22. Suppose (An} are independent events such thatΣ(P(An) (1 - P(An))) =n=18Show P is non-atomic.=∞.

Given::- Suppose \{A_{n}\} are independent events such that sum n = 1 to ∞ (P * (A_{n}) ^ (1 -…

Answered: 22. Suppose (An) are independent events…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

See similar textbooks

Similar questions

•Suppose xau (-54,60) and F(t) is the Cumulative distribution function. which is the probability that x is in the interval E-51.-21] and in the interval [36,57] A. F(57)-F(51) B.(F(-21)- F($1)) +(F(57)-F(-36)) C. (F (-21)-F(-51) × (F(57)-F(-36)) D. F(-21) - F(-36) (6)
For two events, M and N, P(M)= 0.6, P(N|M) = 0.2, and P(N|M')=0.7. Find P(MIN'). P(MIN') = (Simplify your answer. Type an integer or a fraction.)
7
Let {e} be a zero-mean, unit-variance white noise process. Consider a process {Y} which starts at time t = 0, and is defined recursively as follows: Let Yo = cieo, Y₁ = c₂Yo + e₁, and Y₁ = ¢₁Yt-1 + 2Yt-2 + et for t≥ 2 (as in an AR(2) process). a) Show that the process has zero mean. 1 b) For values of 01, 02 in the stationarity region for an AR (2) process, show how to choose C₁, C₂ such that both Var(Y) = Var(Y₁) and the lag 1 autocorrelation cov(Yo, Y₁) equals that of a stationary AR(2) process with parameters 01, 02. c) Once the process {Y} has been generated, show how one can transform in into a new process which has any desired mean and variance. (Remark. This gives a method for simu- lating stationary AR(2) processes.)
4) Let X and X1, X2, ... be random variables. Then - Xn 4 X as n → ∞ + E → 0 as n → 0. 1+|Х, — |x - "x|
23. Suppose (An) are events. (a) If for each k show (d) Show iff n-1 Σ P(An! ΠA) = 00, n=k i=k (b) What is the relevance to the Borel Zero-One Law? (c) Is it enough to assume P(lim sup An) = 1. 8x 8 n=1 n- P(A₂A) = ∞o? i=1 P (lim sup An) = 1 81x 8 Σ P(AAn) = 00 n=1 for all events A such that P(A) > 0.
If A and B are events for which P(A U B) = 1,then P(A' U B') = a) 1 b) Р(А') + P(В") c) P(A) + P(Β') - 1 d) 0 e) P(A') + P(B') – P(A') · P(B')
Example 7-36. Let {Xx} be mutually independent, each assuming the values 0.12 . ..., a - 1 with probability . Let S, = X1 + X2 + ... + Xp. Show that the probability generatino (6) = {; P (s) : 1- sa a(1 – s) function of S, is : P (S„ = j) = £ (-1)"*j*av (4)(;=u) and hence %3D - av v=0
Let X = (Y, Z) be the fair dice roll with E(Y ) = {1, . . . , 6} with PY (y) = 1/6, andZ be an independent fair coin toss with E(Z) = {0, 1} and PZ(z) = 1/2.a) What is PX(x)?b) What is PX(Y) for the abstract event Y = {(1, 1),(2, 0)}?c) What is the probability of the event that z + y = 2?d) What is PZ|Y (1|6)?
Example 17.15. Let X1, X2, ..., X, be a random sample from a distribution with p.d.f. : ••.. (x)- f(x, 8) = e¯*-®),0
Suppose that {u¡ t = ...,-2,-1,0,1,2,...} is an independent time series with mean 0 variance o = 4.0. Suppose that the time series {xj: t = ...,-2,-1,0,1,2,...} satisfies the equation.: x=0.5 x_₁ + 1.0+ u, -0.4 -1. Determine the 1. mean, 2. autocovariance function, 3. variance, 4. autocorrelation function, 5. Partial autocorrelation function of the time series x₁.

Recommended textbooks for you

A First Course in Probability (10th Edition)

Probability

ISBN:

9780134753119

Author:

Sheldon Ross

Publisher:

PEARSON

A First Course in Probability

Probability

ISBN:

9780321794772

Author:

Sheldon Ross

Publisher:

PEARSON

A First Course in Probability (10th Edition)

Probability

ISBN:

9780134753119

Author:

Sheldon Ross

Publisher:

PEARSON

A First Course in Probability

Probability

ISBN:

9780321794772

Author:

Sheldon Ross

Publisher:

PEARSON

SEE MORE TEXTBOOKS

22. Suppose (An) are independent events such that {An} Σ(P(An) (1 - P(A₂)) = x n=1 Show P is non-atomic.