New Semester
Started
Get
50% OFF
Study Help!
--h --m --s
Claim Now
Question Answers
Textbooks
Find textbooks, questions and answers
Oops, something went wrong!
Change your search query and then try again
S
Books
FREE
Study Help
Expert Questions
Accounting
General Management
Mathematics
Finance
Organizational Behaviour
Law
Physics
Operating System
Management Leadership
Sociology
Programming
Marketing
Database
Computer Network
Economics
Textbooks Solutions
Accounting
Managerial Accounting
Management Leadership
Cost Accounting
Statistics
Business Law
Corporate Finance
Finance
Economics
Auditing
Tutors
Online Tutors
Find a Tutor
Hire a Tutor
Become a Tutor
AI Tutor
AI Study Planner
NEW
Sell Books
Search
Search
Sign In
Register
study help
mathematics
statistics
Probability and Stochastic Processes A Friendly Introduction for Electrical and Computer Engineers 3rd edition Roy D. Yates, David J. Goodman - Solutions
X and Y have joint PDF(a) Are X and Y independent?(b) Let U = min(X,Y). Find the CDF and PDF of U.(c) Let V = max(X, Y). Find the CDF and PDF of V.
Random variables X and Y have joint PDFLet W = Y - X.(a) what is Sw, the range of W?(b) Find Fw(w) and fw(w).
Random variables X and Y have joint PDFLet W = X/Y(a) What is Sw, the range of W?(b) Find Fw(w), fw(w), and E[w].
X and Y are independent identically distributed Gaussain (0,1) random variables. Find the CDF of W = X2 + Y2.
Let X and Y be independent discreate random variables such that Px(k) = Py(k) =0 for all non-integer k. Show that the PMF of W = X + Y satisfies
Find the PDF of W = X + Y when X and Y have the joint PDF
Random variables X and Y are independent exponential random variables with expected values E[X] = 1/ λ and E[Y] = 1/μ. If μ ≠ λ, what is the PDF of W = X + Y? If μ = λ what is fw(w)?
Continuous random variables X and Y have joint PDF fx,y(x,y). Show that W = X-Y has PDFUse a variable substitution to show
Use icdfrv.m to write a function w=wrv1(m) that generates m samples of random variable W from Problem 4.2.4 Not the F-1W (u) does not exist for u = ¼: however, you must define a function icdfw(0.25). Does it matter what value you return for u = 0.25?
For random variable W of Example 6.10, we can generate random samples in two different ways: 1. Generate samples of X and Y and calculate W = Y/W. 2. Find the CDF Fw(w) and generate sample using Theorem 6.5. Write MATLAB functions w=wrv1(m) and w=wrv2(m) to implement these methods Does one method
Random variable X has CDFFind the conditional CDF Fx|x > 0(x) and PMF Px|x >0 (x).
X has PMFFind Px|B(x) where B = {X ‰ 0}.
Every day you consider going joging. Before each tulle including the first, you will quit with probability q, independent of the number of miles you have already run. However, you are sufficiently decisive that you never run a fraction of a mile. Also, we say you have run a marathon whenever you
7.1.7i A test for diabetes is a measurement X of a person's blood sugar level following an overnight fast. For a healthy person, a blood sugar level X in the range of 70- 110 rng/dl is considered normal. When a measurement X is used as a test for diabetes, the result is called positive (event T-
For the quantizer of Example 7.6, we showed in Problem 7.1.8 that the quantization noise Z is a uniform random variable. If X is not uniform, show that Z is nonuniform by calculating the PDF of Z for a simple example.
X is the binomial (5,1/2) random variable. Find Px|B(x), where the condition B = {X > μx}. What are E[X|B] and Var [X|B]?
For the distance D of a shot-put toss in Problem 4.7.8, find the conditional PDFs FD|D >(d) and fD|D
X is the continuous uniform (-5,5) random variable, Given the event B = {|X} < 3}, find the (a) Conditional PDF , fx|B(x) (b) Conditional expected value, E[X|B], (c) Conditional variance, Var[X|B]
For the experiment of spinning the pointer three times and observing the maxi mum pointer position, Example 4.5, find the conditional PDF given the event R that the maximum position is on the right side of the circle. 'What are the conditional expected value and the conditional variance?
Select integrated circuits, test them in sequence until you find the first failure, and then stop. Let N be the number of tests. All tests are independent, with probability of failure p = 0. 1. consider the condition B = {N > 20}. (a) Find the PMF PN(fl). (b) Find PN|B(n), the conditional PMF of N
The time between telephone calls at a telephone switch is the exponential random variable T with expected value 0.01. (a) What is E[TIT > 0.02], the conditional expected value of T? (b) What is Var[TIT > 0.02], the conditional variance of T?
