By Geiss

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**Extra resources for An Introduction to Probability Theory**

**Example text**

1) The family (fi )i∈I is independent. (2) For all families (Bi )i∈I of Borel sets Bi ∈ B(❘) one has that the events ({ω ∈ Ω : fi (ω) ∈ Bi })i∈I are independent. Sometimes we need to group independent random variables. In this respect the following proposition turns out to be useful. For the following we say that g : ❘n → ❘ is Borel-measurable provided that g is (B(❘n ), B(❘))measurable. 5 [Grouping of independent random variables] Let fk : Ω → ❘, k = 1, 2, 3, ... be independent random variables.

Then, for all λ > 0, P({ω : f (ω) ≥ λ}) ≤ ❊λf . Proof. We simply have λP({ω : f (ω) ≥ λ}) = λ❊1I{f ≥λ} ≤ ❊f 1I{f ≥λ} ≤ ❊f. 2 [convexity] A function g : ❘ → ❘ is convex if and only if g(px + (1 − p)y) ≤ pg(x) + (1 − p)g(y) for all 0 ≤ p ≤ 1 and all x, y ∈ ❘. Every convex function g : ❘ → ❘ is (B(❘), B(❘))-measurable. 3 [Jensen’s inequality] If g : f : Ω → ❘ a random variable with ❊|f | < ∞, then ❘ → ❘ is convex and g(❊f ) ≤ ❊g(f ) where the expected value on the right-hand side might be infinity.

So we take the probability space ([0, 1], B([0, 1]), λ) and define for p ∈ (0, 1) the random variable f (ω) := 1I[0,p) (ω). Then it holds µ({1}) := µ({0}) := P (ω1 ∈ Ω1 : f (ω1) = 1) = λ([0, p)) = p, P (ω1 ∈ Ω1 : f (ω1) = 0) = λ([p, 1]) = 1 − p. Assume the random number generator gives out the number x. If we would write a program such that ”output” = ”heads” in case x ∈ [0, p) and ”output” = ”tails” in case x ∈ [p, 1], ”output” would simulate the flipping of an (unfair) coin, or in other words, ”output” has binomial distribution µ1,p .