Question

Note: Please someone answer this question.

We conduct an experiment of counting the number of tosses of a fair coin until getting three consecutive heads. We record the outcomes (for example, HT T T T HHH), until we get three heads. Let’s define a geometric random variable X that represents the number of tosses of a fair coin until we get three consecutive heads. Possible values for X are positive integers, such as x ∈ {1, 2, 3, 4...}.

(a) Enumerate all possible records whose length is 3, 4, 5, and 6.

(b) These records give us a sample space Ω, and we can decompose Ω into four sets: · A1 = {(H, H, H)} · A2 = {(T, ∗, ∗, ..., ∗)} (all records starting with T) · A3 = {(H, T, ∗, ∗, ..., ∗)} · A4 = {(H, H, T, ∗, ∗, ..., ∗)} Please compute P[A1], P[A2], P[A3], and P[A4].

(c) Please find E[X] using these four decompositions. Hint: You could write an equation by using E[X], E[X +1], E[X +2], and E[X +3].

Answer 1

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