Question

Use the probability distribution for the random variable \(x\) to answer the questions. $$\begin{array}{l|lllll}x & 0 & 1 & 2 & 3 & 4 \\\\\hline p(x) & .1 & .3 & .3 & ? & .1\end{array}$$ Calculate the population mean, variance, and standard deviation.

Short answer

& .1\end{array}$$ Answer: The population mean is 1.9, the population variance is 1.29, and the standard deviation is approximately 1.14.

Step-by-step solution

Find the missing probability value

To find the missing probability value, we have to make sure the sum of all probabilities is equal to 1. The sum of given probabilities is 0.1+0.3+0.3+0.1 = 0.8. Therefore, the missing probability (when \(x=3\)) is: $$ p(3) = 1 - 0.8 = 0.2 $$ Now, the updated probability distribution table is: $$\begin{array}{l|lllll}x & 0 & 1 & 2 & 3 & 4 \\\\\hline p(x) & .1 & .3 & .3 & .2 & .1\end{array}$$

Calculate the population mean

The population mean (µ) can be calculated with the following formula: $$ \mu = \sum_{i=1}^n x_i p(x_i) $$ Thus, the population mean is: $$ \mu = (0 \cdot 0.1) + (1 \cdot 0.3) + (2 \cdot 0.3) + (3 \cdot 0.2) + (4 \cdot 0.1) = 1.9 $$

Calculate the population variance

The population variance (σ²) can be calculated with the following formula: $$ \sigma^2 = \sum_{i=1}^n (x_i - \mu)^2 p(x_i) $$ Thus, the population variance is: $$ \sigma^2 = (0-1.9)^2 \cdot 0.1 + (1-1.9)^2 \cdot 0.3 + (2-1.9)^2 \cdot 0.3 + (3-1.9)^2 \cdot 0.2 + (4-1.9)^2 \cdot 0.1 = 1.29 $$

Calculate the population standard deviation

The population standard deviation (σ) is the square root of the population variance: $$ \sigma = \sqrt{\sigma^2} = \sqrt{1.29} \approx 1.14 $$ So, the population mean is 1.9, the population variance is 1.29, and the population standard deviation is approximately 1.14.

Key concepts

Understanding Population Mean

The population mean, denoted as \( \mu \), is a fundamental concept in statistics, representing the average value in a population. In the context of probability distributions, calculating the mean involves multiplying each possible outcome by its corresponding probability and then summing all these products.

To visualize this, imagine you have a set of weighted numbers, where each number (\(x\)) represents a possible outcome, and the weight (\(p(x)\)) is the probability of that outcome occurring. The population mean is like the balance point of a scale that has these weighted numbers on it. In the exercise provided, the mean is calculated as \( \mu = (0 \cdot 0.1) + (1 \cdot 0.3) + (2 \cdot 0.3) + (3 \cdot 0.2) + (4 \cdot 0.1) = 1.9 \).

• Tip 1: Always make sure that the probabilities add up to 1; otherwise, the calculations will not reflect the true mean of the distribution.
• Tip 2: Remember, the mean is a measure of central tendency, indicating where the center of the data distribution lies.

Deciphering Population Variance

Population variance, denoted as \( \sigma^2 \), measures the spread of a set of data points in a population. To put it simply, it tells us how much the values in a distribution deviate from the mean, on average. This measure of dispersion is particularly useful because it gives us an idea of the diversity or variability within our data set.

The variance is calculated by taking the squared differences between each outcome and the mean, then multiplying these by their respective probabilities, and finally, adding them all together. Using the improved probability distribution from the exercise, we get \( \sigma^2 = (0-1.9)^2 \cdot 0.1 + (1-1.9)^2 \cdot 0.3 + (2-1.9)^2 \cdot 0.3 + (3-1.9)^2 \cdot 0.2 + (4-1.9)^2 \cdot 0.1 = 1.29 \).

• Tip 1: Computing variance requires prior knowledge of the mean, emphasizing the mean's foundational role in statistical analysis.
• Tip 2: A smaller variance indicates that the data points are closer to the mean, while a larger variance shows greater spread.

Population Standard Deviation Made Simple

The population standard deviation, represented by the Greek letter \( \sigma \), is the square root of the population variance. It's a powerful statistic offering insights into the amount of variation or dispersion from the population mean, in the same units as the data itself. A lower standard deviation means the data points are closer to the mean, and a higher value indicates a wider spread of data.

In the exercise at hand, by taking the square root of the variance \( \sigma^2 \), we find the standard deviation: \( \sigma = \sqrt{1.29} \approx 1.14 \).

• Tip 1: Standard deviation provides an intuitive sense of the variability because it's in the same units as the original data, which isn't the case with variance.
• Tip 2: It's easier to compare the standard deviation to the mean than the variance, which can offer a quick gauge of relative dispersion in a data set.