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- Question : EX1 - (i) If Fi, i ? I are ? -?elds, then ?i?I Fi is. Here I ?= ? is an arbitrary index set (i.e., possibly uncountable). (ii) Use the result in (i) to show that if we are given a set ? and a collection A of subsets of ?, then there is a smallest ?-?eld containing A. We will call this the ?-?eld generated by A and denote it by ?(A).
- Question : EX2 - Let ? = R, F = all subsets so that A or Ac is countable, P (A) = 0 in the ?rst case and = 1 in the second. Show that (?, F, P ) is a probability space.
- Question : EX3 - Recall the de?nition of Sd from Example 1.1.3. Show that ?(Sd) = Rd, the Borel subsets of Rd.
- Question : EX4 - A ? -?eld F is said to be countably generated if there is a countable collection C ? F so that ?(C) = F. Show that Rd is countably generated.
- Question : EX5 - (i) Show that if F1 ? F2 ? . . . are ? -algebras, then ?iFi is an algebra. (ii) Give an example to show that ?iFi need not be a ?-algebra.
- Question : EX6 - A set A ? {1, 2, . . .} is said to have asymptotic density ? if lim n??|A ? {1,2,...,n}|/n = ? Let A be the collection of sets for which the asymptotic density exists. Is A a
- Question : EX2.1 - Suppose X and Y are random variables on (?, F, P ) and let A ? F. Show that if we let Z(?) = X(?) for ? ? A and Z(?) = Y(?) for ? ? Ac, then Z is a random variable.
- Question : EX2.2 - Let ? have the standard normal distribution. Use Theorem 1.2.3 to get upper and lower bounds on P (? ? 4).
- Question : EX2.3 - Show that a distribution function has at most countably many discontinuities.
- Question : EX2.4 - Show that a distribution function has at most countably many discontinuities.
- Question : EX2.5 - Suppose X has continuous density f , P (? ? X ? ?) = 1 and g is a function that is strictly increasing and differentiable on (?, ?). Then g(X) has density f (g?1(y))/g?(g?1(y)) for y ? (g(?), g(?)) and 0 otherwise. When g(x) = ax + b with a > 0, g?1(y) = (y ? b)/a, so the answer is (1/a)f ((y ? b)/a).
- Question : EX2.6 - Suppose X has a normal distribution. Use the previous exercise to compute the density of exp(X). (The answer is called the lognormal distribution.)
- Question : EX2.7 - (i) Suppose X has density function f . Compute the distribution function of X2 and then differentiate to ?nd its density function. (ii) Work out the answer when X has a standard normal distribution to ?nd the density of the chi-square distribution.
- Question : EX3.1 - Show that if A generates S, then X?1(A) ? {{X ? A} : A ? A} generates ?(X) = {{X ? B} : B ? S}.
- Question : EX3.2 - Prove Theorem 1.3.4 when n = 2 by checking {X1 + X2 < x} ? F.
- Question : EX3.3 - Show that if f is continuous and Xn ? X almost surely, then f (Xn) ? f (X) almost surely.
- Question : EX3.4 - (i) Show that a continuous function from Rd ? R is a measurable map from (Rd, Rd) to (R, R). (ii) Show that Rd is the smallest ?-?eld that makes all the continuous functions measurable.
- Question : EX3.5 - A function f is said to be lower semicontinuous or l.s.c. if lim inf y?x f (y) ? f (x) and upper semicontinuous (u.s.c.) if ?f is l.s.c. Show that f is l.s.c. if and only if {x : f (x) ? a} is closed for each a ? R and conclude that semicontinuous
- Question : EX3.6 - Let f : Rd ? R be an arbitrary function and let f ?(x) = sup{f (y) : |y ? x| < ?} and f?(x) = inf{f (y) : |y ? x| < ?} where |z| = (z12 +
- Question : EX3.7 - A function ? : ? ? R is said to be simple if ?(?) = n ? m=1 cm1Am(?)
- Question : EX3.8 - Use the previous exercise to conclude that Y is measurable with respect to ?(X) if and only if Y = f (X) where f : R ? R is measurable.
- Question : EX3.9 - To get a constructive proof of the last result, note that {? : m2?n ? Y < (m + 1)2?n} = {X ? Bm,n} for some Bm,n ? R and set fn(x) = m2?n for x ? Bm,n and show that as n ? ? fn(x) ? f (x) and Y = f (X).
