficient ways to calculate Pr ( K = k ) {\displaystyle \Pr(K=k)} . </p><p>As long as none of the success probabilities are equal to one, one can calculate the probability of <i>k</i> successes using the recursive formula <a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> <a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> Pr ( K = k ) = { ∏ i = 1 n ( 1 − p i ) k = 0 1 k ∑ i = 1 k ( − 1 ) i − 1 Pr ( K = k − i ) T ( i ) k > 0 {\displaystyle \Pr(K=k)={\begin{cases}\prod \limits _{i=1}^{n}(1-p_{i})&k=0\\{\frac {1}{k}}\sum \limits _{i=1}^{k}(-1)^{i-1}\Pr(K=k-i)T(i)&k>0\\\end{cases}}} <p>where </p> T ( i ) = ∑ j = 1 n ( p j 1 − p j ) i . {\displaystyle T(i)=\sum \limits _{j=1}^{n}\left({\frac {p_{j}}{1-p_{j}}}\right)^{i}.} <p>The recursive formula is not <a href="/facts/Numerically_stable/jRDdo8zV">numerically stable</a>, and should be avoided if n {\displaystyle n} is greater than approximately 20. </p><p>An alternative is to use a <a href="/facts/Divide-and-conquer_algorithm/pC1X7Ws7">divide-and-conquer algorithm</a>: if we assume n = 2 b {\displaystyle n=2^{b}} is a power of two, denoting by f ( p i : j ) {\displaystyle f(p_{i:j})} the Poisson binomial of p i , … , p j {\displaystyle p_{i},\dots ,p_{j}} and ∗ {\displaystyle *} the <a href="/facts/Convolution_of_probability_distributions/l467HgVK">convolution</a> operator, we have f ( p 1 : 2 b ) = f ( p 1 : 2 b − 1 ) ∗ f ( p 2 b − 1 + 1 : 2 b ) {\displaystyle f(p_{1:2^{b}})=f(p_{1:2^{b-1}})*f(p_{2^{b-1}+1:2^{b}})} . </p><p>More generally, the probability mass function of a Poisson binomial can be expressed as the convolution of the vectors P 1 , … , P n {\displaystyle P_{1},\dots ,P_{n}} where P i = [ 1 − p i , p i ] {\displaystyle P_{i}=[1-p_{i},p_{i}]} . This observation leads to the direct convolution (DC) algorithm for computing Pr ( K = 0 ) {\displaystyle \Pr(K=0)} through Pr ( K = n ) {\displaystyle \Pr(K=n)} : </p> <i>// PMF and nextPMF begin at index 0</i> function DC( p 1 , … , p n {\displaystyle p_{1},\dots ,p_{n}} ) is declare new PMF array of size 1 PMF[0] = [1] for i = 1 to n {\displaystyle n} do declare new nextPMF array of size i + 1 nextPMF[0] = (1 - p i {\displaystyle p_{i}} ) * PMF[0] nextPMF[i] = p i {\displaystyle p_{i}} * PMF[i - 1] for k = 1 to i - 1 do nextPMF[k] = p i {\displaystyle p_{i}} * PMF[k - 1] + (1 - p i {\displaystyle p_{i}} ) * PMF[k] repeat PMF = nextPMF repeat return PMF end function <p> Pr ( K = k ) {\displaystyle \Pr(K=k)} will be found in PMF[k]. DC is numerically stable, exact, and, when implemented as a software routine, exceptionally fast for n ≤ 2000 {\displaystyle n\leq 2000} . It can also be quite fast for larger n {\displaystyle n} , depending on the distribution of the p i {\displaystyle p_{i}} .<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p><p>Another possibility is using the <a href="/facts/Discrete_Fourier_transform/uK9Tjsbl">discrete Fourier transform</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p> Pr ( K = k ) = 1 n + 1 ∑ ℓ = 0 n C − l k ∏ m = 1 n ( 1 + ( C ℓ − 1 ) p m ) {\displaystyle \Pr(K=k)={\frac {1}{n+1}}\sum _{\ell =0}^{n}C^{-lk}\prod _{m=1}^{n}\left(1+(C^{\ell }-1)p_{m}\right)} <p>where C = exp ( 2 i π n + 1 ) {\displaystyle C=\exp \left({\frac {2i\pi }{n+1}}\right)} and i = − 1 {\displaystyle i={\sqrt {-1}}} . </p><p>Still other methods are described in "Statistical Applications of the Poisson-Binomial and conditional Bernoulli distributions" by Chen and Liu<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> and in "A simple and fast method for computing the Poisson binomial distribution function" by Biscarri et al.