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Probability and Stochastic Processes A Friendly Introduction for Electrical and Computer Engineers 3rd edition Roy D. Yates, David J. Goodman - Solutions
For independent events A and B, prove that (a) A and Bc are independent. (b) Ac and B are independent. (c) Ac and Bc are independent.
Use a Venn diagram in which event areas are in proportion to their probabilities to illustrate events A, B, and C that are pair-wise independent but not independent.
At a Phone-smart store, each phone sold is twice as likely to be an Apricot as a Banana. Also each phone sale is independent of any other phone sale. If you monitor the sale of two phones, what is the probability that the two phones sold are the same?
In an experiment, A and B are mutually exclusive events with probabilities P[A] = 1/4 and P[B] = 1/8. (a) Find P[A ∩ B], P[A ∪ B], P[A ∩ Bc], and P[A ∪ Bc]. (b) Are A and B independent?
In an experiment, A and B are mutually exclusive events with probabilities P[A ∪ B] = 5/8 and P[A] = 3/8. (a) Find P[B], P[A ∩ Bc], and P[A ∪ Bc]. (b) Are A and B independent?
In an experiment with equiprobable outcomes, the sample space is S = {1,2,3,4} and P[s] = 1/4 for all s ∈ S. Find three events in S that are pair-wise independent but are not independent.
Following Quiz 1.3, use MATLAB, but not the randi function, to generate a vector T of 200 independent test scores such that all scores between 51 and 100 are equally likely.
Suppose you flip a coin twice. On any flip, the coin comes up heads with probability 1/4. Use Hi and Ti to denote the result of flip i. (a) What is the probability, P[H1|H2], that the first flip is heads given that the second flip is heads? (b) What is the probability that the first flip is heads
In Steven Strogatz's New York Times blog opinionator.blogs. nytimes.com/2010/04/25/chances-are/?ref=opinion, the following problem was posed to highlight the confusing character of conditional probabilities. Before going on vacation for a week, you ask your spacey friend to water your ailing plant.
At the end of regulation time, a basketball team is trailing by one point and a player goes to the line for two free throws. If the player makes exactly one free throw, the game goes into overtime. The probability that the first free throw is good is 1/2. However, if the first attempt is good, the
Suppose that for the general population, 1 in 5000 people carries the human immunodeficiency virus (HIV). A test for the presence of HIV yields either a positive (+) or negative (-) response. Suppose the test gives the correct answer 99% of the time. What is P[-|H], the conditional probability that
You have two biased coins. Coin A comes up heads with probability 1/4. Coin B comes up heads with probability 3/4. However, you are not sure which is which so you flip each coin once, choosing the first coin randomly. Use Hi and Ti to denote the result of flip i. Let A1 be the event that coin A was
Suppose Dagwood (Blondie's husband) wants to eat a sandwich but needs to go on a diet. Dagwood decides to let the flip of a coin determine whether he eats. Using an unbiased coin, Dagwood will postpone the diet (and go directly to the refrigerator) if either (a) He flips heads on his first flip
On each turn of the knob, a gum-ball machine is equally likely to dispense a red, yellow, green or blue gumball, independent from turn to turn. After eight turns, what is the probability P[R2Y2G2B2] that you have received 2 red, 2 yellow, 2 green and 2 blue gumballs?
At a casino, the only game is numberless roulette. On a spin of the wheel, the ball lands in a space with color red (r), green (g), or black (b). The wheel has 19 red spaces, 19 green spaces and 2 black spaces. (a) In 40 spins of the wheel, find the probability of the event A = {19 reds, 19 greens,
An instant lottery ticket consists of a collection of boxes covered with gray wax. For a subset of the boxes, the gray wax hides a special mark. If a player scratches off the correct number of the marked boxes (and no boxes without the mark), then that ticket is a winner. Design an instant lottery
Your Starburst candy has 12 pieces, three pieces of each of four flavors: berry, lemon, orange, and cherry, arranged in a random order in the pack. You draw the first three pieces from the pack. (a) What is the probability they are all the same flavor? (b) What is the probability they are all
In a game of rummy, you are dealt a seven-card hand.(a) What is the probability P[R7] that your hand has only red cards?(b) What is the probability P[F] that your hand has only face cards?(c) What is the probability P[R7F] that your hand has only red face cards? (The face cards are jack, queen, and
Consider a binary code with 5 bits (0 or 1) in each code word. An example of a code word is 01010. How many different code words are there? How many code words have exactly three 0's?
