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elementary probability for applications

Probability Theory Independence Interchangeability Martingales 3rd Edition Yuan Shih Chow, Henry Teicher - Solutions

Show that it if the Poisson r.v.s S, have pdf.pk, ni nz 1, then E(S) - Q Hint: Recall Exercise 2.2.2 (ii) Any sequence of rvs Y
If the rxs X., are ui, so are S21, where S, if X. X, in particular, 1, are identically distributed, random variables, then (5,1
Let P{X,=> 0} -1/-1-PEX, 0, 1 (X) if 4-10>07 6IX, 21) are rxa with sup. EX, for some > 0, then [...] is ui. for 0 <
If X, and X, are independent rvs and X, +X,, for some pe (000), then X, 1.2. H: For all large >0
Let (4,21] be events with X for any p>0, (ii) XX for all P4). If there exist events (8,21) and (D)>L21) such that for all large i and some positive integer & (resp. infinitely many k > 0) LAAD < PUD PUB i. the classes (4) and (DDDDDD.] are independent, then PB-1 (resp. P[8,..) = 1).
such that (0 X, X but X. all p>0 but XX.
Prove that if rvs X, X and X, Y, then X-Y, ac. Construct rxs X. X.
Improve the inequality of Lemma 3 by showing that (a+by sta+b)-maxi1, 2) for a > 0, &>>>Q also, ifa, 20, then (ar of (resp. s) for p1 (resp si
If (X1) are Lid. random variables then (i) 1/masif EX 0.
If X and Y are measurable functions on (Q,, P) with EXI, EYI, for all AF. then XY, a.c. Hier: Consider A,,( Xor>22-0).
@(X) is a sequence of nonnegative, integrable r.v.s with S, X, +- X, and (ii) EX, co, then &, converges ac. Hence, if (i) obtains, ES, > 0, 21, and EXES,, then S/E &, converges ac
I{X, X, 1) is a sequence of r.v.x on some probability space such that EIX-XT < somer> 0, then X, X.
Construct a sequence of discrete rvs X, such that X,
If X is an integrable rv, n P(X 2) (11 but the converse does not hold.
Utilize Exercise 1.2.6 to give an alternative proof of Corollary 2.1.3. 4 P(X-1) P(X=0)-4-1-p prove that E(X-EXY-- (-12) if X is a rv. with P(X)-1, then EX- but E X. +0.
If X is an integrable rv, for every >0 there is a simple function X, with EIX-X,
Let S, X, where (X.2 11 are independent rsx and suppose that lim PS, 2-> all >0. Then SimSsx, ac implies PIX,> C
If (X. X. 1) are lid. symmetric rvs and S, -x, then (i) PIS>x]P[X>2x)P(X 2x) for x>0, and (ii) P(&>x2PX>x[1-(n-1)P(X>x]]. Part() is untrue if all "two" are deleted (Taken 2x-1 and PX-11 Hi Apropos of (i) define T-inf{1: X,>x)
then
Let S, X, where (X.2 11 are independent rsx and suppose that lim PS, 2-> all >0. Then SimSsx, ac implies PIX,> C 15 rvs X, X.finite, prove that for every > Othere is a set 4, with PI4)
If (X. X. 1) are lid. symmetric rvs and S, -x, then (i) PIS>x]P[X>2x)P(X 2x) for x>0, and (ii) P(&>x2PX>x[1-(n-1)P(X>x]]. Part() is untrue if all "two" are deleted (Taken 2x-1 and PX-11 Hi Apropos of (i) define T-inf{1: X,>x)
then
14. Let (X, 1) be rva such that PIX,l2>0) 2021 fare finite constants for whicha, X, 0, then aa Copyrighted material 15 rvs X, X.finite, prove that for every > Othere is a set 4, with PI4)
IX. X. 11 are finite measurable functions on a measure space (S. L.) with atX.-X> (1)>0, then supX-X> () > 0, and there exists a subsequence X, with allim, X., X] =
IX. XX, are independent, symmetric rv with PX, X, X, M)=1. then PIXM)-1.
If the r.. W, on (P) is X. measurable. 21, where, is a decreasing sequence of sub-e-algebras of F and WW, then Wis, measurable.
