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

Result Not Found

Books FREE
Study Help
Tutors
AI Study Planner NEW
Sell Books

Search
Sign In
Register

study help
business
introduction to probability statistics

Introduction To Probability Statistics And Random Processes 1st Edition Hossein Pishro-Nik - Solutions

Let X and Y be two independent N(0, 1) random variable andFind ρ(Z,W). Z = 7+X+Y, W = 1+ Y.
Let X ∼ Uniform(1, 3) and Y |X ∼ Exponential(X). Find Cov(X, Y).
Let X and Y be two independent N(0, 1) random variables andFind Cov(Z,W). Z = 1+X+XY2, W = 1 + X.
Let X and Y be two random variables. Suppose that σ2X = 4, and σ2Y = 9. If we know that the two random variables Z = 2X −Y and W = X + Y are independent, find Cov(X, Y) and ρ(X, Y).
Let X1, X2,⋯,Xn be i.i.d. random variables, where Xi ∼ Bernoulli(p). Define1. E[Y],2. Var(Y). If Y = Y₁ + Y₂ + + Yn. find Y₁ = X₁X₂, Y2 = X2 X3, E Yn-1 = Xn-1Xn, Yn = XnX₁.
Let X and Y be jointly normal random variables with parameters μX = 2, σ2X = 4, μY = 1, σ2Y = 9, and ρ = −1/2.a. Find E[Y |X = 3].b. Find V ar(Y |X = 2).c. Find P(X +2Y ≤ 5|X +Y = 3).
Let X and Y be jointly normal random variables with parameters μX, σ2X, μY, σ2Y, and ρ. Find the conditional distribution of Y given X = x.
Let X ∼ Exponential(λ). Find the MGF of X, MX(s), and all of its moments, E[Xk].
In this problem, our goal is to find the variance of the hypergeometric distribution. Let's remember the random experiment behind the hypergeometric distribution. You have a bag that contains b blue marbles and r red marbles. You choose k ≤ b + r marbles at random (without replacement) and let X
Let X ∼ Poisson(λ). Find the MGF of X, MX(s).
If X ∼ Geometric(p), find the MGF of X.
For a random variable X, we know thatFind the distribution of X. Mx(s) 2 2-s for s € (-2,2).
Iffind EX and Var(X). 1 Mx(s) = =+ =/e³ + 1e²s
Using MGF s show that if X ∼ N(μX, σ2X) and Y ∼ N(μY, σ2Y) are independent, then X + Y ~ N] μχ + μY, on tôi
Using MGFs prove that if X ∼ Binomial(m, p) and Y ∼ Binomial(n, p) are independent, then X + Y ∼ Binomial(m + n, p).
Let X be a continuous random variable with the following PDFFind the MGF of X, MX(s). fx(x) = +) == 12e-²1|²2|1 e-A/²).
If X ∼ Exponential (λ), show that ox(w) = T X -jw
Let X, Y, and Z be three independent N(1, 1) random variables. Find E[XY |Y +Z = 1].
For each of the following random variables, find the MGF.a. X is a discrete random variable, with PMFb. Y is a Uniform(0, 1) random variable. Px(k) = 3 k = 1 k = 2
Suppose that X, Y, and Z are three independent random variables. If X, Y ∼ N(0, 1) and Z ∼ Exponential(1), find1. E[XY |Z = 1],2. E[X2Y2Z2|Z = 1].
Let X, Y and Z be three jointly continuous random variables with joint PDF1. Find the joint PDF of X and Y.2. Find the marginal PDF of X.3. Find the conditional PDF of fXY |Z(x, y|z) using4. Are X and Y independent of Z? fxyz(x, y, z) = x+y 0 0≤x, y, z≤1 otherwise
Let X, Y and Z be three jointly continuous random variables with joint PDF1. Find the constant c.2. Find the marginal PDF of X. fxyz(x, y, z) = c(x + 2y + 3z) 0 0≤x, y, z ≤ 1 otherwise
Let X and Y be jointly normal random variables with parametersa. Find P(2X + Y ≤ 3).b. Find Cov(X + Y, 2X −Y).c. Find P(Y > 1|X = 2). x = 1, o=1, y = 0, o = 4, and p py =
Remember that a continuous random variable X is said to have a Gamma distribution with parameters α > 0 and λ > 0, shown as X ∼ Gamma(α, λ), if its PDF is given byIf X ∼ Gamma(α, λ), find the MGF of X.Remember that fx(x) = 1a-1e-xx I(a) 0 x > 0 otherwise
Using the M GFs show that if Y = X1 +X2 + ⋯ + Xn, where the Xi's are independent Exponential(λ) random variables, then Y ∼ Gamma(n,λ).
For a random v ector X, show CX = RX −EXEXT.
A sensor net work consists of n sensors that are distributed randomly on the unit square. Each node's location is uniform over the unit square and is independent of the locations of the other node. A node is isolated if there are no nodes that are within distance r of that node, where 0 1. Show
