Question

Die Physiker RUTHERFORD und GEIGER untersuchten die Emission von \(\alpha\) Teilchen aus einer radioaktiven Substanz. Die Anzahl \(X\) der \(\alpha\)-Teilchen, die in einem bestimmten Zeitintervall emittiert werden, ist eine diskrete Zufallsgröße. RUTHERFORD und GEIGER stellten fest, dass die Zufallsgröße \(X\) für Zeitintervalle der Länge 7,5 Sekunden die 11 Werte \(0,1, \ldots, 10\) annehmen kann. Es wurde eine Stichprobe vom Umfang \(n=2608\) untersucht, d.h. es wurden die Werte von \(X\) in 26087,5 -Sekunden-Intervallen experimentell ermittelt. Die Anzahl der Zeitintervalle, in denen \(X\) den Wert \(i(i=0,1, \ldots, 10)\) angenommen hat, sei \(n_{i} .\) Es ist \(\sum_{i=0}^{10} n_{i}=n=2608\). \begin{tabular}{lcccccccccccc} \hline\(i\) & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & \\ \hline\(n_{i}\) & 57 & 203 & 383 & 525 & 532 & 408 & 273 & 139 & 45 & 27 & 16 & \(n=2608\) \\ \hline \end{tabular} (a) Man bestimme die empirische Häufigkeitsverteilung \(\left(i, \frac{n_{i}}{n}\right)\). (b) Man berechne die Summenhäufigkeiten \(s_{i}\) und die empirische Wahrscheinlichkeitsverteilungsfunktion \(f(x)\) (Schätzung für die Wahrscheinlichkeitsverteilungsfunktion \(P\\{X

Short answer

Calculate empirical frequencies and median, find \(\lambda\), compare Poisson probabilities with data, test hypothesis.

Step-by-step solution

Calculate the Empirical Relative Frequencies

To calculate the empirical relative frequencies for each value of \( i \), use the formula \( \frac{n_i}{n} \), where \( n_i \) is the number of intervals with value \( i \) and \( n = 2608 \). Perform this calculation for the entire range of \( i \), \( i = 0, 1, \ldots, 10 \).

Calculate the Cumulative Frequencies \( s_i \)

To obtain the cumulative frequencies, which represent the sum of relative frequencies up to a certain \( i \), sum the relative frequencies calculated in Step 1 cumulatively starting from \( i = 0 \) to each \( i \).

Determine the Empirical Median and Mode

The median is the value of \( i \) where the cumulative frequency crosses 0.5. Find this \( i \) by inspecting the cumulative frequencies. The mode is the \( i \) value with the highest frequency \( n_i \).

Calculate the Poisson Parameter \( \lambda \)

Compute the expected value \( \lambda \) using the formula \( \lambda = \frac{1}{n} \sum_{i=1}^{10} i \cdot n_i \). Substitute the given \( n_i \) values into this equation to find \( \lambda \).

Calculate Poisson Probabilities

For each \( i \) from 0 to 10, calculate the Poisson probability \( P\{Y=i\} = \frac{\lambda^i}{i!} e^{-\lambda} \). Use the \( \lambda \) value obtained in Step 4.

Compare Poisson with Empirical Frequencies

Compare the Poisson probabilities obtained in Step 5 with the empirical relative frequencies \( \frac{n_i}{n} \) obtained in Step 1 to observe any differences or similarities.

Perform the \( \chi^2 \)-Goodness-of-Fit Test

Formulate the null hypothesis that the observed frequencies come from a Poisson distribution. Use the \( \chi^2 \) goodness-of-fit test to compare the empirical frequencies \( n_i \) with the expected frequencies from the Poisson distribution, accounting for the estimated \( \lambda \).

Key concepts

Empirical Frequency Distribution

In statistics, an empirical frequency distribution is vital for understanding observed data patterns. It shows how often each value occurs within a dataset, providing insight into the data's structure. For Rutherford and Geiger's experiment, they gathered data on the emission counts of \(\alpha\) particles over multiple intervals. The empirical frequency distribution was calculated by determining the relative frequencies for each potential value \(i\) that \(X\), the random variable representing particle counts, can take.

Here is how you calculate these frequencies step-by-step:

• First, note how often each particle count \(i\) (from 0 to 10) appears across all measured intervals.
• Divide the count \(n_i\) for each \(i\) by the total count \(n\), which is 2608 overall observations.

By doing this, you create a proportional view of how the data is distributed across the possible values, forming the empirical frequency distribution. This is a key step in analyzing the data and understanding the behavior of the \(\alpha\) particle emissions.

Chi-Squared Goodness-of-Fit Test

The chi-squared goodness-of-fit test is a statistical method used to determine if there's a significant difference between observed and expected frequencies. In Rutherford and Geiger's study, this test helps to assess whether the distribution of \(\alpha\) particle emissions fits a Poisson distribution shape.

Here's how this process works in practical terms:

• Set up the null hypothesis, stating that the observed data follows the expected Poisson distribution with an estimated parameter \(\lambda\).
• Calculate the expected frequencies based on this Poisson distribution.
• Compare these expected frequencies to the observed frequencies using the chi-squared statistic: \(\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}\), where \(O_i\) are the observed values, and \(E_i\) are the expected values.
• Determine the degrees of freedom, typically the number of values minus one, and use it to find the critical value from a chi-squared distribution table.
• If your test statistic exceeds this critical value, the null hypothesis is rejected.

This test provides a formal mechanism to check how well a theoretical distribution aligns with observed real-world data, guiding scientists in correctly interpreting their experimental results.

Expected Value Calculation

Expected value is a fundamental concept in probability, central to understanding random variables and their long-run behavior. In this exercise, the expected value is used to find the parameter \(\lambda\) for the Poisson distribution, which is then compared to the observed data of \(\alpha\) particle counts.

Here’s how expected value calculation helps in this context:

• The expected value of a Poisson-distributed random variable can be denoted as \(\lambda\). It's calculated as the sample mean of the data.
• For Rutherford and Geiger’s \(\alpha\) emissions, this is computed using: \(\lambda = \frac{1}{n} \sum_{i=0}^{10} i \cdot n_i\)
• This calculation produces a numerical value for \(\lambda\), representing the average count of \(\alpha\) emissions per interval. This value is integral in calculating the Poisson probabilities and serves as a bridge between empirical data and theoretical models.

Thus, expected value calculation not only facilitates the transition from empirical data to statistical modeling, but it also underpins the entire process of fitting the observed data to a theoretical distribution and deriving insights from it.