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Chapter 32: Problem 12

Prove that the relative standard deviation of the counting rate \(\sigma_{R} / R\) is simply \(M^{-1 / 2},\) where \(M\) is the number of counts.

Short Answer

Expert verified
RSD is proved to be \(M^{-1/2}\) for counting rates.

Step by step solution

01

Understand Relative Standard Deviation

The relative standard deviation (RSD) is defined as the standard deviation (\(\sigma_R\)) divided by the mean (\(R\)) of a data set. For counting measurements, this relative variance helps determine the variability as a proportion of the mean.
02

Relationship Between Standard Deviation and Count

In counting experiments, the number of counts recorded (\(M\)) follows a Poisson distribution. For a Poisson distribution, the standard deviation is the square root of the mean, so \(\sigma_R = \sqrt{M}\).
03

Calculating the Relative Standard Deviation

Substitute the standard deviation into the formula for relative standard deviation: \(\frac{\sigma_R}{R} = \frac{\sqrt{M}}{M}\). Simplify the expression to get \(\frac{1}{\sqrt{M}}\) as required.
04

Conclusion of the Proof

Based on the relationship between standard deviation and the Poisson distribution's mean, and simplifying the relative standard deviation formula, we have shown that \(\frac{\sigma_R}{R} = M^{-1/2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Poisson Distribution
The Poisson distribution is a probability distribution that describes events occurring independently at a constant average rate. It is particularly useful in scenarios where we want to model the frequency of rare events over a fixed interval of time or space. Some typical examples include:
  • Phone calls received at a call center in an hour.
  • Number of decay events per minute from a radioactive source.
  • Number of typing errors a person makes per page.
The defining characteristic of a Poisson distribution is that the mean (average rate of occurrence) is equal to its variance. This property greatly simplifies our calculations when the Poisson distribution is used in statistical modeling, particularly involving counting statistics.
Counting Rate
Counting rate refers to the number of events counted per unit of time or another interval measure. It is an important concept when dealing with phenomena that are naturally described by discrete events鈥攕uch as radioactive decay, particle detection, or any similar process.
  • In counting experiments, data are often collected over a specified duration, and the counting rate is used to describe the event frequency.
  • It acts as the mean (\(R\)) in situations modeled by a Poisson distribution, where the actual counts might slightly vary around this central value.
Understanding the counting rate is essential in achieving accurate measurements and analysis, especially in fields such as nuclear physics and engineering.
Standard Deviation
The standard deviation is a measure of the amount of variation or dispersion in a set of values. In the context of counting measurements, it helps quantify the level of uncertainty or noise present in the data.
  • For a dataset following a Poisson distribution, the standard deviation has a special property: it is equal to the square root of the mean (\(\sigma_R = \sqrt{M}\)), where \(M\) is the number of counts.
  • This relationship makes it easy to assess the probable error in measurements involving counts, as it directly connects to the central value without requiring additional complex calculations.
The standard deviation is instrumental in determining how much a set of data points deviate from the mean, which is crucial for accurate experimental data analysis.
Mean
The mean, also known as the average, is a central measure used to summarize a set of data points. In a Poisson distribution, the mean (\(R\)) holds particular significance due to its unique properties.
  • For counting experiments, the mean represents the expected rate at which events occur鈥攅ssentially acting as the predicted outcome over a set period or across an interval.
  • As we often consider the mean to be equal to the count number (\(M\)) in a Poisson distribution, this link simplifies understanding and calculations involving probabilistic models.
Overall, the mean provides a snapshot of the data trend and serves as the foundation for calculating other statistical metrics like variance and standard deviation.

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Most popular questions from this chapter

Calculate the fraction of each of the following radionuclides that remains after 1 day, 2 days, 3 days, and 4 days (half-lives are given in parentheses): iron-59 (44.51 days), titanium-45 (3.078 h), calcium-47 (4.536 days), and phosphorus-33 (25.3 days).

The streptomycin in 500 g of a broth was determined by addition of \(1.25 \mathrm{mg}\) of the pure antibiotic containing "C. The specific activity of this preparation was found to be \(240 \mathrm{cpm} / \mathrm{mg}\) for a 30 -min count. From the mixture, \(0.112 \mathrm{mg}\) of purified streptomycin was isolated, which produced 675 counts in \(60.0 \mathrm{min}\). Calculate the concentration in parts per million streptomycin in the sample.

One-half of the total activity in a particular sample is due to \(^{38} \mathrm{Cl}\left(t_{1 / 2}=87.2 \mathrm{min}\right)\) The other half of the activity is due to \(^{35} \mathrm{S}\) \(\left(t_{1 / 2}=\right.\)days). The beta emission of \(^{35}\) S must be measured because this nuclide emits no gamma photons. Therefore, it is desirable to wait until the activity of the \(^{38} \mathrm{Cl}\) has decreased to a negligible level. How much time must elapse before the activity of the \(^{3 s}\) Cl has decreased to only \(0.1 \%\) of the remaining activity because of \(^{35} \mathrm{S}\) ?

Show, via a calculation, that the average kinetic energy of a population of thermal neutrons is approximately 0.04 ev.

For a particular radioactive sample, the total counting rate (sample plus background) was \(450 \mathrm{cpm}\), and this value was obtained over a 15.0 -min counting period. The background was counted for 2.0 min and gave \(7 \mathrm{cpm} .\) Estimate (a) the corrected counting rate \(R_{c}\) (b) the standard deviation associated with the corrected counting rate \(\sigma_{R_{i}}\) (c) the \(95 \%\) confidence interval associated with the corrected counting rate.
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