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p> <h3>Cumulative distribution function</h3> <p>The <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a> (CDF) can be expressed as: </p> Pr ( K ≤ k ) = ∑ ℓ = 0 k ∑ A ∈ F ℓ ∏ i ∈ A p i ∏ j ∈ A c ( 1 − p j ) , {\displaystyle \Pr(K\leq k)=\sum _{\ell =0}^{k}\sum _{A\in F_{\ell }}\prod _{i\in A}p_{i}\prod _{j\in A^{c}}(1-p_{j}),} <p>where F ℓ {\displaystyle F_{\ell }} is the set of all subsets of size ℓ {\displaystyle \ell } that can be selected from { 1 , 2 , 3 , … , n } {\displaystyle \{1,2,3,\ldots ,n\}} . </p><p>It can be computed by invoking the DC function above, and then adding elements 0 {\displaystyle 0} through k {\displaystyle k} of the returned PMF array. </p> <h2 id="properties">Properties</h2> <h3>Mean and variance</h3> <p>Since a Poisson binomial distributed variable is a sum of <i>n</i> independent Bernoulli distributed variables, its mean and variance will simply be sums of the mean and variance of the <i>n</i> Bernoulli distributions: </p> μ = ∑ i = 1 n p i {\displaystyle \mu =\sum \limits _{i=1}^{n}p_{i}} σ 2 = ∑ i = 1 n ( 1 − p i ) p i {\displaystyle \sigma ^{2}=\sum \limits _{i=1}^{n}(1-p_{i})p_{i}} <h3>Entropy</h3> <p>There is no simple formula for the entropy of a Poisson binomial distribution, but the entropy is bounded above by the entropy of a binomial distribution with the same number parameter and the same mean. Therefore, the entropy is also bounded above by the entropy of a Poisson distribution with the same mean.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p><p>The Shepp–Olkin concavity conjecture, due to <a href="/facts/Lawrence_Shepp/un54gtmS">Lawrence Shepp</a> and <a href="/facts/Ingram_Olkin/omLnnzI2">Ingram Olkin</a> in 1981, states that the entropy of a Poisson binomial distribution is a concave function of the success probabilities p 1 , p 2 , … , p n {\displaystyle p_{1},p_{2},\dots ,p_{n}} .<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> This conjecture was proved by Erwan Hillion and Oliver Johnson in 2015.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> The Shepp–Olkin monotonicity conjecture, also from the same 1981 paper, is that the entropy is monotone increasing in p i {\displaystyle p_{i}} , if all p i ≤ 1 / 2 {\displaystyle p_{i}\leq 1/2} . This conjecture was also proved by Hillion and Johnson, in 2019.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> </p> <h3>Chernoff bound</h3> <p>The probability that a Poisson binomial distribution gets large, can be bounded using its moment generating function as follows (valid when s ≥ μ {\displaystyle s\geq \mu } and for any t > 0 {\displaystyle t>0} ): </p> Pr [ S ≥ s ] ≤ exp ( − s t ) E [ exp [ t ∑ i X i ] ] = exp ( − s t ) ∏ i ( 1 − p i + e t p i ) = exp ( − s t + ∑ i log ( p i ( e t − 1 ) + 1 ) ) ≤ exp ( − s t + ∑ i log ( exp ( p i ( e t − 1 ) ) ) ) = exp ( − s t + ∑ i p i ( e t − 1 ) ) = exp ( s − μ − s log s μ ) , {\displaystyle {\begin{aligned}\Pr[S\geq s]&\leq \exp(-st)\operatorname {E} \left[\exp \left[t\sum _{i}X_{i}\right]\right]\\&=\exp(-st)\prod _{i}(1-p_{i}+e^{t}p_{i})\\&=\exp \left(-st+\sum _{i}\log \left(p_{i}(e^{t}-1)+1\right)\right)\\&\leq \exp \left(-st+\sum _{i}\log \left(\exp(p_{i}(e^{t}-1))\right)\right)\\&=\exp \left(-st+\sum _{i}p_{i}(e^{t}-1)\right)\\&=\exp \left(s-\mu -s\log {\frac {s}{\mu }}\right),\end{aligned}}} <p>where we took t = log ( s / μ ) {\textstyle t=\log \left(s/\mu \right)} . This is similar to the <a href="/facts/Binomial_distribution/UMoFMjDj">tail bounds of a binomial distribution</a>. </p> <h2 id="related-distribution">Related