On an American League baseball team with 15 field players and 10 pitchers, the manager selects a starting lineup with 8 field players, 1 pitcher, and 1 designated hitter. The lineup specifies the players for these positions and the positions in a batting order for the 8 field players and designated
Consider a binary code with 5 bits (0 or 1) in each code word. An example of a code word is 01010. In each code word, a bit is a zero with probability 0.8, independent of any other bit. (a) What is the probability of the code word 00111? (b) What is the probability that a code word contains exactly
Suppose each day that you drive to work a traffic light that you encounter is either green with probability 7/16, red with probability 7/16, or yellow with probability 1/8, independent of the status of the light on any other day. If over the course of five days, G, Y, and R denote the number of
A collection of field goal kickers are divided into groups 1 and 2. Group i has 3i kickers. On any kick, a kicker from group i will kick a field goal with probability + independent of the outcome of any other kicks. (a) A kicker is selected at random from among all the kickers and attempts one
A particular operation has six components. Each component has a failure probability q, independent of any other component. A successful operation requires both of the following conditions: • Components 1, 2, and 3 all work, or component 4 works. • Component 5 or component 6 works. Draw a block
Suppose a 10-digit phone number is transmitted by a cellular phone using four binary symbols for each digit, using the model of binary symbol errors and deletions given in Problem 2.4.2. Let C denote the number of bits sent correctly, D the number of deletions, and E the number of errors. Find P[C
Build a MATLAB simulation of 50 trials of the experiment of Example 2.3. Your output should be a pair of 50 × 1 vectors C and H. For the ith trial, Hi will record whether it was heads (Hi = 1) or tails (Hi = 0), and Ci ∈ {1,2} will record which coin was picked.
For a failure probability q = 0.2, simulate 100 trials of the six-component test of Problem 2.4.1. How many devices were found to work? Perform 10 repetitions of the 100 trials. What do you learn from 10 repetitions of 100 trials compared to a simulated experiment with 100 trials?
In this problem, we use a MATLAB simulation to "solve" Problem 2.4.4. Recall that a particular operation has six components. Each component has a failure probability q independent of any other component. The operation is successful if both • Components 1, 2, and 3 all work, or component 4
Random variable X and Y have the joint CDF(a) what is P[X (b) what is the marginal CDF, Fx(x)? (c) what is the marginal CDF, FY(y)?
For continuous random variables X, Y with joint CDF Fx,y{x,y) and marginal CDFs Fx(x) and Fy(y), find P[x1
Every laptop returned to a repair center is classified according its needed repairs: (1) LCD screen, (2) Motherboard, (3) Keyboard, or (4) Other. A random broken laptop needs a type i repair with probability pi = 24-i/15. Let N equal the number of type i broken laptops returned on a day in which
Given the set {U1,.........,Un} of iid uniform (0;T) random variables, we definceXk = smallk(U1,...........,Un)As the kth "smallest" element of the set. That is , X1 is the minimum element, x2 is the smallest, and so on, up to Xn, which is the maximum element of {U1,......,Un}. That X1,....Xn are
The random variable X1,…………Xn have the joint PDFFind(a) The joint CDF, Fx1…….xn(x1……………..,xn)(b) P[min] (X1,X2,X3)
In a compressed data file of 10,000 bytes, each byte is equally likely to be any one of 256 possible characters b0,.........., b255 independent of any other byte. In Ni is the number of times bi appear in the file, find the joint PMF of N0,...................,N255. Also what is the joint PMF of N0
X1,X2,X3 are iid exponential (λ) random variable Find:(a) The PDF of V = min (X1,X2,X3) (b) The PDF of W = max (X1,X2,X3)
Random variable X1,X2....Xn are iid; each Xj has CDF Fx(x) and PDF fx(x) ConsiderLn = min(X1,.......Xn)Un = max (X1,............,Xn)In terms of Fx(x) and /or fx(x):(a) Find the CDF FUn(u).(b) Find the CDF FLn(l)(c) Find the joint CDF FLn,Un(l,u)
For random variable X and Y in Example 5.26, use MATLAB to generate a list of the formThat includes all possible pairs (x,y).
