Questions: Groups of adults are randomly selected and arranged in groups of three. The random variable x is the number in the group who say that they would feel comfortable in a self-driving vehicle. Determine whether a probability distribution is given. If a probability distribution is given, find its mean and standard deviation. If a probability distribution is not given, identify the requirements that are not satisfied. x P(x) ------ 0 0.364 1 0.447 2 0.167 3 0.022 Does the table show a probability distribution? Select all that apply. A. Yes, the table shows a probability distribution. B. No, not every probability is between 0 and 1 inclusive. C. No, the random variable x's number values are not associated with probabilities. D. No, the sum of all the probabilities is not equal to 1 . E. No, the random variable x is categorical instead of numerical. Find the mean of the random variable x. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

Solution Steps

To determine if the table represents a probability distribution, we need to verify two conditions:

1. The sum of all probabilities must equal 1.
2. Each probability must be between 0 and 1 inclusive.

Calculating the sum of the probabilities:

\[ \text{Sum of probabilities} = 0.364 + 0.447 + 0.167 + 0.022 = 1.0 \]

Since the sum equals 1 and all probabilities are between 0 and 1, we conclude that the table does show a probability distribution.

The mean \( \mu \) of the random variable \( x \) is calculated using the formula:

\[ \mu = \sum (x \cdot P(x)) = 0 \times 0.364 + 1 \times 0.447 + 2 \times 0.167 + 3 \times 0.022 \]

Calculating each term:

\[ \mu = 0 + 0.447 + 0.334 + 0.066 = 0.847 \]

The variance \( \sigma^2 \) is calculated using the formula:

\[ \sigma^2 = \sum ((x - \mu)^2 \cdot P(x)) \]

Calculating each term:

\[ \sigma^2 = (0 - 0.847)^2 \times 0.364 + (1 - 0.847)^2 \times 0.447 + (2 - 0.847)^2 \times 0.167 + (3 - 0.847)^2 \times 0.022 \]

Calculating each squared difference:

\[ \sigma^2 = (0.718) \times 0.364 + (0.023) \times 0.447 + (1.313) \times 0.167 + (4.617) \times 0.022 \]

\[ \sigma^2 = 0.261 + 0.010 + 0.219 + 0.102 = 0.596 \]

The standard deviation \( \sigma \) is the square root of the variance:

\[ \sigma = \sqrt{0.596} \approx 0.772 \]

Final Answer