Questions: Suit Sales The number of suits sold per day at a retail store is shown in the table, with the corresponding probabilities. Number of suits sold X: 21, 22, 23, 24, 25 Probability P(X): 0.1, 0.2, 0.3, 0.1, 0.3

Suit Sales The number of suits sold per day at a retail store is shown in the table, with the corresponding probabilities.
Number of suits sold X: 21, 22, 23, 24, 25
Probability P(X): 0.1, 0.2, 0.3, 0.1, 0.3

Transcript text: Suit Sales The number of suits sold per day at a retail store is shown in the table, with the corresponding probabilities. \begin{tabular}{c|ccccc} Number of suits sold $\boldsymbol{X}$ & 21 & 22 & 23 & 24 & 25 \\ \hline Probability $\boldsymbol{P}(\boldsymbol{X})$ & 0.1 & 0.2 & 0.3 & 0.1 & 0.3 \end{tabular}

Solution

Solution Steps

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

To calculate the mean ($\mu$) of the discrete random variable $X$, we use the formula $\mu = E[X] = \sum_{i=1}^{n} x_i P(X=x_i)$. Given the values and their probabilities, we perform the following calculation: $\mu = (21 * 0.1) + (22 * 0.2) + (23 * 0.3) + (24 * 0.1) + (25 * 0.3) = 23.3$.

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

To calculate the variance ($\sigma^2$), we use the formula $\sigma^2 = E[X^2] - (E[X])^2$, where $E[X^2] = \sum_{{i=1}}^{{n}} x_i^2 P(X=x_i)$. Given the values and their probabilities, we perform the following calculation for $E[X^2]$: $E[X^2] = (21^2 * 0.1) + (22^2 * 0.2) + (23^2 * 0.3) + (24^2 * 0.1) + (25^2 * 0.3) = 544.7$.

Thus, $\sigma^2 = E[X^2] - (E[X])^2 = 544.7 - (23.3)^2 = 1.8$.