91Ó°ÊÓ

The mean of a probability distribution gives us a sense of the "center" of the distribution. It tells us the expected value or the average of our random variable. In our exercise, the random variable, \( x \), represents the number of lots ordered by customers. To calculate the mean, we multiply each value of \( x \) by its corresponding probability \( p(x) \) and sum up these products. Mathematically, it is expressed as:\[ \mu = \Sigma (x \cdot p(x)) \]In the given probability distribution table:

• When \( x = 1 \), \( p(x) = 0.2 \)
• When \( x = 2 \), \( p(x) = 0.4 \)
• When \( x = 3 \), \( p(x) = 0.3 \)
• When \( x = 4 \), \( p(x) = 0.1 \)

Plug these into the formula:\[ \mu = (1 \times 0.2) + (2 \times 0.4) + (3 \times 0.3) + (4 \times 0.1) = 2.3 \]This means, on average, a randomly chosen customer orders 2.3 lots of the chemical. Interpretively, we expect most customers to order around 2 to 3 lots.

Variance measures the spread of the probability distribution, or how much the values of the random variable \( x \) deviate from the mean. It helps in understanding the variability or dispersion of the data points. To compute the variance, use the formula:\[ \sigma^2 = \Sigma ((x- \mu)^2 \cdot p(x)) \]Here, \( \mu \) is the mean that we've already calculated, which is 2.3. For each possible value of \( x \), we:

• Calculate \((x-\mu)^2\)
• Multiply the result by \(p(x)\)
• Then sum these products up

Using our probability distribution:\[ \sigma^2 = ((1-2.3)^2 \times 0.2) + ((2-2.3)^2 \times 0.4) + ((3-2.3)^2 \times 0.3) + ((4-2.3)^2 \times 0.1) \]Carrying out these calculations:\[ \sigma^2 = 1.69 \times 0.2 + 0.09 \times 0.4 + 0.49 \times 0.3 + 2.89 \times 0.1 = 0.61 \]Thus, the variance is 0.61, indicating a moderate dispersion from the mean.

Standard deviation provides a measure of the average distance between each data point and the mean. It's essentially the square root of the variance, making it a useful indicator of the spread in the same units as the data itself. Calculating standard deviation helps us understand how much variation or "spread" exists within a set of data.Since we have calculated the variance \( \sigma^2 \) to be 0.61, the standard deviation \( \sigma \) is obtained by:\[ \sigma = \sqrt{0.61} \approx 0.78 \]Therefore, the standard deviation is approximately 0.78. This means that the number of lots ordered by customers typically varies by about 0.78 lots from the mean. It suggests a relatively consistent ordering behavior among most customers.