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Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section: Chapter Questions

Problem 63RE

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i-0¹¹
for a - 1, 2; but ₁0¹i < ∞ for d ≥ 3. By Theorem 2
the drunkard's walk in certain recurrent in dimensions 1 and 2; but uncertain
recurrent in dimensions 3 and higher.
Exercise 3. Much more is known about all this. Some things for you to research
and report on.
What if the drunkard's walk is asymmetric? That is the probability of a
move to the left is not the same as a move to the right?
• For the symmetric walk, what is the value of the recurrence probability for
d> 3?
• In the case d = 1,2 what is the expected length of a path returning to the
origin?
. In the case d = 1,2, what is the probability of two returns? Three returns?
An infinity of returns to the origin?
7

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22. The networks in parts (a) and (b) of Figure 2.23 show two resistors coupled in series and in parallel, respectively. We wish to find a general formula for the effective resistance of each network—that is, find such that . (a) Show that the effective resistance of a network with two resistors coupled in series [Figure 2.23(a)] is given by (b) Show that the effective resistance of a network with two resistors coupled in parallel [Figure 2.23(b)] is given by
3.) Suppose X has probability generating function GX(t) = 0.2 + 0.3t + 0.1t2 + 0.4t3. What is P(X = 2)? What is P(X = 0)?
Suppose N = 10 and r = 3. Compute the hypergeometric probabilities for the following values of n and x. (Round your answers to four decimal places.) (a) n = 6, x = 1 (b) n = 3, x = 3 (c) n = 3, x = 0 (d) n = 6, x = 3 (e) n = 6, x = 6
4.3.2 Let Y ∼ Uniform[0, 1], and let Xn = Yn. Prove that Xn → 0 with probability 1.
Show that for the simple random walk starting at zero,P(first visit to S^(2n) occurs at t = 2k) = P(S_(2k) = 0) · P(S_(2n−2k) = 0)for 0 ≤ k ≤ n.
How to solve this with cohens d using Microsoft Excel Use alpha of 0.05 2 tail. Free Association (in seconds) Partially Controlled (in seconds)5.84 4.844.64 5.643.56 8.565.00 7.002.35 4.353.67 4.677.87 6.677.74 5.444.48 5.586.56 6.26
Everyday, James reverses his car from his driveway on to the road in such a way that there is a very small probability P that his car will be involved in a collision. ( i) Show that the probability that there will be no collision in a five-day week is (1−P)^5 and state one assumption that is made in your answer. (iii) Let P = 0.001. Check that the Poisson approximation can be used, and find the approximate probability that James will avoid a collision in 500 days.
4. The lifetime, in years, of a type of small electric motor operating under adverse conditions is T ~Γ(6, 3.6). Whenever the motor fails, it is replaced with another of the same type. Find theprobability that fewer than 6 motors fail within one year.
Take the Gamblers ruin on [0, N] with a fair coin, i.e., a simple random walk with absorbing boundaries at 0 and N. Let T be the time which elapses before the SRW is absorbed by one of the boundaries (i.e., the duration of the game. Show that P(T < ∞) = 1 and Epecation (T^k) < ∞ for all k.
We now need to attempt to find an exact value for n such that P(r ≥ 1) = 0.99. Clearly, if the bank has any chance of detecting a burglar they will need to have at least one alarm installed. So, we know that, minimally, n must be at least 1.Since n must be at least 1, suppose we start by trying n = 2, where there will be two alarms. When there are two alarms, r ≥ 1 means either one or both alarms could detect the burglar. If neither alarm detects the burglar it is considered a failure. Therefore, P(r ≥ 1) can be calculated using either of the following formulas. P(r ≥ 1) = P(1) + P(2) P(r ≥ 1) = 1 − P ( ___ ) fill in the blank

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