X and Y are independent identical discrete uniform (1,10) random variables. Let A denote the event htat min(X,Y) > 5. Find the Conditional PMF Px,y|A(x,y).
Random variable X and Y have joint PDFLet A be the event that X + Y
X and Y have joint PDFLet A = {Y (a) what i, P[A]?(b) Find fX,Y|A(x,y).(c) Find fX|A(x) and fy|Ay).
A study examined whether there was correlation between how much football a person watched and how bald the person was. The time T watching football was measured on a 0, 1. 2 scale such that T = 0 if a person never watched football, T = 1 if a person watched football occasionally, and T = 2 if a
X and Y are independent random variable with PDFsLet A = {X > Y}. (a) What are E[X] and E[Y]? (b) What are E[X|A] and E[Y|A]?
Flip a coin twice. On each flip, the probability of head equal p. Let X1 equal the number of heads (either (0 or I) on flip i. Let W = X1 - X2 and Y = X1 + X2. Find Pw,y(w.y), Pw y(w|y), and PY|W y|w).
Packets arriving at an Internet router are either voice packets (v) or data packets (d). Each packet is a voice packet with probability p. independent of any other packet. Observe packets at the Internet router until you see two voice packet. Let M equal the number of packets up to and including
Each millisecond at an Internet router, a packet independently arrives with probability p. Each packet is either a data packet (d) with probability q or a video packet (v). Each data packet belongs to an email with probability r. Let. N equal the number of milliseconds required to observe the first
X is the continuous uniform (0, 1) random variable. Given X = x, Y is a continuous uniform (0, 1 + x) random variable. What is the joint PDF fX,Y(x, y) of X and Y?
A business trip is equally likely to take 2, 3, or 4 (lays. After a d-day trip, the change in the traveler's weight, measured as au integer number of pounds, is a uniform (-d, d) random variable. For one much trip, denote the number of days by D and the change in weight by 11'. Fluid the joint PMF
A student's final exam grade depends on how close the student sits to the center of the classroom during lectures. If a student sits r feet from the center of the room, the grade is a Gaussian random variable with expected value 80 - r and standard deviation r. If r is a sample variable of random
At the One Top Pizza Shop, mushrooms are the only topping. Curiously, a pizza sold before noon has mushrooms with probability p = 1/3 while a pizza sold after noon never has mushrooms. Also, a pizza is equally likely to be sold before noon as after noon. On a day in which 100 pizzas are sold, let N
X and Y have joint PDFFind the PDF fy(y), the conditional PDF fX|Y(x|y), and the conditional expected value E[X|Y = y].
The probability model for random variable A isThe conditional probability model for random variable B given A is:(a) What is the probability model for random variables A and B? Write the joint PMF PA,B(a,b) as a table.(b) If A = 1, what is the conditional expected value E{B|A = 1]?(c) If B 1, what
Random variables N and K have the joint PMF(a) Find the marginal PMF PN(n) and the conditional PMF PK|N( k|n). (b) Find the conditional expected value E[K|N = n. (c) Express the random variable E[K|N] as a function of N and use the iterated expectation to find E[K].
Over the circle X2 + Y2(a) What is fY|X(y|x)? (b) What is E[Y|X = x]?
In a weekly lottery, each $1 ticket sold adds 50 cents to the jackpot that starts at $1 million before any tickets are sold. The jackpot is announced each morning to encourage people to play. On the morning of the ith day before the drawing, the current value of the jackpot J: is announced. On that
You wish to measure random variable X with expected value EIX] = 1 and variance Var[X] = 1, but your measurement procedure yields the noisy observation Y = X + Z, where Z is the Gaussian (0,2) noise that is independent of X. (a) Find the conditional PDF fZ|X(z|x) of Z given X = x. (b) Find the
A study of bicycle riders found that a male cyclist's speed X (in miles per hour over a 100 mile 'century" ride) and weight 1' (kg) could be modeled by a bivariate Gaussian PDF fx,y(x, y) with parameters μx = 20, σx = 2, ,μy = 75, σy = 5 and pX,Y = -0.6. In addition, a female cyclist's speed X'
Use the iterated expectation for a proof of Theorem 5.19 without integrals.
For random variables X1,.... Xn Problem 5.10.3, let X = [X1 ... Xn]. What is fx(x)?
