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Exercise 4. (8p) In this exercise we consider the Bienaymé Galton Watson processes ($Z_n$) with $p_0 = \frac{7}{10}$, $p_3 = \frac{2}{10}$, $p_{10} = \frac{9}{100}$, $p_{50} = \frac{1}{100}$. For example Covid (SARS Cov 2) was roughly similar to this in 2020-2021: people had a high probability $p_0$ to contaminate no one, but also a not-so-small probability to contaminate many people (a small number of people contributed to a significant proportion of the contaminations). (a) (2p) What is $G_1$ in this case? (b) (2p) Compute $\mu$ and $\sigma$. (c) (4p) Make a drawing representing the graph of $G_1$ and the first diagonal "y = x". Indicate on the drawing how to find $\eta$ and use a calculator to find an approximation of $\eta$. (Give 2 characteristic digits, for example writing the result as $\eta = 4.5 \cdot 10^{-1}$ or $8.2 \cdot 10^{-2}$).

          Exercise 4. (8p) In this exercise we consider the Bienaymé Galton Watson processes ($Z_n$) with $p_0 = \frac{7}{10}$, $p_3 = \frac{2}{10}$, $p_{10} = \frac{9}{100}$, $p_{50} = \frac{1}{100}$. For example Covid (SARS Cov 2) was roughly similar to this in 2020-2021: people had a high probability $p_0$ to contaminate no one, but also a not-so-small probability to contaminate many people (a small number of people contributed to a significant proportion of the contaminations).
(a) (2p) What is $G_1$ in this case?
(b) (2p) Compute $\mu$ and $\sigma$.
(c) (4p) Make a drawing representing the graph of $G_1$ and the first diagonal "y = x". Indicate on the drawing how to find $\eta$ and use a calculator to find an approximation of $\eta$. (Give 2 characteristic digits, for example writing the result as $\eta = 4.5 \cdot 10^{-1}$ or $8.2 \cdot 10^{-2}$).

Exercise 4. (8p) In this exercise we consider the Bienaymé Galton Watson processes (Zn) with p0 = (7)/(10), p3 = (2)/(10), p10 = (9)/(100), p50 = (1)/(100). For example Covid (SARS Cov 2) was roughly similar to this in 2020-2021: people had a high probability p0 to contaminate no one, but also a not-so-small probability to contaminate many people (a small number of people contributed to a significant proportion of the contaminations).
(a) (2p) What is G1 in this case?
(b) (2p) Compute μ and σ.
(c) (4p) Make a drawing representing the graph of G1 and the first diagonal "y = x". Indicate on the drawing how to find η and use a calculator to find an approximation of η. (Give 2 characteristic digits, for example writing the result as η = 4.5 · 10^-1 or 8.2 · 10^-2).

Added by Antonio J.

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