Two independent frequency counters each report the frequency of a separate audio…

PHYSICS · Subjective · Fermi University Physics · Scientific Practice & Experimental Methods · Mathematical Methods for Physics · Uncertainty in measurement

Mathematical Methods for Physics, Ch Probability, random variables and distributions

Problem

Two independent frequency counters each report the frequency of a separate audio oscillator. Counter 1 reports f1=1000.0 Hzf_1 = 1000.0\,\mathrm{Hz} with a one-sigma uncertainty σ1=0.5 Hz\sigma_1 = 0.5\,\mathrm{Hz}; counter 2 reports f2=997.0 Hzf_2 = 997.0\,\mathrm{Hz} with σ2=0.3 Hz\sigma_2 = 0.3\,\mathrm{Hz}. The two readings are independent random variables, each with a well-defined mean and variance, and the beat frequency is defined as Δf=f1−f2\Delta f = f_1 - f_2. (a) For independent random variables XX and YY with means μX,μY\mu_X,\mu_Y and variances σX2,σY2\sigma_X^2,\sigma_Y^2, derive var(X−Y)=σX2+σY2\mathrm{var}(X-Y) = \sigma_X^2 + \sigma_Y^2 starting from the definition var(D)=⟨(D−μD)2⟩\mathrm{var}(D) = \langle (D-\mu_D)^2\rangle with D=X−YD = X - Y, and using independence to show the covariance term vanishes. (b) Generalise part (a): for constants a,ba,b, derive var(aX+bY)=a2σX2+b2σY2\mathrm{var}(aX+bY) = a^2\sigma_X^2 + b^2\sigma_Y^2 for independent X,YX,Y, and check that setting a=1, b=−1a=1,\,b=-1 recovers part (a). (c) Using the result of part (a), find the beat frequency Δf\Delta f and its standard deviation σΔf\sigma_{\Delta f}.

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