distribution</h2> <h3>Approximation by binomial distribution</h3> <p>A Poisson binomial distribution P B {\displaystyle PB} can be approximated by a binomial distribution B {\displaystyle B} where μ {\displaystyle \mu } , the mean of the p i {\displaystyle p_{i}} , is the success probability of B {\displaystyle B} . The variances of P B {\displaystyle PB} and B {\displaystyle B} are related by the formula </p> Var ( P B ) = Var ( B ) − ∑ i = 1 n ( p i − μ ) 2 {\displaystyle \operatorname {Var} (PB)=\operatorname {Var} (B)-\sum _{i=1}^{n}(p_{i}-\mu )^{2}} <p>As can be seen, the closer the p i {\displaystyle p_{i}} are to μ {\displaystyle \mu } , that is, the more the p i {\displaystyle p_{i}} tend to homogeneity, the larger P B {\displaystyle PB} 's variance. When all the p i {\displaystyle p_{i}} are equal to μ {\displaystyle \mu } , P B {\displaystyle PB} becomes B {\displaystyle B} , Var ( P B ) = Var ( B ) {\displaystyle \operatorname {Var} (PB)=\operatorname {Var} (B)} , and the variance is at its maximum.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> </p><p>Ehm has determined bounds for the <a href="/facts/Total_variation_distance_of_probability_measures/agdAvDuW">total variation distance</a> of P B {\displaystyle PB} and B {\displaystyle B} , in effect providing bounds on the error introduced when approximating P B {\displaystyle PB} with B {\displaystyle B} . Let ν = 1 − μ {\displaystyle \nu =1-\mu } and d ( P B , B ) {\displaystyle d(PB,B)} be the total variation distance of P B {\displaystyle PB} and B {\displaystyle B} . Then </p> d ( P B , B ) ≤ ( 1 − μ n + 1 − ν n + 1 ) ∑ i = 1 n ( p i − μ ) 2 ( ( n + 1 ) μ ν ) {\displaystyle d(PB,B)\leq (1-\mu ^{n+1}-\nu ^{n+1}){\frac {\sum _{i=1}^{n}(p_{i}-\mu )^{2}}{((n+1)\mu \nu )}}} d ( P B , B ) ≥ C min { 1 , 1 n μ ν } ∑ i = 1 n ( p i − μ ) 2 {\displaystyle d(PB,B)\geq C\min \left\{\,1,{\frac {1}{n\mu \nu }}\,\right\}\sum _{i=1}^{n}(p_{i}-\mu )^{2}} <p>where C ≥ 1 124 {\displaystyle C\geq {\frac {1}{124}}} . </p><p> d ( P B , B ) {\displaystyle d(PB,B)} tends to 0 if and only if Var ( P B ) / Var ( B ) {\displaystyle \operatorname {Var} (PB)/\operatorname {Var} (B)} tends to 1.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> </p> <h3>Approximation by Poisson distribution</h3> <p>A Poisson binomial distribution P B {\displaystyle PB} can also be approximated by a <a href="/facts/Poisson_distribution/CvCzXkHr">Poisson distribution</a> P o {\displaystyle Po} with mean λ = ∑ i = 1 n p i {\displaystyle \lambda =\sum _{i=1}^{n}p_{i}} . Barbour and Hall have shown that </p> 1 32 min { 1 λ , 1 } ∑ i = 1 n p i 2 ≤ d ( P B , P o ) ≤ 1 − ε − λ λ ∑ i = 1 n p i 2 {\displaystyle {\frac {1}{32}}\min \left\{\,{\frac {1}{\lambda }},1\,\right\}\sum _{i=1}^{n}p_{i}^{2}\leq d(PB,Po)\leq {\frac {1-\varepsilon ^{-\lambda }}{\lambda }}\sum _{i=1}^{n}p_{i}^{2}} <p>where d ( P B , B ) {\displaystyle d(PB,B)} is the total variation distance of P B {\displaystyle PB} and P o {\displaystyle Po} .<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> It can be seen that the smaller the p i {\displaystyle p_{i}} , the better P o {\displaystyle Po} approximates P B {\displaystyle PB} . </p><p>As Var ( P o ) = λ = ∑ i = 1 n p i {\displaystyle \operatorname {Var} (Po)=\lambda =\sum _{i=1}^{n}p_{i}} and Var ( P B ) = ∑ i = 1 n p i − ∑ i = 1 n p i 2 {\displaystyle \operatorname {Var} (PB)=\sum \limits _{i=1}^{n}p_{i}-\sum \limits _{i=1}^{n}p_{i}^{2}} , Var ( P o ) > Var ( P B ) {\displaystyle \operatorname {Var} (\mathrm {Po} )>\operatorname {Var} (PB)} ; so a Poisson binomial distribution's variance is bounded above