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Exam Probability Theory TUM...

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Probability Theory Summer Term 2020 Gantert/Criens

Technical University of Munich Department of Mathematics Chair for Probability

One-Time Exercise Every step (also computations) in your solution has to be reasoned. Problem E.1 (8 Points) Let (X, Y ) be an R2 -valued random variable with density f (x, y) =

2 I{x≥0} I{x2 +y2 ≤1} . π

(i) Compute E[X|Y ]. (ii) Compute E[Y |X ]. Problem E.2 (10 Points) Let p ∈ [0, 1] and let X0 , X1 , X2 , . . . be random variables. Suppose that X0 has values in [0, 1] and a.s. for all n = 0, 1, . . . P (Xn+1 = 1 − p + pXn |X0 , X1 , . . . , Xn ) = Xn , P (Xn+1 = pXn |X0 , X1 , . . . , Xn ) = 1 − Xn . Show that Xn converges almost surely and in L1 as n → ∞. Problem E.3 (10 Points) Let X1 , X2 , . . . be i.i.d. random variables and c ∈ (0, 1). Show that ∞ X

 <

∞ a.s. if E[X1 ] < ∞, e c  = ∞ a.s. if E[X1 ] = ∞. k=1 Xk k

Problem E.4 (12 Points) Let X1 , X2 , . . . be independent random variables such that Xk is exponentially distributed with parameter k!1 , i.e. Xk has density fk (x) = Set Sn :=

Pn

k=1

1 1 − k! e x I{x≥0} . k!

Xk . Show that Sn → exp(1) in law as n → ∞, n!

where exp(1) is the exponential distribution with parameter 1. Hint: What is the distribution of Xn /n!?...