A circular garden bed of radius \(1 \mathrm{m}\) is to be planted so that \(N\) seeds are uniformly distributed over the circular area. Then we can talk about the number \(n\) of seeds in some particular area \(A,\) or we can call \(n / N\) the probability for any one particular seed to be in the area \(A\). Find the probability \(F(r)\) that a seed (that is, some particular seed) is within \(r\) of the center. (Hint: What is \(\mathrm{F}(1) ?\) ) Find \(f(r) d r,\) the probability for a seed to be between \(r\) and \(r+d r\) from the center. Find \(\bar{r}\) and \(\sigma\).

The concept of a circular area is crucial in solving this problem. To calculate the area of a circle, we use the formula: \ A = \pi r^2 \ where \( r \) is the radius of the circle. In this exercise, the radius is given as 1 meter. Plugging this value into the formula, we get: \ A = \pi (1)^2 = \pi \text{ square meters} \ Thus, the total area of the garden bed is \( \pi \text{ square meters} \). This area serves as the basis for determining the probability distribution of the seeds planted within the garden bed.

In this problem, the seeds are uniformly distributed across the circular garden bed. A uniform distribution means that every point in the area has an equal chance of having a seed. When dealing with a uniform distribution in a circle, each seed is equally likely to be found anywhere in the circular area. This assumption allows us to calculate probabilities by comparing areas. For example, if we want to find the probability that a seed is within a smaller circle of radius \( r \), we compare the area of this smaller circle to the total area of the garden. Hence, the probability function \( F(r) \) that a seed is within radius \( r \) of the center is: \ F(r) = \frac{\pi r^2}{\pi} = r^2 \

The expectation value, or mean, is a key statistical measure that tells us the average value of a random variable, in this case, the average distance of a seed from the center. To find the expectation value \( \bar{r} \), we use the formula: \ \bar{r} = \int_0^1 r f(r) dr \ Given the probability density function \( f(r) = 2r \), the calculation becomes: \ \bar{r} = \int_0^1 r (2r) dr = 2 \int_0^1 r^2 dr = 2 \left[ \frac{r^3}{3} \right]_0^1 = 2 \times \frac{1}{3} = \frac{2}{3} \ Thus, the mean distance of a seed from the center of the circular garden bed is \( \frac{2}{3} \) meters.

Standard deviation helps us understand the variability or spread of the seeds' distances from the center. It is calculated using the variance: \ \sigma^2 = \int_0^1 r^2 f(r) dr - (\bar{r})^2 \ Firstly, we find the second moment: \ \int_0^1 r^2 (2r) dr = 2 \int_0^1 r^3 dr = 2 \left[ \frac{r^4}{4} \right]_0^1 = 2 \times \frac{1}{4} = \frac{1}{2} \ Then, we subtract the square of the mean: \ \sigma^2 = \frac{1}{2} - \left(\frac{2}{3}\right)^2 = \frac{1}{2} - \frac{4}{9} = \frac{9}{18} - \frac{8}{18} = \frac{1}{18} \ Finally, taking the square root gives us the standard deviation: \ \sigma = \sqrt{\frac{1}{18}} = \frac{1}{\sqrt{18}} = \frac{1}{3\sqrt{2}} \ The standard deviation is therefore \( \frac{1}{3\sqrt{2}} \) meters, indicating the spread of seed distances around the mean.