Given fx(x) with c = 2/3 and a1 = a2 = a3 = 1 in Problem 8.1.2, find the marginal PDF fx3(x3).
A wireless data terminal has three messages waiting for transmission. After sending a message, it expects an acknowledgement from the receiver. When it receives the acknowledgement, it transmits the next message. If the acknowledgement does not arrive, it sends the message again. The probability of
Let N be the r-dimensional random vector with the multinomial PMF given in Example 5.21 with n > r > 2:(a) What is the joint PMF of N1 and N2? (b) Let Ti = N1 + ... + Ni. What is the PMF of Ti? (c) What is the joint PMF of T1 and T2?
As a generalization of the message transmission system in Problem 8.1.5, consider a terminal that has n messages to transmit. The components ki of the n-dimensional random vector K are the total number of messages transmitted when message i is received successfully. (a) Find the PMF of K. (b) For
The n components Xi of random vector X have E[Xi] = 0 Var[Xi] = σ2. What is the covariance matrix CX?
As in Example 8.1, the random vector X has PDFwhere a = [1 2 3]ʹ. Are the components of X independent random variables?
The random vector X has PDFFind the marginal PDFs fX1(x1), fX2(x2), and fX3(x3).
Discrete random vector X has PMF Px(x). Prove that for an invertible matrix A, Y = AX + b has PMF PY(y) = PX(A-1(y - b)).
In an automatic geo-location system, a dispatcher sends a message to six trucks in a fleet asking their locations. The waiting times for responses from the six trucks are iid exponential random variables, each with expected value 2 seconds. (a) What is the probability that all six responses will
Random variables X1 and X2 have zero expected value and variances Var[X1] = 4 and Var[X2] = 9. Their co-variance is Cov[X1, X2] = 3. (a) Find the covariance matrix of X = [X1 X2]ʹ. (b) Find the covariance matrix of Y = [Y1 Y2]ʹ given by Y1 = X1 - 2X2, Y2 = 3X1 + 4X2.
The two-dimensional random vector Y has PDFFind the expected value vector E[Y], the correlation matrix RY, and the covariance matrix CY.
The two-dimensional random vector X and the three-dimensional random vector Y are independent and E[Y] = 0. What is the vector cross-correlation RXY?
The random vector Y = [Y1 Y2]ʹ has covariance matrix where γ is a constant. In terms of γ, what is the correlation coefficient ÏY1, Y2 of Y1 and Y2? For what values of γ is CY a valid covariance matrix?
In the message transmission system in Problem 8.1.5,For p = 0.8, find the expected value vector E[K], the covariance matrix CK, and the correlation matrix RK.
As in Quiz 5.10 and Example 5.23, the 4-dimensional random vector Y has PDFFind the expected value vector E[Y], the correlation matrix RY, and the covariance matrix CY.
X is the 3-dimensional Gaussian random vector with expected value μX = [4 8 6] and covarianceCalculate (a) The correlation matrix, RX, (b) The PDF of the first two components of X, fX1,X2(x1, x2), (c) The probability that X1 > 8.
The 2 × 2 matrixis called a rotation matrix because y = QX is the rotation of x by the angle θ. Suppose X = [X1 X2] is a Gaussian (0, CX) vector where CX = diag[σ21, σ22] and Let Y = QX. (a) Find the covariance of Y1 and Y2. Show that Y1 and Y2 are
An n-dimensional Gaussian vector W has a block diagonal covariance matrixwhere CX is m × m, CY is (n €“ m) × (n €“ m). Show that W can be written in terms of component vectors X and Y in the formsuch that X and Y are independent Gaussian random vectors.
Given the Gaussian random vector X in Problem 8.5.1, Y = AX + b, whereand b = [- 4 - 4]ʹ. Calculate (a) The expected value μY, (b) The covariance CY, (c) The correlation RY. (d) The probability that - 1
Random variables X1 and X2 have zero expected value. The random vector X = [X1 X2] has a covariance matrix of the form(a) For what values of α and β is C a valid covariance matrix? (b) For what values of α and β can X be a Gaussian random
The Gaussian random vector X = [X1X2] has expected value E[X] = 0 and covariance matrix(a) Find the PDF of W = X1 + 2X2.(b) Find the PDF fY(y) of Y = AX where
(a) What conditions must a, b, c, and d satisfy?(b) Under what conditions (in addition to those in part (a)) are X1 and X2 independent?(c) Under what conditions (in addition to those in part (a)) are X1 and X2 identical?
Consider the vector X in Problem 8.5.1 and define Y = (X1 + X2 + X3)/3. What is the probability that Y > 4?
