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mathematics
statistics

Questions and Answers of Statistics

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
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
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
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.
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
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
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
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]
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
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
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)
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
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
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
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
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
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: •
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
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]
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
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
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
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^
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
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
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
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
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
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
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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