by a Poisson distribution with λ = ∑ i = 1 n p i {\displaystyle \lambda =\sum _{i=1}^{n}p_{i}} , and the smaller the p i {\displaystyle p_{i}} , the closer Var ( P o ) {\displaystyle \operatorname {Var} (\mathrm {Po} )} will be to Var ( P B ) {\displaystyle \operatorname {Var} (PB)} . </p> <h2 id="computational-methods">Computational methods</h2> <p>The reference <a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> discusses techniques of evaluating the probability mass function of the Poisson binomial distribution. The following software implementations are based on it: </p> <ul><li>An R package <a href="https://CRAN.R-project.org/package=poibin"><i>poibin</i></a> was provided along with the paper,<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> which is available for the computing of the cdf, pmf, quantile function, and random number generation of the Poisson binomial distribution. For computing the PMF, a DFT algorithm or a recursive algorithm can be specified to compute the exact PMF, and approximation methods using the normal and Poisson distribution can also be specified.</li> <li><a href="https://github.com/tsakim/poibin"><i>poibin</i> – Python implementation</a> – can compute the PMF and CDF, uses the DFT method described in the paper for doing so.</li></ul> <h2 id="see-also">See also</h2> <ul> <li>Mathematics portal</li></ul> <ul><li><a href="/facts/Le_Cam%2527s_theorem/lN6AW5Cu">Le Cam's theorem</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Wang, Y. H. (1993). "On the number of successes in independent trials" (PDF). Statistica Sinica. 3 (2): 295–312. <a href="http://www3.stat.sinica.edu.tw/statistica/oldpdf/A3n23.pdf" target="_blank">http://www3.stat.sinica.edu.tw/statistica/oldpdf/A3n23.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Shah, B. K. (1994). "On the distribution of the sum of independent integer valued random variables". American Statistician. 27 (3): 123–124. JSTOR 2683639. <a href="/wiki/JSTOR_(identifier)" target="_blank">/wiki/JSTOR_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Chen, X. H.; A. P. Dempster; J. S. Liu (1994). "Weighted finite population sampling to maximize entropy" (PDF). Biometrika. 81 (3): 457. doi:10.1093/biomet/81.3.457. <a href="http://www.people.fas.harvard.edu/~junliu/TechRept/94folder/cdl94.pdf" target="_blank">http://www.people.fas.harvard.edu/~junliu/TechRept/94folder/cdl94.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Biscarri, William; Zhao, Sihai Dave; Brunner, Robert J. (2018-06-01). "A simple and fast method for computing the Poisson binomial distribution function". Computational Statistics & Data Analysis. 122: 92–100. doi:10.1016/j.csda.2018.01.007. ISSN 0167-9473. <a href="https://www.sciencedirect.com/science/article/pii/S0167947318300082" target="_blank">https://www.sciencedirect.com/science/article/pii/S0167947318300082</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Fernandez, M.; S. Williams (2010). "Closed-Form Expression for the Poisson-Binomial Probability Density Function". IEEE Transactions on Aerospace and Electronic Systems. 46 (2): 803–817. Bibcode:2010ITAES..46..803F. doi:10.1109/TAES.2010.5461658. S2CID 1456258. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Chen, S. X.; J. S. Liu (1997). "Statistical Applications of the Poisson–Binomial and conditional Bernoulli distributions". Statistica Sinica. 7: 875–892. <a href="http://www3.stat.sinica.edu.tw/statistica/password.asp?vol=7&num=4&art=4" target="_blank">http://www3.stat.sinica.edu.tw/statistica/password.asp?vol=7&num=4&art=4</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Biscarri, William; Zhao, Sihai Dave; Brunner, Robert J. (2018-06-01). "A simple and fast method for computing the Poisson binomial distribution function". Computational Statistics & Data Analysis. 