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中文题目A Galton-Watson branching process starts with one individual. Each individual independently has 0 offspring with probability 1/4, 1 offspring with probability 1/4, and 2 offspring with probability 1/2. Let q be the extinction probability. Using that q^{Z_n} is a martingale (where
打开 →A branching process has offspring PGF $\phi(s)=\frac12+\frac12 s^2$. What is the extinction probability?
打开 →Let $Z_t$ be a branching process with immigration. Each individual in generation $t$ produces offspring with PGF $\phi(s)$, independently, and the number of immigrants arriving at the next generation has PGF $\psi(s)$, independently of everything else. Express the PGF of $Z_{t+1}
打开 →Suppose each individual has $0$ children with probability $1/2$ and $2$ children with probability $1/2$. What is the extinction probability?
打开 →Let $m>0$ be the offspring mean in a Galton-Watson process. Show that \[ M_n=\frac{Z_n}{m^n} \] is a martingale with respect to the natural filtration.
打开 →Why can two testing systems with the same maximum branch count per question have very different identification power once branch-specific follow-up rules are imposed?
打开 →A first yes/no question is asked. If yes, you may then ask one ternary question followed by one binary question. If no, you may ask only one ternary question. What is the maximum number of distinguishable states?
打开 →In a branching process, let the offspring distribution have variance $\sigma^2$. Express \[ \mathrm{Var}(Z_{n+1}\mid Z_n) \] in terms of $Z_n$.
打开 →A branching process starts from $10$ ancestors and has mean offspring count $m=0.8$. What is $E[Z_5]$?
打开 →A branching process starts from $Z_0=2$ ancestors and has mean offspring count $m=1.3$. What is $E[Z_4]$?
打开 →A branching process starts from one ancestor and has mean offspring count $m=1$. What is $E[Z_n]$?
打开 →A subcritical branching process with offspring mean $m<1$ starts from $k$ ancestors. What is the expected total progeny?
打开 →A Galton-Watson branching process has offspring PGF $\phi(s)=0.3+0.7s^2$. Compute the extinction probability.
打开 →A branching process started from one ancestor has extinction probability $q=2/3$. If it starts instead from three independent ancestors, what is the probability that at least one lineage survives forever?
打开 →A branching process starts from $Z_0=k$ ancestors and has offspring mean $m$. What is $E[Z_n]$?
打开 →Suppose a Galton-Watson branching process starts from one ancestor and has offspring PGF $\phi$ with mean $m=\phi'(1)<1$. Use the total-progeny PGF equation to derive $E[T]$.
打开 →Suppose every individual has at least one child almost surely. What is the extinction probability of the branching process started from one ancestor?
打开 →A branching process has mean offspring count $1.1$ but extinction probability $q=2/3$. Explain, in one or two sentences, why there is no contradiction between these two facts.
打开 →A branching process has extinction probability $q=3/7$ from one ancestor. If the process starts from five independent ancestors, what is the probability that at least one lineage survives forever?
打开 →Let $\phi(s)$ be the offspring PGF of a Galton-Watson branching process started from one ancestor, and let $T$ be the total progeny. Show that the PGF of $T$ satisfies \[ G_T(s)=s\,\phi(G_T(s)). \]
打开 →Let a branching process have offspring mean $m$ and variance $\sigma^2$. Show that \[ \mathrm{Var}(Z_{n+1})=m^2\mathrm{Var}(Z_n)+\sigma^2 E[Z_n]. \]
打开 →Each individual has $0$ children with probability $1/2$ and $3$ children with probability $1/2$. Starting from one ancestor, compute the extinction probability.
打开 →A Galton-Watson process has offspring mean $m<1$. What is its extinction probability?
打开 →Each individual has offspring distribution \[ P(\xi=0)=0.2,\qquad P(\xi=1)=0.5,\qquad P(\xi=2)=0.3. \] Starting from one ancestor, compute the extinction probability.
打开 →Each individual has $0$ children with probability $0.4$ and $2$ children with probability $0.6$. Compute the extinction probability.
打开 →Suppose each individual has one child with probability $p$ and zero children with probability $1-p$. What is the extinction probability?
打开 →Each individual has $0$ children with probability $0.3$ and $2$ children with probability $0.7$. Starting from one ancestor, compute the extinction probability.
打开 →A Galton-Watson process starts from one ancestor and has offspring mean $m<1$. Let \[ T=\sum_{n\ge 0} Z_n \] be the total progeny. Compute $E[T]$.
打开 →If the extinction probability from one ancestor is $q$, what is the extinction probability from $k$ independent ancestors?
打开 →Suppose each individual has one child with probability $p$ and two children with probability $1-p$. What is the extinction probability?
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