Question

X 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. Does the table show a probability distribution? Select all that apply. 0 1 2 3 P(x) 0.357 0.446 0.168 0.029 A. Yes, the table shows a probability distribution B. No, the sum of all the probabilities is not equal to 1. C. No, the random variable x is categorical instead of numerical D. No, the random variable x's number values are not associated with probabilities E. No, not every probability is between 0 and 1 inclusive.

Answer 1

x	P(x)	x * P(x)
0	0.357	0	0
1	0.446	0.446	0.446
2	0.168	0.336	0.672
3	0.029	0.087	0.261
		$\sum x * P(x) = 0.869$	$\sum x^2 * P(x) = 1.379$

Since, all probabilities are between 0 and 1 inclusive, and sum of all probabilities equal to 1, so this is a valid probabilitiy distribution.

Answer : A) Yes, the table shows a probability distribution.

The mean and standard deviation of the probabilitiy distribution are as follows :

$\\ \mu = E(X) = \sum x * P(x) = 0.869 \\\\ => \mu = \bold {0.9} \text { (rounded to one decimal place)}$

$\\ \sigma= \sqrt {\sum x^2 * P(x) -\mu^2} = \sqrt {1.379 -(0.869)^2} = 0.7898 \\\\ => \sigma= \bold {0.8} \text { (rounded to one decimal place)}$