题目内容

For each of the following statements, determine whether it is true (meaning, always true) or false (meaning, not always true). Here, we assume all random variables are discrete, and that all expectations are well-defined and finite.1.Let X and Y be two binomial random variables.a) If X and Y are independent, then X+Y is also a binomial random variable.

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(c) Let Yₖ denote the result of the kᵗʰ roll. Let X₁=Y₁, and for k≥2, let Xₖ=Yₖ+Yₖ₋₁. Does the sequence X₁,X₂,… satisfy the Markov property?

A. Yes
B. No

(b) Let Xₖ denote the number of 6's obtained in the first k rolls, up to a maximum of ten. (That is, if ten or more 6's are obtained in the first k rolls, then Xₖ=10.) Does the sequence X₁,X₂,… satisfy the Markov property?

A. Yes
B. No

PROBLEM 3: CHECKING THE MARKOV PROPERTY For each one of the following definitions of the state Xₖ at time k (for k=1,2,…), determine whether the Markov property is satisfied by the sequence X₁,X₂,….A fair six-sided die (with sides labelled 1,2,…,6) is rolled repeatedly and independently.(a) Let Xₖ denote the largest number obtained in the first k rolls. Does the sequence X₁,X₂,… satisfy the Markov property?

A. Yes
B. No

p₀₀=______