Numerical Analysis
3rd Edition
ISBN: 9780134696454
Author: Sauer, Tim
Publisher: Pearson,
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Chapter 9.3, Problem 4CP
a.
To determine
To find: The probability of the random walk reaching the top a of the given interval
b.
To determine
To find: The probability of the random walk reaching the top a of the given interval
c.
To determine
To find: The probability of the random walk reaching the top a of the given interval
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1. Suppose that Y₁ = A + Bt + Xt, where {Xt} is a random walk. First suppose that A
and B are constants.
(a) Is {Y} stationary?
(b) Is {VY} stationary?
Now suppose that A and B are random variables that are independent of the random
walk {Xt}.
(c) Is {Y} stationary?
(d) Is {VY} stationary?
2. A bank operates both a drive-up facility and a walk-up window. On a
randomly selected day, let X = the proportion of time that the drive-up facility
is in use (at least one customer is being served or waiting to be served) and Y
= the proportion of the time that the walk-up window is in use. Then the set
of possible values for (X, Y) is the rectangle
D = {(x, y): 0 ≤ x ≤ 1,0 ≤ y ≤ 1}. Suppose the joint probability density
function of (X, Y) is given by:
fx,y(x,y) = { / (x + y ² )
²²(x + y²) 0≤x≤ 1,0 ≤ y ≤ 1
Otherwise
0
(a) Show that fx,y is legitimate
(b) Find the probability that neither facility is busy more than one-quarter of
the time.
(c) Find fx and fy, the marginal probability density function of X and Y
respectively.
(d) Construct the conditional probability density function of Y given that
X=0.8.
(e) Evaluate the probability that the walk-up facility is busy at most half of the
time given that X=0.8
(f) Calculate the expected proportion of the time that the walk-up facility…
4. Find the least-squares solution of Ax = b for A
-
012
12
and b
=
|
Chapter 9 Solutions
Numerical Analysis
Ch. 9.1 - Find the period of the linear congruential...Ch. 9.1 - Find the period of the LCG defined by a=4,b=0,m=9...Ch. 9.1 - Approximate the area under the curve y=x2 for 0x1,...Ch. 9.1 - Approximate the area under the curve y=1x for 0x1,...Ch. 9.1 - Prob. 5ECh. 9.1 - Prove that u1=x21+x22 in the Box-Muller Rejection...Ch. 9.1 - Implement the Minimal Standard random number...Ch. 9.1 - Implement randu and find the Monte Carlo...Ch. 9.1 - (a) Using calculus, find the area bounded by the...Ch. 9.1 - Carry out the steps of Computer Problem 3 for the...
Ch. 9.1 - Use n=104 pseudo-random points to estimate the...Ch. 9.1 - Use n=104 pseudo-random points to estimate the...Ch. 9.1 - (a) Use calculus to evaluate the integral 01x2x,...Ch. 9.1 - Prob. 8CPCh. 9.1 - Prob. 9CPCh. 9.1 - Devise a Monte Carlo approximation problem that...Ch. 9.2 - Prob. 1CPCh. 9.2 - Prob. 2CPCh. 9.2 - Prob. 3CPCh. 9.2 - Prob. 4CPCh. 9.2 - Prob. 5CPCh. 9.2 - One of the best-known Monte Carlo problems is the...Ch. 9.2 - Prob. 7CPCh. 9.2 - Prob. 8CPCh. 9.2 - Prob. 9CPCh. 9.3 - Design a Monte Carlo simulation to estimate the...Ch. 9.3 - Calculate the mean escape time for the random...Ch. 9.3 - In a biased random walk, the probability of going...Ch. 9.3 - Prob. 4CPCh. 9.3 - Design a Monte Carlo simulation to estimate the...Ch. 9.3 - Calculate the mean escape time for Brownian motion...Ch. 9.3 - Prob. 7CPCh. 9.4 - Use Itos formula to show that the solutions of the...Ch. 9.4 - Use Itos formula to show that the solutions of the...Ch. 9.4 - Use Itos formula to show that the solutions of the...Ch. 9.4 - Prob. 4ECh. 9.4 - Prob. 5ECh. 9.4 - Prob. 6ECh. 9.4 - Use the Euler-Maruyama Method to find approximate...Ch. 9.4 - Use the Euler-Maruyama Method to find approximate...Ch. 9.4 - Apply the Euler-Maruyama Method with step size...Ch. 9.4 - Prob. 4CPCh. 9.4 - Prob. 5CPCh. 9.4 - Prob. 6CPCh. 9.4 - Use the Milstein Method to find approximate...Ch. 9.4 - Prob. 8CPCh. 9.4 - Prob. 9CPCh. 9.4 - Prob. 10CPCh. 9.4 - Prob. 11CPCh. 9.4 - Prob. 12CPCh. 9.4 - Prob. 1SACh. 9.4 - Prob. 2SACh. 9.4 - Prob. 3SACh. 9.4 - Prob. 4SACh. 9.4 - Compare your approximation in step 4 with the...Ch. 9.4 - Prob. 6SA
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