Let (Xn)neN be independent and identically distributed random variables satisfying P(Xn = 1) = P(Xn Consider...

    

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Let (Xn)neN be independent and identically distributed random variables satisfying P(Xn = 1) = P(Xn Consider the stochastic process (Sn)neN defined by setting So each n 1 setting Sn = 1 X₁. Moreover, define Then we have that E[ST] = 0. YES - -1) =1/4 2 Tinf{n > 0: Sn=1} NO = 0 and for No answer Question 10 -(1) Consider two branching processes (X¹))neN and (X²))neN, where the branch- ing processes are defined as in (8.1.1) in the script on page 310. v(1) For the first branching process (X)neN, we assume X(¹) = 1 and that the corresponding ((¹))neN satisfy P(Y(¹) = 0) = P(Y(¹) = 1) = P(Y(¹) = 2) = -(2) For the second branching process (X²))neN, we assume that X(²) = 1 and that the corresponding (Y²))neN satisfy P(Y(²) = 0) = P(Y(²) = 2) = Then the extinction probability of the two branching processes is the same. YES 1 NO No answer Question 11 There exists a branching process (Xn)neN defined as in (8.1.1) in the script on page 310 which satisfies all the following: • Xo = 1, The extinction probability is strictly positive (meaning that a₁ > 0), • One has limn→∞ E[Xn] =∞. YES NO No answer Let (Xn)neN be independent and identically distributed random variables satisfying P(Xn = 1) = P(Xn Consider the stochastic process (Sn)neN defined by setting So each n 1 setting Sn = 1 X₁. Moreover, define Then we have that E[ST] = 0. YES - -1) =1/4 2 Tinf{n > 0: Sn=1} NO = 0 and for No answer Question 10 -(1) Consider two branching processes (X¹))neN and (X²))neN, where the branch- ing processes are defined as in (8.1.1) in the script on page 310. v(1) For the first branching process (X)neN, we assume X(¹) = 1 and that the corresponding ((¹))neN satisfy P(Y(¹) = 0) = P(Y(¹) = 1) = P(Y(¹) = 2) = -(2) For the second branching process (X²))neN, we assume that X(²) = 1 and that the corresponding (Y²))neN satisfy P(Y(²) = 0) = P(Y(²) = 2) = Then the extinction probability of the two branching processes is the same. YES 1 NO No answer Question 11 There exists a branching process (Xn)neN defined as in (8.1.1) in the script on page 310 which satisfies all the following: • Xo = 1, The extinction probability is strictly positive (meaning that a₁ > 0), • One has limn→∞ E[Xn] =∞. YES NO No answer

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