Branching Process - Extinction probability geometricConsider a branching process with offspring distribution Geometric( α); that is, p k = α ( 1 − α ) k for k ≥ 0.a) For what values of α ∈ ( 0 , 1 ) is the extinction probability q = 1.Let { Z n } n ≥ 0 be a branching process with p 0 &gt; 0. Let μ = ∑ k ≥ 0 k p k be the mean of the offspring distribution and let g ( s ) = ∑ k ≥ 0 s k p k be the probability generating function of the offspring distribution.- If μ ≤ 1, then the extinction probability q = 1.- If μ &gt; 1, then the extinction probability q is the unique solution to the equation s = g ( s ) with s ∈ ( 0 , 1 ).b) Use the following proposition to give a formula for the extinction probability of the branching process for any value of the parameter α ∈ ( 0 , 1 ).

Question

Branching Process - Extinction probability geometricConsider a branching process with offspring distribution Geometric(  α); that is,       p          k        =      α    (  1  −      α        )    k   for   k  ≥  0.a) For what values of       α    ∈  (  0  ,  1  ) is the extinction probability   q  =  1.Let   {      Z    n        }          n      ≥      0       be a branching process with       p    0    &amp;gt;  0. Let   μ  =      ∑          k      ≥      0        k      p    k   be the mean of the offspring distribution and let   g  (  s  )  =      ∑          k      ≥      0            s          k            p          k       be the probability generating function of the offspring distribution.- If   μ  ≤  1, then the extinction probability   q  =  1.- If   μ  &amp;gt;  1, then the extinction probability q is the unique solution to the equation   s  =  g  (  s  ) with   s  ∈  (  0  ,  1  ).b) Use the following proposition to give a formula for the extinction probability of the branching process for any value of the parameter       α    ∈  (  0  ,  1  ).

Carmelo Peck · Accepted Answer

Step 1The extinction probability is the smaller of the two roots in [0,1] of the equation       ∑          k      =      0              ∞            p    k        z          k        =  s. In this equation this equation becomes       α          1      −      (      1      −      α      )      s        =  s or   (  1  −  α  )      s          2        −  s  +  α  =  0.Step 2You can write this as   (  1  −  s  )  (  α  −  (  1  −  α  )  s  )  =  0 so the extinction probability is       α          1      −      α      .

Consider a branching process with offspring distribution Geometric( alpha ); that is, p_k=alpha(1-alpha)^k for k geq 0.

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