Let Bn be the event that a graph randomly generated according to G(n, p) model has at least one isolated node. Show thatAnd conclude that for any P(Bn) ≤n(1-p)n-1.
A system consists of 4 components in a series, so the system works properly if all of the components are functional. In other words, the system fails if and only if at least one of its components fails. Suppose the probability that the component i fails is less than or equal to pf = 1/100, for i
Letbe a normal random vector with the following mean and covarianceFind the MGF of X defined as X= X1 X₂ X3.
Let X and Y be two jointly normal random variables with X ∼ N(μX,σX), Y ∼ N(μY, σY), and ρ(X, Y) = ρ. Show that the above PDF formula for PDF of [X Y] is the same as fX,Y (x, y) given in Definition 5.4 in Section 5.3.2. That is, fxr(x,y) 1 2V1 - 1 - My pra-p|(***)* +(*=#r)* 2(1 p) exp
Letbe a normal random vector with the following mean and covariance matricesLet also1. Find P(X2 > 0).2. Find expected value vector of Y, mY = EY.3. Find the covariance matrix of Y, CY.4. Find P(Y2 ≤ 2). X = X1 [X₂] X2
Let X be an n-dimensional random vector. Let A be a fixed (non-random) invertible n by n matrix, and b be a fixed n-dimensional vector. Define the random vector Y as Y = AX +b.Find the PDF of Y in terms of PDF of X.
Let X ∼ Uniform (0, 1). Suppose that given X = x, Y and Z are independent and Y |X = x ∼ Uniform(0,x) and Z|X = x ∼ Uniform(0, 2x). Define the random vector U as1. Find the PDFs of Y and Z.2. Find the PDF of U, fU(u), by using U = X Y Z
Let X be an n-dimensional random vector and the random vector Y be defined aswhere A is a fixed m by n matrix and b is a fixed m-dimensional vector. Show that Y = AX +b,
Let X ∼ Geometric (p). Using Markov's inequality find an upper bound for P(X ≥ a), for a positive integer a. Compare the upper bound with the real value of P(X ≥ a).
A bank teller serves customers standing in the queue one by one. Suppose that the service time Xi for customer i has mean EXi = 2 (minutes) and Var(Xi) = 1. We assume that service times for different bank customers are independent. Let Y be the total time the bank teller spends serving 50
The number of accidents in a certain city is modeled by a Poisson random variable with an average rate of 10 accidents per day. Suppose that the number of accidents on different days are independent. Use the central limit theorem to find the probability that there will be more than 3800 accidents
In a communication system each data packet consists of 1000 bits. Due to the noise, each bit may be received in error with probability 0.1. It is assumed bit errors occur independently. Find the probability that there are more than 120 errors in a certain data packet.
Consider the following random experiment: A fair coin is tossed repeatedly forever. Here, the sample space S consists of all possible sequences of heads and tails. We define the sequence of random variables X1, X2, X3, ⋯ as follows: Xn = 1 if the nth coin toss results in a heads if the nth coin
In a communication system, each codeword consists of 1000 bits. Due to the noise, each bit may be received in error with probability 0.1. It is assumed bit errors occur independently. Since error correcting codes are used in this system, each codeword can be decoded reliably if there are less than
If X1, X2, X3, ⋯ is a sequence of i.i.d. random variables with CDF FX(x), then Xn d→ X. This is because Fx₂(x) = Fx(x), for all x.
The amount of time needed for a certain machine to process a job is a random variable with mean EXi = 10 minutes and Var(Xi) = 2 minutes2. The times needed for different jobs are independent from each other. Find the probability that the machine processes less than or equal to 40 jobs in 7 hours.
