Chapter 4 Special Distributions - 4.2 The Poisson Distribution - Questions - Page 231: 11

$\color{blue}{p_X(k) = \dfrac{e^{-0.435}0.435^k}{k!},\ k=0,1,2,3,\ldots. \\ \begin{array}{|c|c|c|c|} \hline k & \rm Frequency & p_X(k) & \rm Predicted \\ \hline 0 & 237 & 0.647 & 230.3 \\ \hline 1 & 90 & 0.282 & 100.4 \\ \hline 2 & 22 & 0.061 & 21.7 \\ \hline 3+ & 7 & 0.010 & 3.6 \\ \hline \rm Total & 356 & 1.000 & 356 \\ \hline \end{array}}$ Corresponding observed and predicted frequencies in columns 2 and 4 agree, showing that the data can be modelled by using a Poisson pdf. See explanation.

Let $X$ be the number of times a senior student has changed majors at the University of West Florida (UWF). An estimate of the mean of the number of times that a senior has changed majors at UWF is given by $\overline{k} = \dfrac{(0)(237) + (1)(90) + (2)(22) + (3)(7)}{237+90+22+7} = \dfrac{155}{356} \approx 0.435$. If we assume that $X$ is Poisson, its pdf would approximately be $p_X(k) = \dfrac{e^{-0.435}0.435^k}{k!},\ k=0,1,2,3,\ldots.$ The number of times a senior student changed majors $k$, the observed frequencies for each/combined $k$ categories, the estimated proportions for each/combined category, and the predicted frequencies (obtained as $356\cdot p_X(k)$ for each $k$ or combined category ) are summarized in the table below. $\begin{array}{|c|c|c|c|} \hline k & \rm Frequency & p_X(k) & \rm Predicted \\ \hline 0 & 237 & 0.647 & 230.3 \\ \hline 1 & 90 & 0.282 & 100.4 \\ \hline 2 & 22 & 0.061 & 21.7 \\ \hline 3+ & 7 & 0.010 & 3.6 \\ \hline \rm Total & 356 & 1.000 & 356 \\ \hline \end{array}$ Note that the predicted frequencies in column 4 agree with the corresponding observed frequencies in column 2, showing that the data can be modelled by using a Poisson pdf.