Consider the αpsrticles emitted by a radiosctive source...

Consider the $\alpha$ psrticles emitted by a radiosctive source during some time interval $t .$ One can insgine this time intervsl to be subdivided into meny small intervals of length $\Delta e$. Since the $\alpha$ particles are emitted st random times, the probability of a radioactive disintegration oecurring during suy such tixne $\Delta t$ is completely independent of whatever disintegrations occur at other times. Furthermore, $\Delta t$ can be imsgined to be chosen smsll enough so that the probability of more than one disintegration occurring in a time $\Delta t$ is negligibly smsll. This means that there is some probability $p$ of one disintegration pecurring during s time $\Delta t$ (with $p \ll 1$, since $\Delta t$ was chosen small enough) snd probsbility $1-p$ of no disintegration occurring during this time. Each such time interval $\Delta t$ can then be regarded as an independent trial, there being a total of $N=t / \Delta t$ such trisls turing a time $\ell$ (a) Shew that the probablitr $\mathrm{II}^{\top}(n)$ of $n$ devintegrations occurring in a time $t$ is given by a Poisgon distribution. (b) Suppose that the streagth of the radioactive source is such that the mesn number of disintegrations per minute is 24. What is the probability of obtaining $n$ counts in a time interval of 10 seconds? Ohtain numericsl values for all integral values of $n$ from 0 to 8 .

Consider the $\alpha$ psrticles emitted by a radiosctive source during some time interval $t .$ One can insgine this time intervsl to be subdivided into meny small intervals of length $\Delta e$. Since the $\alpha$ particles are emitted st random times, the probability of a radioactive disintegration oecurring during suy such tixne $\Delta t$ is completely independent of whatever disintegrations occur at other times. Furthermore, $\Delta t$ can be imsgined to be chosen smsll enough so that the probability of more than one disintegration occurring in a time $\Delta t$ is negligibly smsll. This means that there is some probability $p$ of one disintegration pecurring during s time $\Delta t$ (with $p \ll 1$, since $\Delta t$ was chosen small enough) snd probsbility $1-p$ of no disintegration occurring during this time. Each such time interval $\Delta t$ can then be regarded as an independent trial, there being a total of $N=t / \Delta t$ such trisls turing a time $\ell$
(a) Shew that the probablitr $\mathrm{II}^{\top}(n)$ of $n$ devintegrations occurring in a time $t$ is given by a Poisgon distribution.
(b) Suppose that the streagth of the radioactive source is such that the mesn number of disintegrations per minute is 24. What is the probability of obtaining $n$ counts in a time interval of 10 seconds? Ohtain numericsl values for all integral values of $n$ from 0 to 8 .

Key Concepts

Poisson Distribution

The Poisson distribution models the probability of a given number of events occurring in a fixed interval of time when these events happen with a constant mean rate and independently of the time since the last event. It is succinctly expressed by the formula P(n; ?) = (e^(??) * ?^n) / n!, where ? is the average rate of occurrence and n is the number of events.

Independent Trials and the Binomial to Poisson Limit

When a time interval is divided into a large number of very small subintervals, each with a tiny probability of an event occurring, the assumption of independence between events in each subinterval leads to a binomial model. In the limit where the number of trials approaches infinity and the probability in each trial approaches zero such that their product remains finite (?), the binomial distribution converges to the Poisson distribution.

Stochastic Nature of Radioactive Decay

Radioactive decay is inherently a stochastic process where each disintegration event occurs randomly and independently. The lack of memory (each event is independent of the previous ones) and the constant probability per unit time for each decay are key reasons why the process is accurately modeled using a Poisson distribution.

Scaling of the Rate Parameter with Time

The rate parameter in the Poisson distribution (?) represents the expected number of events in a specified time interval. When the time interval changes, ? scales linearly with time. This concept allows for the adjustment of predictions for different observational periods by multiplying the base rate by the time duration.

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