91Ó°ÊÓ

The mean, often referred to as the average, provides a central value for a distribution. When we're dealing with a uniform distribution, which is evenly distributed across a range, calculating the mean is straightforward.

For a continuous uniform distribution defined from a to b, the mean is calculated using the formula:

• \[ \text{Mean} = \frac{a + b}{2} \]

This indicates the midpoint of the interval over which the distribution is defined.

In our given problem, where the distribution ranges from 2 to 5, substituting these values into the formula gives:

• \[ \text{Mean} = \frac{2 + 5}{2} = \frac{7}{2} = 3.5 \]

Thus, the mean of this uniform distribution is 3.5. This represents the balance point of the distribution, where observations are equally spread out on either side.

Standard deviation helps in understanding the spread or dispersion of values in a distribution. In a uniform distribution, this value is not simply calculated like in a normal distribution but through a specific formula involving the endpoints a and b.

For a uniform distribution, the standard deviation σ is calculated by the following formula:

• \[ \sigma = \frac{b-a}{\sqrt{12}} \]

This formula takes the difference between the maximum and minimum bounds of the interval and divides it by the square root of 12.

Applying this formula to our interval from 2 to 5:

• \[ \sigma = \frac{5 - 2}{\sqrt{12}} = \frac{3}{\sqrt{12}} = \frac{3}{2\sqrt{3}} = \frac{\sqrt{3}}{2} \]

Thus, the standard deviation of our uniform distribution is \(\frac{\sqrt{3}}{2}\), which explains the variability of data points around the mean.

Probability calculations in a uniform distribution are quite straightforward, owing to its equal spreading of values across the interval. This characteristic leads to simple expressions for finding the probability of values falling above or between certain points.

### Probability of a Value Greater Than a Specific Point
The probability of selecting a value greater than a specific point c is given by the formula:

• \[ P(x > c) = \frac{b - c}{b - a} \]

In the problem, to find the probability of choosing a value greater than 2.6, substitute c = 2.6:

• \[ P(x > 2.6) = \frac{5 - 2.6}{5 - 2} = \frac{2.4}{3} = 0.8 \]

This indicates an 80% chance that a randomly chosen value will be more than 2.6.

### Probability of a Value Between Two Points
To find the probability of a value falling between two points, c and d, the formula used is:

• \[ P(c < x < d) = \frac{d - c}{b - a} \]

For values between 2.9 and 3.7:

• \[ P(2.9 < x < 3.7) = \frac{3.7 - 2.9}{5 - 2} = \frac{0.8}{3} \approx 0.267 \]

This calculation shows about a 26.7% probability of selecting a value falling within this range.