(15) A powerful detector measures the frequency of radiation over virtually the entire electromagnetic spectrum, ranging from 1 kHz (extremely low frequency radio waves used to communicate with submarines) through the visible spectrum, and all the way up to frequencies of over 1020 Hz (Gamma rays). Let random variable Z be the frequency measured in kHz. Its PDF is: 2 fz(z) = u(z − 1), where u() is the standard unit step function. a) The wavelength is given by random variable Y = cZ-¹, where constant c denotes the speed of light (in appropriate units). Find its PDF, fy(y). b) Given the extensive electromagnetic spectrum, it is often useful to plot the frequency measurements in log scale. Let X = loge (Z), and find its PDF, fx(x). c) Are X and Y uncorrelated and/or independent? Prove your answer mathematically, and explain why it should have been expected.
(15) A powerful detector measures the frequency of radiation over virtually the entire electromagnetic spectrum, ranging from 1 kHz (extremely low frequency radio waves used to communicate with submarines) through the visible spectrum, and all the way up to frequencies of over 1020 Hz (Gamma rays). Let random variable Z be the frequency measured in kHz. Its PDF is: 2 fz(z) = u(z − 1), where u() is the standard unit step function. a) The wavelength is given by random variable Y = cZ-¹, where constant c denotes the speed of light (in appropriate units). Find its PDF, fy(y). b) Given the extensive electromagnetic spectrum, it is often useful to plot the frequency measurements in log scale. Let X = loge (Z), and find its PDF, fx(x). c) Are X and Y uncorrelated and/or independent? Prove your answer mathematically, and explain why it should have been expected.
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![### Electromagnetic Spectrum Frequency Measurement
**Problem Statement:**
A powerful detector measures the frequency of radiation over virtually the entire electromagnetic spectrum, ranging from 1 kHz (extremely low-frequency radio waves used to communicate with submarines) through the visible spectrum, and all the way up to frequencies of over 10²⁰ Hz (Gamma rays). Let the random variable \( Z \) be the frequency measured in kHz. Its Probability Density Function (PDF) is:
\[ f_Z(z) = \frac{2}{z^3} u(z - 1), \]
where \( u(\cdot) \) is the standard unit step function.
**Tasks:**
a) The wavelength is given by the random variable \( Y = cZ^{-1} \), where constant \( c \) denotes the speed of light (in appropriate units). Find its PDF, \( f_Y(y) \).
b) Given the extensive electromagnetic spectrum, it is often useful to plot the frequency measurements in the log scale. Let \( X = \log_e(Z) \), and find its PDF, \( f_X(x) \).
c) Are \( X \) and \( Y \) uncorrelated and/or independent? Prove your answer mathematically, and explain why it should have been expected.
### Detailed Explanations:
**Part (a): Finding the PDF of \( Y \)**
The transformation \( Y = cZ^{-1} \) can be used to find the PDF of \( Y \):
1. Start by expressing \( Z \) in terms of \( Y \):
\[ Z = \frac{c}{Y} \]
2. Calculate the absolute value of the derivative of \( Z \) with respect to \( Y \):
\[ \left| \frac{dZ}{dY} \right| = \left| \frac{d}{dY} \left( \frac{c}{Y} \right) \right| = \frac{c}{Y^2} \]
3. Using the change of variables formula for PDFs:
\[ f_Y(y) = f_Z\left(\frac{c}{y}\right) \left| \frac{d}{dy} \left( \frac{c}{y} \right) \right| \]
Substitute \( f_Z(z) \) and the derivative computed above:
\[ f_Y(y) = f_Z\left(\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4a579a7f-99c6-465f-8352-06e32f8837ce%2Fdcadf634-6078-4467-9c60-a6b61af1493b%2Fw7oz0a5_processed.png&w=3840&q=75)
Transcribed Image Text:### Electromagnetic Spectrum Frequency Measurement
**Problem Statement:**
A powerful detector measures the frequency of radiation over virtually the entire electromagnetic spectrum, ranging from 1 kHz (extremely low-frequency radio waves used to communicate with submarines) through the visible spectrum, and all the way up to frequencies of over 10²⁰ Hz (Gamma rays). Let the random variable \( Z \) be the frequency measured in kHz. Its Probability Density Function (PDF) is:
\[ f_Z(z) = \frac{2}{z^3} u(z - 1), \]
where \( u(\cdot) \) is the standard unit step function.
**Tasks:**
a) The wavelength is given by the random variable \( Y = cZ^{-1} \), where constant \( c \) denotes the speed of light (in appropriate units). Find its PDF, \( f_Y(y) \).
b) Given the extensive electromagnetic spectrum, it is often useful to plot the frequency measurements in the log scale. Let \( X = \log_e(Z) \), and find its PDF, \( f_X(x) \).
c) Are \( X \) and \( Y \) uncorrelated and/or independent? Prove your answer mathematically, and explain why it should have been expected.
### Detailed Explanations:
**Part (a): Finding the PDF of \( Y \)**
The transformation \( Y = cZ^{-1} \) can be used to find the PDF of \( Y \):
1. Start by expressing \( Z \) in terms of \( Y \):
\[ Z = \frac{c}{Y} \]
2. Calculate the absolute value of the derivative of \( Z \) with respect to \( Y \):
\[ \left| \frac{dZ}{dY} \right| = \left| \frac{d}{dY} \left( \frac{c}{Y} \right) \right| = \frac{c}{Y^2} \]
3. Using the change of variables formula for PDFs:
\[ f_Y(y) = f_Z\left(\frac{c}{y}\right) \left| \frac{d}{dy} \left( \frac{c}{y} \right) \right| \]
Substitute \( f_Z(z) \) and the derivative computed above:
\[ f_Y(y) = f_Z\left(\
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