Questions: From the probability distribution, find the mean and standard deviation for the random variable x, which represents the number of cars per household in a town of 1000 households. x P(x) 0 0.160 1 0.393 2 0.270 3 0.094 4 0.083

From the probability distribution, find the mean and standard deviation for the random variable x, which represents the number of cars per household in a town of 1000 households.

x P(x)
0 0.160
1 0.393
2 0.270
3 0.094
4 0.083

Transcript text: From the probability distribution, find the mean and standard deviation for the random variable $x$, which represents the number of cars per household in a town of 1000 households. \begin{tabular}{|l|l|} \hline $\boldsymbol{x}$ & $\boldsymbol{P}(x)$ \\ \hline 0 & 0.160 \\ \hline 1 & 0.393 \\ \hline 2 & 0.270 \\ \hline 3 & 0.094 \\ \hline 4 & 0.083 \\ \hline \end{tabular}

Solution

Solution Steps

Step 1: Calculate the Mean

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

\[ \mu = \sum_{i=0}^{n} x_i \cdot P(x_i) \]

Substituting the values:

\[ \mu = 0 \times 0.16 + 1 \times 0.393 + 2 \times 0.27 + 3 \times 0.094 + 4 \times 0.083 = 1.547 \]

Step 2: Calculate the Variance

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

\[ \sigma^2 = \sum_{i=0}^{n} (x_i - \mu)^2 \cdot P(x_i) \]

Substituting the values:

\[ \sigma^2 = (0 - 1.547)^2 \times 0.16 + (1 - 1.547)^2 \times 0.393 + (2 - 1.547)^2 \times 0.27 + (3 - 1.547)^2 \times 0.094 + (4 - 1.547)^2 \times 0.083 = 1.254 \]

Step 3: Calculate the Standard Deviation

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

\[ \sigma = \sqrt{\sigma^2} = \sqrt{1.254} \approx 1.12 \]