Mr. X, Xasand (N. 1) are positive-integer valued 1.v.s with (0N,, then X, X. I rather, (ii) N, x, that is, PIN, 0 and X is a rv, then X,
Prove for any rv.ax.) and constants (h) with 0see Lemma 10.1.1) P{max X.-2P max X- 121
If the rvs X, 40, where the constantsb, satisfy them max X, X.-
If (X1) are independent, symmetric rvs sach that (1x, for some positive constants h,, then (bmas, X, H Pmax X>d s
5. For any sequence of rvs (X.20) with XX.. Pim X, X, slim X.-1. x-1 Conversely, if lim X-X (resp. lim. X. Xel ac, then for any > 0) PIXX (1) (resp. PIX, X-4) (1)
If independent rvs X,X, then X is degenerate. Prove for nondegenerate ii.d. rss (X) that PX, converges) -
i. What is wrong with the following "proof" of Corollary 3?ii. A r.v. X is symmetric iff X and X have identical dis If 0 is a median of a rv. X, it is also a median of XIe for any ()
Let (X21) and (Ya 1) be two sequences of rvs with F-F for 1.IX, X. prove that Y Y and that X and Y are identically distributed. Hint: Apply Lemma 3.3.2.
IX. X and X, Y, then P(X-1)-1 i X, X and Y, X, 0 implies Y imply X, Y, X+Y X)-0 iv. X, X. Y, Yandgisa continuous function on Rtheng(X, Yax.n
If (X1) is a sequence of independent finite-valued rvs and (a) is a sequence of finite constants with 0
If (X. n 1) is a sequence of finite-valued tvs and S-x, determine which of the following are tail events of [X, 21: 15, converges): [lim S, > lim S,);
Show that random vectors X-(XX) and (YY) are independent (of one another) if the joint df. of X and Y is the product of the dfs of X and of Y.
Hint: Recall Exercise 2.1.4.
Prove that if (i) (X.21) are Lid. NO,e) random variables, then x. Plim Brasher. (X,,1) are Lid exponential rvs with parameter (Exercise 3.1.15) then Plim, Xlogn-21. f (X1) are iid. Poisson rvs with parameter, then PmXoglogn/logn 1) 1 irrespective of
are independent rxs, prove that Phim X-0-1 all>Q
If x..>
Find a trivial example of dependent events 4, with PA, divergent but
(421) is a sequence of events with P(A), show that lim supP44) - 1 for all m1.
A r.v. with df. Fix; ) is said to have an exponential distribution with parameter
IN, is a geometric rv. with parameter p, that is, PIN, A) pq.k0, prove that lim, PN, 0, and check that FL) (1- adf. for any >
Such a "Lebesgue measure exists and is unique (Section 6.1) Fore, let be the decimal expansion of to (for definineness, no "finite expansion" is permitted). Prove that (21) are independent rvs with P, -1)=1-01....
Let [0,1] and of Borel subsets of 2, and let P be a probability measure on such that P(a, b)-b-a for sass
If 14,) is a sequence of independent events with P4) 1,42 1,and P4.1-1. then P4] L
In the random casting of balls into a cells, let Y-1 if the ith cell is empty and -0 otherwise. For any kn, show that the pdf. of YY, depends only on and not on Hint: It suffices to consider PY, 1-13 for all sn
If Y and Z are independent Binomial (resp. Poison, negative binomial) .v.s with parameters (n, p) and (m, p) (resp. A, and A., and r), then Y+Z is a binomial (resp. Poisson, negative binomial) r.v. with parameter (m + np) (resp. + Ap + Thus if (X.2 1) are Lid Poisson rvs with parameter, the sum
X-0, is where (XX) are Lid with PIX, 11-p-1-PX, 0). Find PIY- -0.1 which for obvious reasons is called the geometric distribution. If Y., are Lid. rvs with a geometric distribution, find the pdf of S.-X. known as the negative binomial distribution (with parameter ) Hint: S, may be envisaged as the
10. In Bernoulli trials with success probability plet Y be the waiting time beyond the 1st trial until the first success occurs, ie, YX,
Let (X,, na 1) be iid. rvs with PIX, X-0 for X- 21.prove that (X,21) are independent rvs with PX-1-1/4-12.... and
H X and Y are independent rva and X + Y is degenerate, then both X and Y are degenerate
IX. Y. Z are ton (2, P) and signifies the relation of independence, prove or counter examples for: give iX-Yi X-y X Y Z imply X-Z . X (X,Z), Y-Z imply X, Y, Z independent rv.s
Copyrighted material
If (X.21] are Lid. rvs with PIX, 0) 0 there exists an integer, such that P||S,l>c>
then (1) obtains for m-3 but the events B,. B, are not independent. 5.) If XX, are independent rvs and g.. Isis, are finite Borel functions on (-) then Y-X) Isis are independent rvs. In particular, -X-X, are independent r.v.a. (n) 8,1sis, are linear Borel sets and Y-X,where, is the indicator
As is a pair of independent events but the events 4, 42, 4, are not inde- pendent. On the other hand, if 8, lol. B-4. B,
A).