You have a fair coin. You toss the coin n times. Let X be the portion of times that you observe heads. How large n has to be so that you are 95% sure that 0.45 ≤ X ≤ 0.55? In other words, how large n has to be so that P(0.45 ≤X ≤0.55) ≥.95 ?
Let X2, X3, X4, ⋯ be a sequence of random variable such thatShow that Xn converges in distribution to Exponential(1). Fxn (x) = nx (1-(1-4) ⁰² 0 x > 0 otherwise
An engineer is measuring a quantity q. It is assumed that there is a random error in each measurement, so the engineer will take n measurements and reports the average of the measurements as the estimated value of q. Specifically, if Yi is the value that is obtained in the i'th measurement, we
Let X1, X2, X3, ⋯ be a sequence of random variable such thatwhere λ > 0 is a constant. Show that Xn converges in distribution to Poisson(λ). (MA). n Xn Binomialn, for n E N, n>d,
Let X2, X3, X4, ⋯ be a sequence of random variables such thatShow that Xn converges in distribution to X = 1. Fx₁, (x) = en(x-1) 1+en (2-1) 0 x>0 otherwise
Let X2, X3, X4, ⋯ be a sequence of random variables such thatShow that Xn converges in distribution to Uniform(0, 1). Fxn(x) = = enx + xen n+. enz + (¹+¹ ) en n enx ten - (+¹²) en enx + 0≤x≤ 1 x > 1
Let Xn ∼ Exponential(n), show that Xn ρ → 0. That is, the sequence X1, X2, X3, ⋯ converges in probability to the zero random variable X.
Let X be a random variable, and Xn = X +Yn, wherewhere σ > 0 is a constant. Show that Xn ρ→ X. EYn n 9 Var (Y) = n 9
Consider a sequence {Xn,n = 1, 2, 3,⋯} such thatShow that Xn = n 0 with probability n2 with probability 1 1 n²
Let Xn ∼ Uniform(0, 1/n). Show that Xn Lr→ 0, for any r ≥ 1.
Consider a sequence {Xn,n = 1, 2, 3,⋯} such thatShow thata. Xn p→ 0.b. Xn does not converge in the rth mean for any r ≥ 1. Xn = n² 0 with probability with probability 1 - n
Consider the following random experiment: A fair coin is tossed once. Here, the sample space has only two elements S = {H,T}. We define a sequence of random variables X1, X2, X3, ⋯ on this sample space as follows:a. For each of the possible outcomes (H or T), determine whether the resulting
We perform the following random experiment. We put n ≥ 10 blue balls and n red balls in a bag. We pick 10 balls at random (without replacement) from the bag. Let Xn be the number of blue balls. We perform this experiment for n = 10, 11, 12,⋯. Prove that d X₂ → Binomial (10,7).
Consider the sample space S = [0, 1] with a probability measure that is uniform on this space, i.e.,Define the sequence {Xn, n = 1, 2,⋯} as follows:Also, define the random variable X on this sample space as follows:Show that Xn a.s.→ X. P([a, b])=b-a, for all 0 < a < b < 1.
Find two sequences of random variables {Xn, n = 1, 2,⋯} and {Yn, n = 1, 2,⋯} such thatbut Xn + Yn does not converge in distribution to X +Y. d Xn → X, and d Y→ Y, n
Let X1, X2, X3, ⋯ be a sequence of continuous random variable such thatShow that Xn converges in probability to 0. n fxn (ax) = "e-미피. -ll. ·e 2
Consider a sequence {Xn,n = 1, 2, 3,⋯} such thatShow that Xn a.s.→ 0. Xn n n with probability with probability
Let X1, X2, X3, ⋯ be a sequence of continuous random variable such thatShow that Xn converges in probability to 0. fxn (x) nx² 0 x > otherwise
Let Y1, Y2, Y3, ⋯ be a sequence of i.i.d. random variables with mean EYi = μ and finite variance Var(Yi) = σ2. Define the sequence {Xn,n = 2, 3, . . . } as Xn = Y₁Y2 + Y2Y3+ Yn-1Yn + Y₂Y₁ Р → μ². Show that X₂ → n for n = 2,3,.
Let Y1, Y2, Y3, ⋯ be a sequence of positive i.i.d. random variables with 0 i] = γ n,n = 1, 2, 3, . . . } as Xn = (Y1Y2Y3 · · · Yn-1Yn) = 7, ... Р Show that Xne. for n = 1,2,3,..
Let X1, X2, X3, ⋯ be a sequence of random variable such thatwhere λ > 0 is a constant. Define a new sequence Yn asShow that Yn converges in mean square to λ, i.e., Yn m.s.→ λ. Xn Poisson (nλ), for n = 1,2,3,...,
