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Business / Statistics for Business and Economics 12 / Chapter 4 / Problem 80E

Statistics for Business and Economics | 12th Edition | ISBN: 9780321826237 | Authors: James T. McClave, P. George Benson, Terry T Sincich

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Noise in laser imaging. Penumbrol imaging is a technique

Chapter 4, Problem 80E

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QUESTION:

Noise in laser imaging. Penumbrol imaging is a technique used by scanning companies for imaging objects (e.g., X-rays and lasers) that emit high-energy photons. In IEICE Transactions on Information & Systems (Apr. 2005), researchers demonstrated that penumbrol images are always degraded by noise, where the number x of noise events occurring in a unit of time follows a Poisson process with mean \(\lambda\). Suppose that \(\lambda =9\) for a particular image.

a. Find and interpret the mean of x.

b. Find the standard deviation of x.

c. The signal-to-noise ratio (SNR) for a penumbrol image is defined as \(SNR=\mu/\sigma\), where \(\mu\) and \(\sigma\) are the mean and standard deviation, respectively, of the noise process. Find the SNR for x.

Questions & Answers

QUESTION:

Noise in laser imaging. Penumbrol imaging is a technique used by scanning companies for imaging objects (e.g., X-rays and lasers) that emit high-energy photons. In IEICE Transactions on Information & Systems (Apr. 2005), researchers demonstrated that penumbrol images are always degraded by noise, where the number x of noise events occurring in a unit of time follows a Poisson process with mean \(\lambda\). Suppose that \(\lambda =9\) for a particular image.

a. Find and interpret the mean of x.

b. Find the standard deviation of x.

c. The signal-to-noise ratio (SNR) for a penumbrol image is defined as \(SNR=\mu/\sigma\), where \(\mu\) and \(\sigma\) are the mean and standard deviation, respectively, of the noise process. Find the SNR for x.

ANSWER:

Answer

Step 1 of 3

(a)

Researchers demonstrated that penumbral images are always degraded by noise.

The number $x$ of noise events occurring in a unit of time follows a Poisson process with mean $1.$

Suppose that $\lambda{}\ =9$ for a particular image.

We are asked to find and interpret the mean of $x.$

A random variable $Y$ is said to have a Poisson probability distribution if and only if

$p(y)\ =\ \frac{{\lambda{}}^{y}\times{}{e}^{-\lambda{}}}{y!},\ \ \ \ \ \ \ \ \ \ \ y\ =\ 0,\ 1,\ 2,......,\ \ \lambda{}>0\$

…………(1)

A random variable $Y$ possessing a Poisson probability distribution with parameter $\lambda{},$ then

$Mean\ \mu{}\ =E(Y)\ =\lambda{}\ \ \ \ \ \ \ \ \ \ \ \ and\ \ \ \ \ Variance\ {\sigma{}}^{2}\ =V(Y)\ =\lambda{}$

Since we have given $\lambda{}\ =9$ for a particular image.

Hence we can write the mean of $x,$

$Mean\ {\mu{}}_{x}\ =E(Y)\ =\lambda{}\ =9$

Hence the mean of $x$ is $9$ in unit of time.

And the meaning of this is that the average of 9 noise events in a unit of time.

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