122: 92–100. doi:10.1016/j.csda.2018.01.007. ISSN 0167-9473. <a href="https://www.sciencedirect.com/science/article/pii/S0167947318300082" target="_blank">https://www.sciencedirect.com/science/article/pii/S0167947318300082</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Harremoës, P. (2001). "Binomial and Poisson distributions as maximum entropy distributions" (PDF). IEEE Transactions on Information Theory. 47 (5): 2039–2041. doi:10.1109/18.930936. <a href="https://web.archive.org/web/20160116154755id_/http://yaroslavvb.com/papers/harremoes-binomial.pdf" target="_blank">https://web.archive.org/web/20160116154755id_/http://yaroslavvb.com/papers/harremoes-binomial.pdf</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Shepp, Lawrence; Olkin, Ingram (1981). "Entropy of the sum of independent Bernoulli random variables and of the multinomial distribution". In Gani, J.; Rohatgi, V.K. (eds.). Contributions to probability: A collection of papers dedicated to Eugene Lukacs. New York: Academic Press. pp. 201–206. ISBN 0-12-274460-8. MR 0618689. <a href="0-12-274460-8" target="_blank">0-12-274460-8</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Hillion, Erwan; Johnson, Oliver (2015-03-05). "A proof of the Shepp–Olkin entropy concavity conjecture". Bernoulli. 23 (4B): 3638–3649. arXiv:1503.01570. doi:10.3150/16-BEJ860. S2CID 8358662. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Hillion, Erwan; Johnson, Oliver (2019-11-09). "A proof of the Shepp–Olkin entropy monotonicity conjecture". Electronic Journal of Probability. 24 (126): 1–14. arXiv:1810.09791. doi:10.1214/19-EJP380. <a href="https://doi.org/10.1214%2F19-EJP380" target="_blank">https://doi.org/10.1214%2F19-EJP380</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Wang, Y. H. (1993). "On the number of successes in independent trials" (PDF). Statistica Sinica. 3 (2): 295–312. <a href="http://www3.stat.sinica.edu.tw/statistica/oldpdf/A3n23.pdf" target="_blank">http://www3.stat.sinica.edu.tw/statistica/oldpdf/A3n23.pdf</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Ehm, Werner (1991-01-01). "Binomial approximation to the Poisson binomial distribution". Statistics & Probability Letters. 11 (1): 7–16. doi:10.1016/0167-7152(91)90170-V. ISSN 0167-7152. <a href="https://dx.doi.org/10.1016/0167-7152%2891%2990170-V" target="_blank">https://dx.doi.org/10.1016/0167-7152%2891%2990170-V</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Barbour, A. D.; Hall, Peter (1984). "On the Rate of Poisson Convergence" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 95 (3): 473–480. doi:10.1017/S0305004100061806 (inactive 24 December 2024).{{cite journal}}: CS1 maint: DOI inactive as of December 2024 (link) <a href="https://www.zora.uzh.ch/id/eprint/23054/11/Barbour_1984V.pdf" target="_blank">https://www.zora.uzh.ch/id/eprint/23054/11/Barbour_1984V.pdf</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Hong, Yili (March 2013). "On computing the distribution function for the Poisson binomial distribution". Computational Statistics & Data Analysis. 59: 41–51. doi:10.1016/j.csda.2012.10.006. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Hong, Yili (March 2013). "On computing the distribution function for the Poisson binomial distribution". Computational Statistics & Data Analysis. 59: 41–51. doi:10.1016/j.csda.2012.10.006. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> </ol>