# Exercises in Probability: A Guided Tour from Measure Theory by L. Chaumont

By L. Chaumont

This set of solved difficulties contains degree conception and likelihood and the extent of hassle is that of the Ph. D. scholar. the issues delve deeply into the speculation of likelihood, independence, Gaussian variables, allotted computations and random procedures. There are nearly a hundred difficulties and approximately whole recommendations to them all are incorporated. there's a assertion at the again conceal that the various difficulties can lead the scholar directly to examine issues in likelihood and that i absolutely consider that.

The bankruptcy headings are:

*) degree conception and probability

*) Independence and conditioning

*) Gaussian variables

*) Distributional computations

*) Convergence of random variables

*) Random processes

This is a superb self-study advisor for the coed that wishes difficulties that would push them to the very fringe of examine in likelihood.

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**Additional resources for Exercises in Probability: A Guided Tour from Measure Theory to Random Processes, via Conditioning**

**Sample text**

Let P (X ∈ dt) = ϕ(t) dt. Find a density ϕ such that: (i) {X} and [X] are independent; (ii) {X} is uniformly distributed on [0, 1]. Hint: Use the previous question and make an adequate change of probability from e−t dt to ϕ(t) dt. 3. We make the same hypothesis and use the same notations as in question 2. Characterize the densities ϕ such that {X} and [X] are independent. 2. 1 1. Let f and g be two bounded measurable functions on (Ω, F, P ). Then, 1 E(f (εX)g(ε)) = [E(f (X)g(1)) + E(f (−X)g(−1))] 2 and the variables εX and ε are independent if and only if E(f (X))g(1) + E(f (−X))g(−1) = E(f (X))(g(1) + g(−1)) for every f and g as above.

3. 2) for some γ > 0. 4. Assume that the equivalent properties stated in question 1 hold. If ν(dt) is a positive σ-ﬁnite measure on IR+ , we deﬁne: ν(dt)e−t Lν ( ) = IR+ and Sν (m) = ν(dt) IR+ m . 1 + tm 42 Exercises in Probability Then, prove that: E[XLν (L)] = Sν (E[X]) . e. P (Z ∈ dt) = e−t dt. Deﬁne {Z} and [Z] to be respectively the fractional part, and the integer part of Z. 1. Prove that {Z} and [Z] are independent, and compute their distributions explicitly. 2. Consider X a positive random variable whose law is absolutely continuous.

25 (67), 107–150 (1949). 2. 1: further statements of independence versus measurability Consider, on a probability space (Ω, F, P ), three sub-σ-ﬁelds A, B, C, which are (F, P ) complete. Assume that: (i) A ⊆ B ∨ C, and (ii) A and C are independent. 1. Show that if the hypotheses (i) A ⊆ B ∨ C, and (ii)’ A ∨ B is independent of C are satisﬁed, then A is included in B. 2. Show that, if (ii)’ is not satisﬁed, it is not always true that A is included in B. 3. Show that if, besides (i) and (ii), the property (iii): B ⊆ A is satisﬁed, then: A = B.