Timeline for Difference equation $A(n,x)=p(x)A(n-1,x-1)+q(x)A(n-1,x)$
Current License: CC BY-SA 3.0
13 events
when toggle format | what | by | license | comment | |
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Jan 11, 2012 at 3:03 | vote | accept | sigma_z_1980 | ||
Nov 8, 2011 at 16:22 | comment | added | Max Alekseyev | It does not matter how you call it. The formulae still hold. | |
Nov 8, 2011 at 10:06 | comment | added | sigma_z_1980 | No, it's not probability. It's a state, size of population. | |
Nov 8, 2011 at 8:02 | comment | added | Max Alekseyev | So $A(n,x)$ is the probability that in the $n$-th iteration the size of population equals $x$. Then $A(1,x) = \delta_{x1}$ (Kronecker's delta). Correspondingly, $\mathcal{A_1}(x,z) = z^{x-1}$. | |
Nov 7, 2011 at 20:36 | comment | added | sigma_z_1980 | OK, it's not very good notation then. The boundary value would be $A(1)=1$, i.e. in the first iteration the size of population is 1. Each iteration the size either increases by 1 w.p. $p(x), x$ being the size of the population in the previous turn, or stays the same w.p. $q(x)$. | |
Nov 7, 2011 at 5:52 | comment | added | Max Alekseyev | "the boundary constraint is A(1,1)" - and what is its value? And what's about $A(1,x)$ for $x$ not equal 1? To express $A(n,x)$ in terms of $A(1,x)$, take $m=n-1$. $P_m(x,z)$ is a polynomial defined via given $p(x)$ and $q(x)$. | |
Nov 6, 2011 at 22:45 | comment | added | sigma_z_1980 | also, where does the expression for $P_{m}(x,z)$ come from? IS this some determinant? | |
Nov 6, 2011 at 21:39 | comment | added | sigma_z_1980 | the boundary constraint is $A(1,1)$, i.e. in the first iteration the size is 1. | |
Nov 6, 2011 at 11:38 | history | edited | Max Alekseyev | CC BY-SA 3.0 |
added 58 characters in body
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Nov 6, 2011 at 11:32 | history | undeleted | Max Alekseyev | ||
Nov 6, 2011 at 11:32 | history | edited | Max Alekseyev | CC BY-SA 3.0 |
added 161 characters in body; edited body
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Nov 6, 2011 at 11:13 | history | deleted | Max Alekseyev | ||
Nov 6, 2011 at 11:01 | history | answered | Max Alekseyev | CC BY-SA 3.0 |