91Ó°ÊÓ

Histograms are a powerful visual tool used to understand and compare probability distributions. They use bars to represent different categories of data, making it easier to see differences in probability distributions at a glance. For variables like household size and family size, histograms can provide clear insight into how frequently each size occurs.
To create a histogram, plot the number of persons on the x-axis and the corresponding probabilities on the y-axis. For household sizes, the highest probability is for a 1-person household (0.25), showing a significant presence of single-person dwellings. The probabilities decrease as the size increases, reflecting fewer larger households.

In contrast, for family sizes, there are no one-person households, and the distribution starts from 2 persons with a peak probability at this size (0.42). This immediately tells us that family structures inherently exclude single occupancy which is why no probability is shown for it in family histograms. When comparing these histograms, the distinction between types of households is evident, with families clustering around slightly larger sizes compared to all household types.

The mean, or expected value, of a probability distribution gives us a measure of the central tendency of that distribution. It tells us the average number of persons we can expect in a household or family.
To calculate this mean, multiply each possible size by its probability and sum these products. For household sizes, the calculation is as follows:
\( \mu_X = (1)(0.25) + (2)(0.32) + (3)(0.17) + (4)(0.15) + (5)(0.07) + (6)(0.03) + (7)(0.01) = 2.38 \).
For families, which can't have less than two persons, it is:
\( \mu_Y = (2)(0.42) + (3)(0.23) + (4)(0.21) + (5)(0.09) + (6)(0.03) + (7)(0.02) = 3.14 \).
The mean household size is 2.38, indicating that a typical household includes roughly two to three people. Conversely, the mean family size of 3.14 logically reflects that families tend to be larger since they must include at least two individuals, and typically involve multiple members living together.

Standard deviation measures the spread of a probability distribution. It tells us how much the sizes of households and families vary around their respective means. A low standard deviation means the data points tend to be very close to the mean, while a higher standard deviation indicates more spread.
To find the standard deviation, subtract the mean from each size, square the result, multiply by its probability, and take the square root of the sum of these values.

For Household Size:

Using \(\mu_X = 2.38\), the calculation is:
\( \sigma_X = \sqrt{(1-2.38)^2(0.25) + (2-2.38)^2(0.32) + (3-2.38)^2(0.17) + (4-2.38)^2(0.15) + (5-2.38)^2(0.07) + (6-2.38)^2(0.03) + (7-2.38)^2(0.01)} \approx 1.26 \).

For Family Size:

With \(\mu_Y = 3.14\), calculate:
\( \sigma_Y = \sqrt{(2-3.14)^2(0.42) + (3-3.14)^2(0.23) + (4-3.14)^2(0.21) + (5-3.14)^2(0.09) + (6-3.14)^2(0.03) + (7-3.14)^2(0.02)} \approx 1.05 \).
The result shows that household sizes have a greater spread (1.26) than family sizes (1.05), reflecting the variety of living arrangements in non-family households, which include single occupants and a mix of unrelated individuals.