Find the expected time of fixation given an initial proportion.
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Consider the Wright-Fisher model with selection. This means that for a given $lambda>-1$ and $forall n$ the probability of any individual to choose parent of type $A$ is $frac{(1+s)p}{1+sp}$ subject to the condition that the $p= frac{X_n}{N},$ where $X_n$ is the number of individuals with type $A$ allele at generation $n.$
Define the event $tau_N = inf{ngeq 0:X_n=0text{ or }X_n=H}.$ My goal is to find or estimate $E[tau_N|p(0)=x]$ where $p(0)$ denotes the initial proportion of $A$ allele and $xin [0,1].$
Say $xin (0,1)$ in order to avoid fixation at $t=0.$ Then, using conditional expectation we get that
$$E[tau_N|p(0)=x]=sum_{t=0}^{infty}tP[tau_N=t|p(0)=x]=
sum_{t=0}^{infty}tP[X_t=0|p(0)=x]+sum_{t=0}^{infty}tP[X_t=N|p(0)=x].
$$
I don't know how to proceed after this step. Any hints/suggestions will be much appreciated.
probability
asked Nov 12 at 17:22
Hello_World
3,69821630
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up vote
0
down vote
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Consider the Wright-Fisher model with selection. This means that for a given $lambda>-1$ and $forall n$ the probability of any individual to choose parent of type $A$ is $frac{(1+s)p}{1+sp}$ subject to the condition that the $p= frac{X_n}{N},$ where $X_n$ is the number of individuals with type $A$ allele at generation $n.$
Define the event $tau_N = inf{ngeq 0:X_n=0text{ or }X_n=H}.$ My goal is to find or estimate $E[tau_N|p(0)=x]$ where $p(0)$ denotes the initial proportion of $A$ allele and $xin [0,1].$
Say $xin (0,1)$ in order to avoid fixation at $t=0.$ Then, using conditional expectation we get that
$$E[tau_N|p(0)=x]=sum_{t=0}^{infty}tP[tau_N=t|p(0)=x]=
sum_{t=0}^{infty}tP[X_t=0|p(0)=x]+sum_{t=0}^{infty}tP[X_t=N|p(0)=x].
$$
I don't know how to proceed after this step. Any hints/suggestions will be much appreciated.
probability
asked Nov 12 at 17:22
Hello_World
3,69821630
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Consider the Wright-Fisher model with selection. This means that for a given $lambda>-1$ and $forall n$ the probability of any individual to choose parent of type $A$ is $frac{(1+s)p}{1+sp}$ subject to the condition that the $p= frac{X_n}{N},$ where $X_n$ is the number of individuals with type $A$ allele at generation $n.$
Define the event $tau_N = inf{ngeq 0:X_n=0text{ or }X_n=H}.$ My goal is to find or estimate $E[tau_N|p(0)=x]$ where $p(0)$ denotes the initial proportion of $A$ allele and $xin [0,1].$
Say $xin (0,1)$ in order to avoid fixation at $t=0.$ Then, using conditional expectation we get that
$$E[tau_N|p(0)=x]=sum_{t=0}^{infty}tP[tau_N=t|p(0)=x]=
sum_{t=0}^{infty}tP[X_t=0|p(0)=x]+sum_{t=0}^{infty}tP[X_t=N|p(0)=x].
$$
I don't know how to proceed after this step. Any hints/suggestions will be much appreciated.
probability
asked Nov 12 at 17:22
Hello_World
3,69821630
Consider the Wright-Fisher model with selection. This means that for a given $lambda>-1$ and $forall n$ the probability of any individual to choose parent of type $A$ is $frac{(1+s)p}{1+sp}$ subject to the condition that the $p= frac{X_n}{N},$ where $X_n$ is the number of individuals with type $A$ allele at generation $n.$
Define the event $tau_N = inf{ngeq 0:X_n=0text{ or }X_n=H}.$ My goal is to find or estimate $E[tau_N|p(0)=x]$ where $p(0)$ denotes the initial proportion of $A$ allele and $xin [0,1].$
Say $xin (0,1)$ in order to avoid fixation at $t=0.$ Then, using conditional expectation we get that
$$E[tau_N|p(0)=x]=sum_{t=0}^{infty}tP[tau_N=t|p(0)=x]=
sum_{t=0}^{infty}tP[X_t=0|p(0)=x]+sum_{t=0}^{infty}tP[X_t=N|p(0)=x].
$$
I don't know how to proceed after this step. Any hints/suggestions will be much appreciated.
probability
probability
asked Nov 12 at 17:22
Hello_World
3,69821630
asked Nov 12 at 17:22
Hello_World
3,69821630
edited Nov 14 at 15:39
asked Nov 12 at 17:22
Hello_World
3,69821630
asked Nov 12 at 17:22
Hello_World
3,69821630
asked Nov 12 at 17:22
Hello_World
3,69821630
3,69821630
add a comment |
add a comment |
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