(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.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section5.6: Exponential And Logarithmic Equations

Problem 64E

See similar textbooks