Take the sample space of infinitely many coin tosses {(W, W2, W3,...) : W E {H,...

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Take the sample space of infinitely many coin tosses {(W₁, W2, W3,...) : W₁ E {H, T} for each i € N}. Let the o-algebra ♬ be generated by all cylinder sets. A cylinder set is of the form {w EN: Wk₁ = e₁ and Wk₂ and Wkm = em}, where m and k₁ <k₂ < < km are natural numbers and e₁, ..., em € {H,T}. Let the probability measure P be such that the probability of a cylinder set of the aforementioned form is. = €₂ and H} ... = (a) (2 points) Let X be the random variable defined by X (w) for every w E. Is X continuous or discrete? = min{i € N: wi = (b) (3 points) Find the pdf or pmf of X depending on what you answered above. (c) (4 points) Let u, v E N be positive integers. What is P [X>u+v | X > u] ? How does it vary with u and v? Take the sample space of infinitely many coin tosses {(W₁, W2, W3,...) : W₁ E {H, T} for each i € N}. Let the o-algebra ♬ be generated by all cylinder sets. A cylinder set is of the form {w EN: Wk₁ = e₁ and Wk₂ and Wkm = em}, where m and k₁ <k₂ < < km are natural numbers and e₁, ..., em € {H,T}. Let the probability measure P be such that the probability of a cylinder set of the aforementioned form is. = €₂ and H} ... = (a) (2 points) Let X be the random variable defined by X (w) for every w E. Is X continuous or discrete? = min{i € N: wi = (b) (3 points) Find the pdf or pmf of X depending on what you answered above. (c) (4 points) Let u, v E N be positive integers. What is P [X>u+v | X > u] ? How does it vary with u and v?

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