Family/Partner Groups of people aged 15–65 are randomly selected and arranged in groups of six. The random variable x is the number in the group who say that their family and/or partner contribute most to their happiness (based on a Coca-Cola survey). The accompanying table lists the values of x along with their corresponding probabilities. Does the table describe a probability distribution? If so, find the mean and standard deviation.

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Step 1: Verify if the table describes a probability distribution. To do this, check two conditions: (a) All probabilities P(x) must be between 0 and 1, inclusive, and (b) The sum of all probabilities P(x) must equal 1.

Step 2: Calculate the sum of all probabilities P(x) from the table: \( P(0) + P(1) + P(2) + P(3) + P(4) + P(5) + P(6) \). Ensure the sum equals 1 to confirm it is a valid probability distribution.

Step 3: To find the mean (expected value), use the formula \( \mu = \sum [x \cdot P(x)] \), where \( x \) is the value of the random variable and \( P(x) \) is its corresponding probability. Multiply each \( x \) by its \( P(x) \), then sum the results.

Step 4: To find the variance, use the formula \( \sigma^2 = \sum [(x - \mu)^2 \cdot P(x)] \). Subtract the mean \( \mu \) from each \( x \), square the result, multiply by \( P(x) \), and sum these values.

Step 5: To find the standard deviation, take the square root of the variance: \( \sigma = \sqrt{\sigma^2} \). This provides the measure of spread for the probability distribution.

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주요 개념

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Probability Distribution

A probability distribution describes how the probabilities are distributed over the values of a random variable. For a valid probability distribution, the sum of all probabilities must equal 1, and each individual probability must be between 0 and 1. In this case, we need to check if the probabilities listed for the values of x (0 to 6) meet these criteria.

Mean of a Probability Distribution

The mean, or expected value, of a probability distribution is calculated by multiplying each value of the random variable by its corresponding probability and then summing these products. This provides a measure of the central tendency of the distribution, indicating the average outcome one can expect if the experiment is repeated many times.

Standard Deviation of a Probability Distribution

The standard deviation measures the dispersion or spread of a probability distribution around its mean. It is calculated by taking the square root of the variance, which is the average of the squared differences between each value and the mean, weighted by their probabilities. A higher standard deviation indicates greater variability in the outcomes.