3a) 3. QR codes: Start by reading the text under Figure 1. We will generally consider X and Y as randomvariables. For the highlighted pixel in the figure, the gray color X = 25 and the true pixel value white,i.e. Y = 0. We assume that QR codes are made so that there are on average as many white as blackpixels, 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 canstill talk about the simultaneous distribution of X and Y. We start by defining the conditional densityof X, given the value of Y:ONOfx|x(x | 0) =fx|x (x | 1) =4100 (1X100.43100 (100) ²DXO3(pixel actually white)(pixel actually black)Figure 1: QR code photographed in poor lighting, making it difficult to distinguish black and whitepixels. The gray color (X) in each pixel is therefore coded on a scale from 0 (white) to 100 (black). Thetrue pixel value (without shadow) is coded Y=0 for white, and Y = 1 for black.P(Y=0 | X = x) =(a) Make a plot of the marginal density fx(x) = Eyfxy(x1y)Py (y) for 0 ≤ x ≤ 100Bayes' formula in this situation takes the form:fxy(x0)Py(0)fx (x)(b) Use Bayes' formula to find the probability of white when x = 25 as in the figure.

3. QR codes: Start by reading the text under Figure 1. We will generally consider X and Y as random variables. For the highlighted pixel in the figure, the gray color X=25 and the true pixel value white, i.e. Y = 0. We assume that QR codes are made so that there are on average as many white as black pixels, which means that pỵ (0)=RY (1) = 1/2. In this situation, X is continuously distributed (0 ≤x≤ 100) and Y is discretely distributed, but we can still talk about the simultaneous distribution of X and Y We start by defining the conditional density

3. QR codes: Start by reading the text under Figure 1. We will generally consider X and Y as random variables. For the highlighted pixel in the figure, the gray color X=25 and the true pixel value white, i.e. Y = 0. We assume that QR codes are made so that there are on average as many white as black pixels, which means that pỵ (0)=RY (1) = 1/2. In this situation, X is continuously distributed (0 ≤x≤ 100) and Y is discretely distributed, but we can still talk about the simultaneous distribution of X and Y We start by defining the conditional density

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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