A transmitter using 3-fold repetition coding uses the vector \(\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}\) to represent a 0 bit and \(\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}\) to send a 1 bit. (Suppose a priori that a 0 bit and a 1 bit are equally likely.) Over the communication channel, the transmitted vector is added to a vector consisting of independent Gaussian random variables with mean 0 and variance 2. Suppose the receiver receives \(\begin{bmatrix} 1.21 \\ 0.18 \\ 0.71 \end{bmatrix}\). Which bit is more likely to have been transmitted?

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Answered: A transmitter using 3-fold repetition…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Consider a signal detection problem involving two hypotheses: H₁ : X = W and where H₁ : X = š+ Ŵ, is a known signal vector, and W is a vector consisting of independent Gaussian random variables with mean 0 and variance 1. Suppose a priori that these two hypotheses are equally likely; that is P(H₁) = P(H₁) = 0.5. Suppose we observe X, but based on X we still determine that the two hypotheses are equally likely. What must be true about X?
5. Consider the following model: Yi Bo+ B1Xi+ B2D2i+ B3D3i+ui Where Y = annual earnings of MBA graduates. X = years of service. D2 = 1 if Harvard MBA, = 0 otherwise. D3 = 1 if Wharton MBA,= 0 otherwise. Find a. What are the expected signs of the various coefficients? b. How would you interpret 32 and 33? c. If ß2 > 33, what conclusion would you draw? (30 per cent) d. Assume that a Harvard MBA graduate with 20 years of service/experience, what is his estimated annual income?
Chapter 6, Section 2-D, Exercise 078 Is a t-Distribution Appropriate?A sample with size n=75 has x¯=18.92, and s=10.1. The dotplot for this sample is given below. Indicate whether or not it is appropriate to use the t-distribution. If it is appropriate, give the degrees of freedom for the t-distribution and give the estimated standard error. If it is not appropriate, enter -1 in both of the answer fields below.Enter the exact answer for the degrees of freedom and round your answer for the standard error to two decimal places.d⁢f= standard error =
A QR code photographed in poor lighting, so that it can be difficult to distinguish black and white pixels. The gray color (X) in each pixel is therefore coded on a scale from 0 (white) to 100 (black). The true pixel value (without shadow) the code is Y = 0 for white, and Y = 1 for black. We treat X and Y as random variables. For the highlighted pixel in the figure is the gray color X = 20 and the true pixel value is white, i.e. Y = 0. We assume that QR codes are designed so that, on average, there are as many white as black pixels, which means that pY (0) = pY (1) = 1/2. In this situation, X is continuously distributed (0 ≤ X ≤ 100) and Y is discretely distributed, but we can still think about the simultaneous distribution of X and Y. We start by defining the conditional density of X, given the value of Y : fX|Y(x|0) = "Pixel is really white" fX|Y(x|1) =" Pixel is really balck " Use Bayes formula as given in the picture and find the probability for x = 20 like in the picture.
show that if X11, X12,..., X1, X21, X22,..., X2n₂ are independent random variables, with the first n constitut- ing a random sample from an infinite population with the mean μ₁ and the variance of and the other n2 constitut- ing a random sample from an infinite population with the mean μ2 and the variance o2, then 1 M2 (a) E(X₁-X₂) = μ₁ −μ2; 07 0 22 (b) var(X₁-X₂) = + n₁ n₂
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7) Suppose that X and Y are two independent measurements. We can treat these measurements as random variables, where X has the mean of 1 and standard deviation of 2, and Y has the mean of 2 and the standard deviation of 3. For the transformations below, find the mean, variance, and the standard deviation. a) X+Y b) X-Y c) 2X+Y d) X-2Y+1
= 3. Propose an algorithm for simulating the occurrence of two dependent events A and B that uses only one uniform random variable U~Unif(0, 1). Assume that the probabilities P(A) ΡΑ, P(B) = PB, and P(AUB) = q are given, where PA, PB, q = (0, 1) and max(PA, PB) < q < PA+PB· Formulate your answer in the form of the following statement: "if U E ..., then event A occurs; if U € .……., then event B occurs."
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