Question 5, 5.1.12 Part 3 of 3 4 correct Close 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. \begin{tabular}{c|c} \hline$x$ & $P(x)$ \\ \hline 0 & 0.353 \\ \hline 1 & 0.426 \\ \hline 2 & 0.200 \\ \hline 3 & 0.021 \\ \hline \end{tabular} Does the table show a probability distribution? Select all that apply. A. Yes, the table shows a probability distribution. B. No, the random variable $x$ 's number values are not associated with probabilities. C. No, the random variable $\mathrm{x}$ is categorical instead of numerical. D. No, the sum of all the probabilities is not equal to 1 . E. No, not every probability is between 0 and 1 inclusive. 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. A. $\mu=0.9$ adult(s) (Round to one decimal place as needed.) B. The table does not show a probability distribution. Find the standard deviation of the random variable $x$. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $\sigma=\square$ adult(s) (Round to one decimal place as needed.) B. The table does not show a probability distribution.
Solution
Step 1 ：First, we need to check if the given table represents a probability distribution. For a table to represent a probability distribution, it must satisfy two conditions: 1. All the probabilities must be between 0 and 1, inclusive. 2. The sum of all the probabilities must be equal to 1. If the table satisfies these conditions, then it represents a probability distribution. If it does not, then it does not represent a probability distribution.
Step 2 ：If the table represents a probability distribution, we can then calculate the mean and standard deviation. The mean of a probability distribution is calculated by multiplying each possible outcome by its probability and then summing these products. The standard deviation is a measure of the dispersion of the probability distribution and is calculated using the formula: \[\sigma = \sqrt{\sum (x - \mu)^2 * P(x)}\] where: - $x$ is a possible outcome, - $\mu$ is the mean of the distribution, - $P(x)$ is the probability of $x$.
Step 3 ：The given table represents a probability distribution. The mean is 0.9 and the standard deviation is 0.9.
Step 4 ：The mean of the random variable x is \(\boxed{0.9}\) adult(s). The standard deviation of the random variable x is \(\boxed{0.9}\) adult(s).