You generate random variable W = w by typing w = sum(4*randn(1,2)) in a MATLAB Command Window. What is Var[W]?
Problem 5.2.6 extended Example 5.3 to a test of n circuits and identified the joint PDF of X, the number of acceptable circuits, and Y, the number of successful tests before the first reject. Write a MATLAB function[SX,SY,PXY] = circuits(n,p)That generates the sample space grid for the n circuit
Random variable X and Y hav the joint PMFa. What is the value of the constant c?b. What is P[y < X]?c. What is P[y > X]?d. What is P[y = X]?e. What is P[y = 3]?
Test two integrated circuits. In each test, the probability of rejecting the circuit is p, independent of the other test. Let X be the number of rejects (either 0 or 1) in the first test and let Y be the number of rejects in the second test. Find the joint PMF PX,Y[x,y).
In Figure 5.2, the axes of the figures are labeled X and Y because the figures depict possible values of the random variables X and y. However, the figure at the end of Example 5.3 depicts Px,y(x,y) on axes labeled with lowercase x and y. Should those axes be labeled with the uppercase X and y?
With two minutes left in a five-minute overtime, the score is 0-0 in a Rutgers soccer match versus Villanova. (That the overtime is NOT sudden-death.) In the next-to-last minute of the game, either (1) Rutgers scores a goal with probability p = 0.2, (2) Villanova scores with probability p = 0.2, or
Each test of an integrated circuit probability p, independent of the outcome of the test of any other circuit. In testing n circuit. In testing n circuits, let K denote the number of circuits rejected and let X denote the number of acceptable circuits.
Given the random variables X and in Problem 5.2.1, finda. The marginal PMFs Px(x) and Py(y),b. The expected values E[X] and E[Y],c. The standard deviations σx and σY.
For n = 0,1,... and 0 PN,K{n,k)Otherwise, PN,K(n,k) = 0, Find the marginal PMFs PN(n) and PK(k).
Random variable N and K have the joint PMFFind the marginal PMFs PN(n) and PK(k).
Random variables X and Y have the joint PDF(a) What is the value of the constant c?(b) What is P[X
Random variable X and Y have joint PDF(a) Find P[X > Y] and P[X + Y (b) Find P[min(X,Y) > 1] (c) Find P[min(X,Y) > 1]
Random variable X and Y have the joint PDFSketch the region of nonzero probability and answer the following question. (a) What is P[X > 0]? (b) What is fx(x)? (c) what is E[X]?
X and y are random variable with the joint PDF(a) What is the marginal PDF fx(x)? (b) What is the marginal PDF fy(y)?
X and Y are random variable with the joint PDF(a) What is the marginal PDF fx(x)? (b) what is the marginal PDF fy(y)?
For a random variable X, let Y = aX + b, Show that if a < 0 than p x,y =1. Also show that if a < 0, then px,y = -1,
Random variable X and Y have joint PDF(a) Find the marginal PDFs fx(x) and fy(y).(b) What are E[x] and Var [X]?(c) What are E[Y] and Var [Y]?
An ice cream company needs to order ingradients from its supplier. Depending on the size of the order, the weight of the shipment can be either1 kg for a small order,2 kg for a big order.The company has three different suppliers. The vanilla supplier is 20 miles away. The chocolate supplier is 100
Observe 100 independent flips of a fair coin. Let X equal the number of heads in the first 75 flips. Let Y equal the number of heads in the remaining 25 flips. Find Px(x) and PY(y). Are X and Y independent? Find Px,y(x,y).
