Question 2, 5.1.8
Part 2 of 3
1 correct
When conducting research on color blindness in males, a researcher forms random groups with five males in each group. The random variable $x$ is the number of males in the group who have a form of color blindness. 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 $\mathbf{x}$ & $\mathbf{P}(\mathbf{x})$ \\
\hline 0 & 0.654 \\
\hline 1 & 0.293 \\
\hline 2 & 0.049 \\
\hline 3 & 0.003 \\
\hline 4 & 0.001 \\
\hline 5 & 0.000 \\
\hline
\end{tabular}
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Does the table show a probability distribution? Select all that apply.
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 $x$ 's number values are not associated with probabilities.
D. No, the random variable $\mathrm{x}$ is categorical instead of numerical.
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. $\mu=\square$ male(s) (Round to one decimal place as needed.)
B. The table does not show a probability distribution.
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Answer
\(\boxed{\text{The table does not represent a probability distribution.}}\)
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. The sum of all probabilities must be equal to 1. 2. Each individual probability must be between 0 and 1, inclusive.
Step 2 ：If the table satisfies these conditions, it represents a probability distribution. In that case, we can calculate the mean and standard deviation of the distribution.
Step 3 ：The mean (or expected value) of a probability distribution is calculated as the sum of the product of each outcome and its corresponding probability. The standard deviation is a measure of the dispersion of the distribution and is calculated as the square root of the variance. The variance is the expected value of the squared deviation of a random variable from its mean.
Step 4 ：Given the values: \(x = [0, 1, 2, 3, 4, 5]\) and \(P_x = [0.654, 0.293, 0.049, 0.003, 0.001, 0]\)
Step 5 ：The table does not represent a probability distribution because the sum of all probabilities is not equal to 1. Therefore, we cannot calculate the mean and standard deviation.
Step 6 ：\(\boxed{\text{The table does not represent a probability distribution.}}\)