4).
If the classes of events, and 9 are independent, 21, so are, and 2 3.) Any r.v. X is independent of a degenerate rv. Y. (i) Two disjoint events are independent iff one of them has probability zero. () P(X +1, Y-11-1 for all four pairs of signs, then X and Y are independent r.v.s. 4 Let (4:40),
are independent classes
are independent rxs iff,-0.04
are independent iff their indicator functions
Events
9. (Reny) Let S, be a binomial r.v. with pdf bik, n, p, where 0
Prove for X, as in Exercise 4 that P(X, 5)>c>0, where A andc, otherwise. Hint: +1, and 4-AA for integer PX, s) for n sic Aimplying 4..> Also- and by Exercise 4 PX, s)- For >0 and -0,1,....let ()() Prove that if j < < j, then where
(Normal approximation to the Poisson distribution.) If X, is a Poisson r.v. with parameter prove that for any- < . S-p (If-1, this yields the Borel strong law of large numbers.)
Use Lemma 2 to show that be; 2, })~(x)-
Prove that as x 1-(x+(x)) 1-x)
Show that the Bernoulli weak law of large numbers is consequence of the DeMoivre-Laplace theorem.
Prove that for
Let (X,2 1,21) be a sequence of t.vs such that P(X all>0. Prove that supo
Show that the Borel-Cantelli lemma is valid on any measure space (2) that is implies ulim, 4,1 0 for A E,
Prove a strong law of large numbers where S, has pdf pk; ni, 21, that is lim(S),ac Hint: Consider P[15,-***
(i) Prove a weak law of large numbers where S, has pdf pik: nl) P(S)-> (1)>0) {X. X.21] are rvs with a common df. and PiX]>) of1), then (1)max, s
Verify that if (0), then pkpl)-1,2. Hint: Recall Exercise 2.1.2 2-pik; 3+) 15 +252 + A show that ) 0 ))=2(d)= 21, that is
Verify that if p-p' in (16) Le, in Exercise 9, the probabilities q coalesce to the binomial and PIA, APA sh
For k-0.1.2.-12.0
A deck of N cards numbered 1.2... N is shuffled. A "match" occurs at position if the card numbered j occupies the jth place in the deck. If denotes the prob ability of exactly m matches. 0 Sms N, prove that --- MEN -) where is the probability of at least matches (-1)* (N-m
Find the df. of X' when EC)(2)-(+) LX is a Poisson r.v. with parameter X is NO. 1). Prove under (0) that! PIX tsk! for 0
Verily for all positive integers n,,, and nonnegative integers that
Verify that kp(k; 2)-2-p(k: 2)-22. kp(k;)=2+32+ kb(k; n. p) np or mpq+(np) as j-1 or 2.
Verify that kp(k; 2)-2-p(k: 2)-22. kp(k;)=2+32+ kb(k; n. p) np or mpq+(np) as j-1 or 2.1. Prove that for n 1.2.... xxxxxx-1-3-(2-1)
Prove that for n 1.2.... xxxxxx-1-3-(2-1)
For a d.f. F. the set S(F)= {x: F(x+2)- F(x-2) > 0 for all > 0) is called the support of F. Show that each jump point of F belongs to the support and that each isolated point of the support is a jump point. Prove that S(F) is a closed set and give an example of a discrete d.f. whose support is (-,
Some authors define such functions to be df.s.
Prove that if H(x) = P(X x), where X is a r.v. on (Q, F, P), then H is a nondecreas- ing right-continuous function with H(-)-0, H(+)
fp. 20,pes,re T. and G. resp. where S and T are countable subsets of (-x, x), define a probability space and random variables X. Y on it with Fx(x) PFMY) - Pr Fly)Pr Hint: Take = Sx T.
Prove that G as defined in (3) is a d.f.; verify that (Q. F. P) as defined thereafter is a probability space and that F,
Prove that A = {(x, y): x + y
8. Show that the class of Borel sets may be generated by 1(x, x]. -x < x < x} or by ={{+}. [-x, x], [a,b).x sash sx).
prove that is the algebra generated by
If Des for every De
Let be a x-class of subsets of 2 and the class of all finite unions of disjoint sets of with
are measurable spaces, the class of all cylinder sets of XQ, with bases in X, for some m 1 is an algebra, but not a -algebra. Moreover, settingX A: A, E, verify that X = (D) = 0(6).
If (2)
The a-algebra generated by a countable class of disjoint, nonempty sets whose union is the class of all unions of these sets.

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