X is the continuous uniform (0,2) random variable. Y has the continuous uniform (0,5) PDF, independent of x. What is the joint PDF Fx,y(x,y)?
X1 and X2 are independent random variables such that X1 has PDFWhat is P[X2 < X1]?
In terms of a positive constant k, random variables X and Y have joint PDF(a) What is k? (b) What is the marginal PDF of X? (c) What is the marginal PDF of Y? (d) Are X and Y independent?
Prove that random variable X and Y are independent if and only if Fx,y(x,y) = Fx(x) FY(y)
Continuing Problem 5.6.1, the price per kilogram for shipping the order is one cent per mile. C cents is the shipping cost of one order. What is E[C]?
X and Y are random variables with E[X] = E[Y] = 0 suce that X has standard deviation σx = 2 while Y has standard deviation σY = 4. (a) For V = X - Y, What are the smallest and largest possible values of Var[V]? (b) For w = X - 2Y, what are the smallest and largest possible values of Var[W]?
Random variable X and Y have joint PDFAnswer the following questions (a) What are E[X] and Var [X]? (b) What are E[Y] and Var [Y]? (c) What is Cov[X + Y]? (d) What is E[X + Y]? (e) What is Var[X + Y]?
A transmitter sends a signal X and a receiver makes the observation Y = X + Z, whare Z is a receiver noise that is independent of X and E[X] = E[Z] = 0.Since the average power of the signal is E[X2] and the average power of the noise is E[Z2], a quality measure for the received signal is the signal
A random ECE sophomore has height X (rounded to the nearest foot) and GPA Y(rounded to the nearest integar).These random variable have joint PMFFind E[X + Y] and Var [X + Y]
X and Y are random variable with E[X] = E[Y] = 0 and Var [X] = 1,Var[Y] = 4 and correlation coefficient p = 1/2. Find Var [X + Y].
Observe independent flips of a fair con until heads occurs twice. Let X1 equal the number of flips up to and including the first H. Let X2 equal the number of additional flips up to and including the section H. Let Y = X1 - X2. Find E[Y] and Var [Y]. Don't try to find PY(y).
X and Y are identically distributed random variables with E[X] = E[Y] = 0 and convariance Cov [X,Y] = 3 and correlation Px,y =1/2. For nonzero constants a and b, U = aX and V = bY.(a) Find Cov[U,v].(b) Find the correlation coefficient pu,v,(c) Let W = U + V. For what values of a and b are x and W
X and Z are independent random variables with E[X] = E[Z] = 0 and variance Var [X] = 1 and Var [Z] = 16. Let Y = X + Z. Find the correlation coefficient p of x and Y, Are X and Y independent?
For the random variable X and Y in Problem 5.2.2 find(a) The expected value of W = 2xy(b) The Correlation, rx,y = E[XY],(c) The covariance, Cov[X,Y],(d) The correlation coefficient, P X,Y(e) the variance of X + Y, Var[X + y] .(Refer to the results of Problem 5.3.2 to answer some of these questions.)
X and Y are idependent random variables with PDFs(a) Find the correlation rX,Y.(b) Find the covariance Cov[X,Y].
Random variable X and Y have joint PDFFind rx,y and E[ex+y]
This problem outlines a proof of Theorem 5.13.(a) Show that(b) Use Part (a) to show that (c) Show that Var[X^] = a2 Var [x] and Var [Y^] = c2 Var[Y]. (d) Combine parts (b) and (c) to relate PX^,Y^ and P x,y.
Random variable X and Y have joint PDF FX,Y(x,y) = ce -(x2/8) - (y2/18). What is the constant c? Are X and Y independent?
Show that the joint Gaussian PDF fx,y(x,y) given by Definition 5.10 satisfiesUse Equation (5.68) and the result of problem 4.6.13.
TRUE OR FALSE: X1 and X2 are Gaussian random variable. For any constant y, there exist a constant a such that P[X1 + aX2 < y] =1/2.
