Questions: Find μ if μ=∑[x ⋅ P(x)]. Then, find σ if σ²=∑[x² ⋅ P(x)]-μ². x P(x) 0 0.133 1 0.382 2 0.367 3 0.118 μ= (Simplify your answer. Round to three decimal places as needed.)

Transcript text: Find $\mu$ if $\mu=\sum[x \cdot P(x)]$. Then, find $\sigma$ if $\sigma^{2}=\sum\left[x^{2} \cdot P(x)\right]-\mu^{2}$. \begin{tabular}{|c|c|} \hline$x$ & $P(x)$ \\ \hline 0 & 0.133 \\ \hline 1 & 0.382 \\ \hline 2 & 0.367 \\ \hline 3 & 0.118 \\ \hline \end{tabular} $\mu=$ $\square$ (Simplify your answer. Round to three decimal places as needed.)

Solution

Solution Steps

Step 1: Calculate the Mean ($\mu$)

The mean of a discrete probability distribution is calculated using the formula:

\[ \mu = \sum [x \cdot P(x)] \]

Given the values and their corresponding probabilities:

\[ \begin{align_} x & : 0, 1, 2, 3 \\ P(x) & : 0.133, 0.382, 0.367, 0.118 \end{align_} \]

The mean is calculated as follows:

\[ \mu = 0 \times 0.133 + 1 \times 0.382 + 2 \times 0.367 + 3 \times 0.118 = 1.47 \]

Step 2: Calculate the Variance ($\sigma^2$)

The variance of a discrete probability distribution is calculated using the formula:

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

Substituting the values and the mean $\mu = 1.47$:

\[ \begin{align_} \sigma^2 & = (0 - 1.47)^2 \times 0.133 + (1 - 1.47)^2 \times 0.382 + (2 - 1.47)^2 \times 0.367 + (3 - 1.47)^2 \times 0.118 \\ & = 0.751 \end{align_} \]

Step 3: Calculate the Standard Deviation ($\sigma$)

The standard deviation is the square root of the variance:

\[ \sigma = \sqrt{\sigma^2} = \sqrt{0.751} = 0.867 \]