For the vector of daily temperatures [T1 ... T31]ʹ and average temperature Y modeled in Quiz 8.6, we wish to estimate the probability of the eventTo form an estimate of A, generate 10,000 independent samples of the vector T and calculate the relative frequency of A in those trials.
Write a MATLAB program that simulates m runs of the weekly lottery of Problem 7.5.9. For m = 1000 sample runs, form a histogram for the jackpot J.
X1 and X2 are iid random variables with variance Var[X]. (a) What is E[X1 - X2]? (b) What is Var [X1 - X2]?
A radio program gives concert tickets to the fourth caller with the right answer to a question. Of the people who call, 25% know the answer. Phone calls are independent of one another. The random variable Nr indicates the number of phone calls taken when the rth correct answer arrives.' (If the
Random variables X and Y have joint PDFWhat is the variance of W = X + V?
For a constant a > 0, a Laplace random variable X has PDFCalculate the MGF Ï• x (s).
X is the continuous uniform (a,b) random variable. Find the MGF ϕX (s). Use the MGF to calculate the first and second moments of X.
Random variable K has discrete uniform (1,n) PMF. Use the MGF ϕK (s) to find E[K] and E [K2]. Use the first and second moments of K to derive well-known expressions for ∑nk=1 k and ∑nk=1 k2.
N is the binomial (100,0.4) random variable. M is the binomial (50,0.4) random variable. M and N are independent. What is the PMF of L = M + N?
Let K1, K2 , . . . denote a sequence of iid Bernoulli (p) random variables. Let M = K1 + ∙ ∙ ∙ Kn. (a) Find the MGF ϕK (s). (b) Find the MGF ϕM (s). (c) Use the MGF ϕM (s) to find E[M] and Var[M].
At tiem t = 0, you begin counting the arrivals of buses at a depot. The number of buses Ki that arrive between time i - 1 minuts and time i minutes has the poisson PMFK1,K2, . . . are an iid random sequence. Let Ri = K1 + K2 + ˆ™ ˆ™ ˆ™ + Ki denote the number of buses
K,K1,K2, . . . are iid random variables. Use the MGF of M = K1 + ∙ ∙ ∙ + Kn to prove that (a) E [M] = nE[K] (b) E[M2] = n(n-1)(E[K])2 + n E[K2].
X1, X2, . . . is a sequence of iid random variables each with exponential PDF(a) Find Ï•X (s).(b) Let K be a geometric random variable with PMFFind the MGF and PDF of V = X1 + ˆ™ ˆ™ ˆ™ + XK.
Suppose we flip a fair coin repeatedly. Let Xi equal 1 if flip i was heads (H) and 0 otherwise. Let N denote the number of flips needed until H has occurred 100 times. Is N independent of the random sequence X1, X2, . . .? Define Y= X1 + ∙ ∙ ∙ + XN. Is Y an ordinary random sum of random
This problem continues the lottery of Problem 3.7.10 in which each ticket has 6 randomly marked numbers out of 1, . . ., 46.A ticket is a winner if the six marked numbers match 6 numbers drawn at random at the end of a week. Suppose that following a week in which the pot carried over waws r
Let X1 , . . ., Xn denote a sequence of iid Bernoulli (p) random variables and let K = X1 + ∙ ∙ ∙ + Xn. In addition, let M denote a binomial (n,) random variable, independent of X1, ∙ ∙ ∙ + Xn. Do the random variables U = X1 + ∙ ∙ ∙ + XK and V = X1 + ∙ ∙ ∙ + XM have the
The waiting time in milliseconds, W, for accessing one record from a computer database is the continuous uniform (0,10) random variable. The read time R (for moving the information from the disk to main memory) is 3 milliseconds. The random v ariable X milliseconds is the total access time (
Suppose your grade in a probability course depends on 10 weekly quizzes. Each quiz has ten yes/no questions, each worth 1 point. The scoring has no partial credit. Your performance is a model of consistency: On each one-point question, you get the right answer with probability p, independent of the
The duration of a cellular telephone call is an exponential random variable with expected value 150 seconds. A subscriber has a calling plan that includes 300 minutes per month at a cost of $30.00 plus $0.40 for each minute that the total calling time exceeds 300 minutes. In a certain month, the
In any one-minute interval, the number of requests for a popular Web page is a Poisson random variable with expected value 300 requests. (a) A Web server has a capacity of C requests per minute. If the number of requests in a one-minute interval is greater than C, the server is overloaded. Use the
Internet packets can be classified as video (V) or as generic data (D). Based on a lot of observations taken by the Internet service provider, we have the following probability model: P[V] = 0.8, P[D] = 0.2. Data packets and video packets occur independently of one another. The random variable Kn
An amplifier circuit has power consumption Y that grows nonlinearly with the input signal voltage X. When the input signal is X volts, the instantaneous power consumed by the amplifier is Y = 20 + 15X2 Watts. The input signal X is the continuous uniform (-1,1) random variable. Sampling the input
Wn is the number of ones in 10n independent transmitted bits, each equiprobably 0 or 1. For n = 3,4,..., use the binomialpmf function to calculate P [0.499 ≤ Wn/10n ≤ 0.501]. What is the largest n for which your MATLAB installation can perform the calculation? Can you perform the exact
Recreate the plots of Figure 9.3. On the same plots, superimpose the PDF of Yn, a Gaussian random variable with the same expected value and variance. If Xn denotes the binomial (n,p) random variable, explain why for most integers k, PXn(k) ≈ fY(k).
