5. Let X, X2,..., Xn be integrable random variables (i.e., EX n} = 0 otherwise. 7....

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5. Let X₁, X2,..., Xn be integrable random variables (i.e., EX <∞), and let y = maxi<k<nXk (a) Prove that Y is also integrable. (b) Prove that EXEY for all ke {1,2,...,n}. (c) Show by means of a counterexample that E|XK| ≤ EY| does not necessarily hold. 6. Let X be a non-negative random variable. Show that lim_ n E ( + I{X > n})= 0 n→∞ and lim ¹ E(XI{X <n})= = 0. n→∞ N Here, I is an indicator function: I{X > n} = 1 if X > n, and I{X > n} = 0 otherwise. 7. Let X₁, X2 be independent exponentially distributed random variables, with densities fi(x) = A₁e-1 and f2(x) = 2e-2 respectively (for a > 0). (a) Determine the density of X₁ + X₂. (b) Determine the density of min{X1, X2}. 8. Consider a Galton-Watson process whose offspring distribution has mean EX = 1 and variance Var X = ² <oo. Determine mean and variance of the number Yn of individuals in generation n. 5. Let X₁, X2,..., Xn be integrable random variables (i.e., EX <∞), and let y = maxi<k<nXk (a) Prove that Y is also integrable. (b) Prove that EXEY for all ke {1,2,...,n}. (c) Show by means of a counterexample that E|XK| ≤ EY| does not necessarily hold. 6. Let X be a non-negative random variable. Show that lim_ n E ( + I{X > n})= 0 n→∞ and lim ¹ E(XI{X <n})= = 0. n→∞ N Here, I is an indicator function: I{X > n} = 1 if X > n, and I{X > n} = 0 otherwise. 7. Let X₁, X2 be independent exponentially distributed random variables, with densities fi(x) = A₁e-1 and f2(x) = 2e-2 respectively (for a > 0). (a) Determine the density of X₁ + X₂. (b) Determine the density of min{X1, X2}. 8. Consider a Galton-Watson process whose offspring distribution has mean EX = 1 and variance Var X = ² <oo. Determine mean and variance of the number Yn of individuals in generation n.

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