- Question : EX4.1 - SHOW THAT IF AND THEN F
- Question : EX4.2 - LET F AND E
- Question : EX4.3 - Let g be an integrable function on R and ? > 0. (i) Use the de?nition of the ?in.t(eigi)raUlsteoEcoxnerccluisdeeAth.2e.r1etios aapspimropxliemfautnectthioenA? = ?k bk1Ak with ? |g ? ?| dx < k by ?nite unions of intervals to get a step function q = k ? j=1 cj1(aj?1,aj)
- Question : EX4.4 - PROVE THE RIEMANN LEBSUGE LEMMA
- Question : EX5.1 - Let ?f ?? = inf{M :
- Question : EX5.2 - Show that if
- Question : EX5.3 - Minkowski
- Question : EX5.4 - If f is integrable and Em are disjoint sets with union E then ? ? m=0 ? Em f d
- Question : EX5.5 - IF GN G AND THEN
- Question : EX5.6 - IF GN G AND THEN
- Question : EX5.7 - de thfat?if g0.is(ii)nStehgorwabltehaatnd? ?f>?0n,dth
- Question : EX5.8 - Show that if f is integrable on [a, b], g(x) = ?[a,x] f (y) dy is continuous on
- Question : EX5.9 - hsoowthtahta?t ?ifnf?hfas??f?p = (? |f|pd
- Question : EX5.10 - Show that if ? n ? |fn|d
- Question : EX6.1 - Suppose ? is strictly convex, that is, > holds for ? ? (0, 1). Show that, under the assumptions of Theorem 1.6.2, ?(EX) = E?(X) implies X = EX a.s.
- Question : EX6.2 - Suppose ? : Rn ? R is convex. Imitate the proof of Theorem 1.5.1 to show E?(X1, . . . , Xn) ? ?(EX1, . . . , EXn) provided E|?(X1, . . . , Xn)| < ? and E|Xi| < ? for all i.
- Question : EX6.3 - Chebyshev
- Question : EX6.4 - One-sided Chebyshev bound. (i) Let a > b > 0, 0 < p < 1, and let X have P(X = a) = p and P(X = ?b) = 1 ? p. Apply Theorem 1.6.4 to ?(x) = (x + b)2 and conclude that if Y is any random variable with EY = EX and var (Y) = var (X), then P (Y ? a) ? p and equality holds when Y = X. (ii) Suppose EY = 0, var (Y) = ?2, and a > 0. Show that P (Y ? a) ? ?2/(a2 + ? 2), and there is a Y for which equality holds.
- Question : EX6.5 - wo nonexistent lower bounds. Show that: (i) if ? > 0, inf{P (|X| > ?) : EX = 0, var (X) = 1} = 0. (ii) if y ? 1, ?2 ? (0, ?), inf{P (|X| > y) : EX = 1, var (X) = ?2} = 0.
- Question : EX6.6 - A useful lower bound. Let Y ? 0 with EY2 < ?. Apply the Cauchy- Schwarz inequality to Y 1(Y>0) and conclude P(Y > 0) ? (EY)2/EY2
- Question : EX6.7 - Let ? = (0, 1) equipped with the Borel sets and Lebesgue measure. Let ? ? (1, 2) and Xn = n?1(1/(n+1),1/n) ? 0 a.s. Show that Theorem 1.6.8 can be
- Question : EX6.8 - StuhpepporsoeotfhtaetcthhneiqpuroeboafbTilhiteyomreemas1u.r6e.9
- Question : EX6.9 - Inclusion-exclusion formula. Let A1, A2, . . . An be events and A = ? n taik=e1eAxip.ePcrtoevdevtahlaute1tAo =con1c?lud?ein=1(1 ? 1Ai). Expand out the right-hand side, then P ??ni=1Ai? = n ? i=1 P(Ai) ? ?i
- Question : EX6.10 - Bonferroni inequalities. Let A1, A2, . . . An be events and A = ?ni=1Ai. Show that 1A ? ?in=1 1Ai, and so forth, and then take expected values to conclude P ??ni=1Ai? ? n ? i=1 P(Ai) P ??ni=1Ai? ? n ? i=1 P(Ai) ? ?i
- Question : EX6.11 - If E|X|k < ? then for 0 < j < k, E|X|j < ?, and furthermore E|X|j ? (E|X|k)j/k
- Question : EX6.12 - Apply Jensen
- Question : EX6.13 - If EX1? < ? and Xn ? X, then EXn ? EX.
- Question : EX6.14 - Let X ? 0 but do NOT assume E(1/X) < ?. Show lim y?? yE(1/X; X > y) = 0, lyi?m0 yE(1/X; X > y) = 0.
- Question : EX6.15 - IF X O AND THEN E
- Question : EX6.16 - IF X IS INTEGRABLE AND A ARE DISJOINT SETS WITH UNION A
- Question : EX7.1 - If ? X ? Y |f (x, y)|
- Question : EX7.2 - Let g ? 0 be a measurable function on (X, A,
- Question : EX7.3 - Let F , G be Stieltjes measure functions, and let
- Question : EX7.4 - Let
- Question : EX7.5 - Show that e?xy sin x is integrable in the strip 0 < x < a, 0 < y. Perform the double integral in the two orders to get ? a 0 sin x x dx = (arctan a) ? (cos a) ?0? 1e+?ayy2 dy ? (sin a) ?0? 1y+e?yay2 dy aan?d 1re.place 1 + y2 by 1 to conclude ???0a(sin x)/x dx ? (arctan a)?? ? 2/a for

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