Use uniforml2.m to estimate the probability of a storm surge greater than 7 feet in Example 10.4 based on: (a) 1000 samples, (b) 10000 samples.
Let X and Y denote independent finite random variables described by the probability and range vectors px, sx and py, sy. Write a MATLAB function [pw,sw]=sumfinitepmf(px,sx,py,sy) such that finite random variable W = X+Y is described by pw and sw.
X1,... ,Xn is an iid sequence of exponential random variables, each with expected value 5. (a) What is Var[M9(X)], the variance of the sample mean based on nine trials? (b) What is p[X1 > 7], the probability that one outcome exceeds 7? (c) Use the central limit theorem to estimate P[M9(X) > 7], the
X is a uniform (0,1) random variable. Y = X2. What is the standard error of the estimate of µY based on 50 independent samples of X?
The weight of a randomly chosen Maine black bear has expected value E[W] = 500 pounds and standard deviation σw = 100 pounds. Use the Chebyshev inequality to upper bound the probability that the weight of a randomly chosen bear is more than 200 pounds from the expected value of the weight.
Elevators arrive randomly at the ground floor of an office building. Because of a large crowd, a person will wait for time W in order to board the third arriving elevator. Let X1 denote the time (in seconds) until the first elevator arrives and let Xi denote the time between the arrival of elevator
In a game with two dice, the event snake eyes refers to both six-sided dice showing one spot. Let R denote the number of dice rolls needed to observe the third occurrence of snake eyes. Find (a) The upper bound to P[R ≥ 250] based on the Markov inequality, (b) The upper bound to P[R ≥ 250]
Use the Chernoff bound to show for a Gaussian (µ,σ) random variable X that P [X ≥ c] ≤ e-(c-μ)2/2σ2
In a subway station, there are exactly enough customers on the platform to fill three trains. The arrival time of the nth train is X1+ ∙ ∙ ∙ + Xn where X1, X2 . . . are iid exponential random variables with E[Xi] = 2 minutes. Let W equal the time required to serve the waiting customers. Find
Let X1,X2,... denote an iid sequence of random variables, each with expected value 75 and standard deviation 15. (a) How many samples n do we need to guarantee that the sample mean Mn(X) is between 74 and 76 with probability 0.99? (b) If each Xi has a Gaussian distribution, how many samples n'
X1, X2, . . . is a sequence of iid Bernoulli (1/2) random variables. Consider the random sequence Yn = X1 + ∙ ∙ ∙ + Xn. (a) What is limn→∞ P[|Y2n - n] ≤ √n/2]? (b) What does the weak law of large numbers say about Y2n?
Showing 72100 - 72200
of 88243
First
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
Last
Step by Step Answers
Get 50% OFF
Claim Now
.png)
.png)
.png)
.png)
.png){kind=small}
-1.png){kind=small}
-2.png){kind=small}
.png)
.png)
.png)
.png)
.png)
.png)
.png)
.png)
.png)
.png)
.png){kind=small}
.png){kind=small}
.png){kind=small}
.png){kind=small}
![The random vector Y = [Y1 Y2]ʹ has covariance matrix](https://dsd5zvtm8ll6.cloudfront.net/si.question.images/image/images13/971-M-S-P(9944).png){kind=small}
.png)
.png)
.png)
.png){kind=small}
.png){kind=small}
.png){kind=small}
![For the vector of daily temperatures [T1 ... T31]ʹ and](https://dsd5zvtm8ll6.cloudfront.net/si.question.images/image/images13/971-M-S-P(9955).png){kind=small}
.png)
.png)
.png)
.png)
.png)