Let {Xn,n = 1, 2,⋯} and {Yn,n = 1, 2,⋯} be two sequences of random variables, defined on the sample space S. Suppose that we knowProve that Xn +Yn Lr→ X +Y . You may want to use Minkowski's inequality which states that for two random variables X and Y with finite moments, and 1 ≤ p
Let X1, X2, X3, ⋯ be a sequence of random variable such that Xn ∼ Rayleigh(1), a.s. Show that X → 0. - {n²z exp{-} {" 0 fx, (x) = x > 0 otherwise
Let Y1,Y2, ⋯ be independent random variables, where Yn ∼ Bernoulli (n + 1) for n = 1, 2, 3,⋯. We define the sequence {Xn,n = 2, 3, 4,⋯} as Show that X → 0. a.s. → Xn+1=Y₁ Y₂Y3 Yn ... for n = 1,2,3,...
Let X be the weight of a randomly chosen individual from a population of adult men. In order to estimate the mean and variance of X, we observe a random sample X1,X2,⋯ ,X10. Thus, the Xi's are i.i.d. and have the same distribution as X. We obtain the following values (in pounds):165.5, 175.4,
Let X1, X2, X3, X4 be a random sample from the Uniform(0, 1) distribution, and let X(1), X(2), X(3) , X(4). Find the PDFs of X(1), X(2), and X(4).
Let X1, X2, X3, . . ., Xn be a random sample with unknown mean EXi = μ, and unknown variance Var(Xi) = σ2. Suppose that we would like to estimate θ = μ2. We define the estimator ^ Θ asto estimate θ. Is ^ Θ an unbiased estimator of θ? Why? 0= (x) n -[Σ] Xk η k=1 2
Let Xi be i.i.d. Uniform(0, 1). We define the sample mean asa. Find E[Mn] and Var(Mn) as a function of n.b. Using Chebyshev's inequality, find an upper bound onc. Using your bound, show that Mn X₁ + X₂+...+Xn n
Let X be a random variable with EX = 1 and RX = (0, 2). If Y = X3 −6X2, show that EY ≤ −5.
Let X be a positive random variable with EX = 10. What can you say about the following quantities?1. E[X −X3]2. E[Xln√X]3. E[|2 −X|]
Prove for two random variables X and Y with finite moments, and 1 ≤ p Note thatThereforeNow, apply Hölder's inequality. |X+YP = |X+Y|P-¹|X+Y| ≤X+YP ¹ (X+Y) ≤X + YP ¹|X| + |X + YP-¹|Y|.
Let X ∼ Binomial(n, p). Using Markov's inequality, find an upper bound on P(X ≥ αn), where p < α < 1. Evaluate the bound for p = 1/2 and α = 3/4.
in Geometric (p). Using Chebyshev's inequality find an upper bound for P(|X −EX| ≥ b).
Let X ∼ Binomial(n, p). Using Chebyshev's inequality, find an upper bound on P(X ≥ αn), where p < α < 1. Evaluate the bound for p = 1/2 and α = 3/4.
Let X be a random variable with EX = 0 and Var(X) = σ2. We would like to prove that for any a > 0, we haveThis inequality is sometimes called the one-sided Chebyshev inequality. One way to show this is to use P(X ≥ a) = P(X + c ≥ a + c) for any constant c ∈ R. P(X ≥a) ≤ 0² σ² + a²
Let X ∼ Binomial (n, p). Using Chernoff bounds, find an upper bound on P(X ≥ αn), where p < α < 1. Evaluate the bound for p = 1/2 and α = 3/4.
The number of customers visiting a store during a day is a random variable with mean EX = 100 and variance Var(X) = 225.1. Using Chebyshev's inequality, find an upper bound for having more than 120 or less than 80 customers in a day. That is, find an upper bound on2. Using the one-sided Chebyshev
Using the Cauchy-Schwarz inequality, show that for any two random variables X and Y |ρ(X,Y)| ≤ 1.Also, |ρ(X,Y )| = 1 if and only if Y = aX +b for some constants a, b ∈ R.
Let Xi be i.i.d. and Xi ∼ Exponential(λ). Using Chernoff bounds find an upper bound for P(X1 +X2 +⋯+Xn ≥ a), where a > n/λ. Show that the bound goes to zero exponentially fast as a function of n.
Let X be a positive random variable. Compare E[Xa] with (E[X])a for all values of a ∈ R.
Let X and Y be two independent Uniform (0, 1) random variables. Let the random vectors U and V be defined asDetermine whether CU and CV are positive definite. U= [X²Y] · V= [X + x] X Y LX+Y +
Let X and Y be two jointly continuous random variables with joint PDFand let the random vector U be defined as1. Find the mean vector of U, EU.2. Find the correlation matrix of U, RU.3. Find the covariance matrix of U, CU. fx,y (x, y) = = 0 (3x+y) 0≤x, y ≤ 1 otherwise