Random variable X and Y have joint PDf Fx,y(x,y) = ce-(2x2 - 4xy + 4y2) (a) What are E[XJ and ELY]? (b) Find the correlation coefficient PX.Y. (c) what are Var[X] and Var[Y]? (d) What is the constant c? (e) Are X and Y independent?
A person's white blood cell (WBC) count W (measured in thousands of cells per microliter of blood) and body temperature T (in degrees Celsius) can be mode led as bivariate Gaussian random variable such that IV is Gaussian (7, 2) and T is Gaussian (37, 1). To determine whether a person is sick,
Your course grade depends on two test scores: X1 and X2. Your score X on test i is Gaussian (μ = 74, σ = 16) random variable, independent of any other test score. (a) With equal weighting, grades are determined by Y = X1/2 + X2/2. You earn an A if Y > 90. What is P[A] = P(Y > 90]? (b) A student
Random variable X and Y have joint PMFFind the PMF of W = X - Y
N is a binomial (n = 100, p = 0.4) random variable. M is a binomial (n = 50,p = 0.4) random variable. Given that M and N are independent, what is the PMF of L = M + N?
Let X and Y be discrete random variable with joint PMFWhat is the PMF of W = min (X,Y)?
The voltage X across a 1 Ω resistor is a uniform random variable with parameters 0 and 1. The instantaneous power is y = X2. Find the CDF Fy(y) and the PDF FY(y) of Y.
For the uniform (0. 1) random variable U, find the CDF and PDF of Y = a + (b - a)U with (I
X is a continuous random variable. Y = aX + b, where a,b ‰ 0, Prove thatConsider the cases a > 0 and a > 0 separately.
In a 50 km Tour de France time trial, a rider's time T, measured in minutes, is the continuous uniform (60,75) random variable. Let V = 3000/T denote the rider's speed over the course in km/hr. Find the PDF of V.
If X has an exponential (λ) PDF what is the PDF of W = X2?
U is the uniform (0,1) random variable and X = -In(1 - U).(a) What is Fx(x)?(b) what is Fx(x)?(c) what is E[X]?
X is the uniform (0,1) random variable. Find a function g(x) such that the PDF of Y = g(X) is
X has CDFY = g(X) where(a) What is Fy(y)?(b) What is fy(y)?(c) What is E[Y]?)
A defective voltmeter measures small voltages as zero . In particular, when the input voltage is V, the measured volt-age isIf V is the continuous uniform(-5,5) random variable, what is the PDF of W?
In this problem prove a generalization of Theorem 6.5. Given a random variable X with CDF Fx(x), defineF-(u) = min {x}Fx(x) > u }.This problem proves that for a continuo us uniform (0, 1) random variable U, X =F-(U) has CDF Fx-(x) Fx(x).(a) Show that when F(x) is a continuous, strictly
The voltage V at the output of a microphone is the continuous uniform (-1, 1) random variable. The microphone voltage is processed by a clipping rectifier with output(a) What is P[L = 0.5]? (b) What is FL(1)? (c) What is E[LI]?
X is random variable with CDF by FX(x) Let Y = g (X) whereExpress FY(Y) in terms of Fx(x)
The input voltage to a rectifier is the continuous uniform(0,1) random variable U. The rectifier output is a random variable W defined byFind the CDF Fw(w) and the expected value E[W].
Given an input voltage V, the output voltage of a half-wave rectifier is givenSuppose the input V is the continuous uniform (-15,15) random variable. Find the PDF of W
Random variable X and Y have joint PDFLet V = Max (X,Y). Find the CDF and PDF of V.
X is the Gaussain (0,1) random variable and Z, independent of X, has PMFFind the PDF of Y = ZX.
For a constant a > 0, random variables X and Y have joint PDFFind the CDF and PDF of random variable Is it possible to observe W
Consider random variables X,Y, and W from Problem 6.4.14?(a) Are W and X independent?(b) Are W and Y independent?
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