Let {X} be a sequence of i.i.d. continuous random variables with probability density function f(x). (a)...

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Let {X₂} be a sequence of i.i.d. continuous random variables with probability density function f(x). (a) Find P(X₁ ≤ X₂). (b) Find P(X₁ ≤ X2, X1 ≤ X3). (c) Let N be a new integer-valued random variable defined as follows. N is the index of the first random variable that is less than X₁, that is, P(N = n) = P(X₁ ≤ X2, X₁ ≤ X3,..., X₁ ≤ Xn-1, X₁ > Xn). Find P(N> n) as a function of n. (d) Show that E(N) = ∞ (5) Let {X₂} be a sequence of i.i.d. continuous random variables with probability density function f(x). (a) Find P(X₁ ≤ X₂). (b) Find P(X₁ ≤ X2, X1 ≤ X3). (c) Let N be a new integer-valued random variable defined as follows. N is the index of the first random variable that is less than X₁, that is, P(N = n) = P(X₁ ≤ X2, X₁ ≤ X3,..., X₁ ≤ Xn-1, X₁ > Xn). Find P(N> n) as a function of n. (d) Show that E(N) = ∞ (5)

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