Let X and Y be two jointly continuous random variables with joint PDFand let the random vector U be defined asFind the correlation and covariance matrices of U. fx,y(x, y) = 3 ت ادا 0 x² + y 0 < x, y < 1 otherwise
Let X be a random variable with characteristic function ϕX(ω). If Y = aX +b, show that ϕY (ω) = ejωbϕX(aω).
Let X1, X2, X3, . . ., Xn be a random sample from a distribution with mean EXi = θ, and variance Var(Xi) = σ2. Consider the following two estimators for θ:Find MSE(^Θ1) and MSE(^Θ2) and show that for n > 1, we have 1. Ôn = X1. X₁. 2. Ô, = X X₁+X₂+...+Xn n
Let X1,…,X4 be a random sample from a Geometric(p) distribution. Suppose we observed (x1,x2,x3,x4) = (2, 3, 3, 5). Find the likelihood function using PXi(xi; p) = p(1 −p)xi−1 as the PMF.
Let X1, X2, X3, . . ., Xn be a random sample with mean EXi = θ, and variance Var(Xi) = σ2. Show that ^Θn = X̅ is a consistent estimator of θ.
Let X1,…,X4 be a random sample from an Exponential (θ) distribution. Suppose we observed (x1,x2,x3,x4) = (2.35, 1.55, 3.25, 2.65). Find the likelihood function usingas the PDF. fx; (x₁; 0) = 0e0x₁, for x ≥ 0
Let X1, X2, X3, . . ., Xn be a random sample with mean EXi = μ, and variance Var(Xi) = σ2. Suppose that we useto estimate σ2. Find the bias of this estimator 1 5--2-xr-|(2x-ux) (X X) = η S η k=1 k=1 nx
Often when working with maximum likelihood functions, out of ease we maximize the log-likelihood rather than the likelihood to find the maximum likelihood estimator. Why is maximizing L(x; θ) as a function of θ equivalent to maximizing log L(x; θ)?
Let T be the time that is needed for a specific task in a factory to be completed. In order to estimate the mean and variance of T, we observe a random sample T1,T2,⋯, T6. Thus, Ti's are i.i.d. and have the same distribution as T. We obtain the following values (in minutes): 18, 21, 17, 16, 24,
Let X be one observation from a N(0,σ2) distribution.a. Find an unbiased estimator of σ2.b. Find the log likelihood, log(L(x;σ2)), usingas the PDF.c. Find the Maximum Likelihood Estimate (MLE) for the standard deviation σ, ^σML. fx(x;o2) = 1 -exp 2по { x2 20²
In this problem, we would like to find the CDFs of the order statistics. Let X1,…,Xn be a random sample from a continuous distribution with CDF FX(x) and PDF fX(x).Define X(1),…,X(n) as the order statistics and show thatFix x ∈ R. Let Y be a random variable that counts the number of Xj's ≤
For the following random samples, find the likelihood function:1. Xi ∼ Binomial(3, θ), and we have observed (x1,x2,x3,x4) = (1, 3, 2, 2).2. Xi ∼ Exponential(θ) and we have observed (x1,x2,x3,x4) = (1.23, 3.32, 1.98, 2.12).
Let X1,…,Xn be a random sample from a Poisson(λ) distribution.a. Find the likelihood equation, L(x1,…,xn;λ), usingas the PMF.b. Find the log likelihood function and use that to obtain the MLE for λ, ^λML. Px, (x₁,...,xn; A) = exi xi!
Suppose that we have observed the random sample X1, X2, X3, . . ., Xn, where Xi ∼ N(θ1, θ2), soFind the maximum likelihood estimators for θ1 and θ2. fx; (xi; 01,02) = 1 2π02 -e (x₁-0₁)² 202
A random sample X1, X2, X3, . . ., X100 is given from a distribution with known variance Var(Xi) = 81. For the observed sample, the sample mean is X̅ = 50.1. Find an approximate 95% confidence interval for θ = EXi.

Showing 7000 - 7100 of 7136

First
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72

Step by Step Answers

Services

Sitemap
Fun
Definitions
Become Tutor
Used Textbooks
Study Help Categories
Recent Questions
Expert Questions
Campus Wear
Sell Your Books

Company Info

Security
Copyrights
Privacy Policy
Terms & Conditions
SolutionInn Fee
Scholarship
Online Quiz
Give Feedback, Get Rewards

Get In Touch

About Us
Contact Us
Career
Jobs
FAQ
Student Discount
Campus Ambassador

Secure Payment

Download Our App

© 2026 SolutionInn. All Rights Reserved