Question

Each individual in a population of size N is, in each period, either active or inactive. If an individual is active in a period then, independent of all else, that individual will be active in the next period with probability α. Similarly, if an individual is inactive in a period then, independent of all else, that individual will be inactive in the next period with probability β. Let Xn denote the number of individuals that are inactive in period n.

Answer 1

The number of individuals that are inactive in each period can be modeled using a binomial probability distribution with a probability of success equal to the probability of an individual being inactive. The expected number of individuals that are inactive in period n is µ = nβ.

Step-by-step explanation:

The number of individuals that are inactive in each period can be modeled using a binomial probability distribution. Let's denote the number of individuals that are inactive in period n as Xn. Each individual can either be active or inactive, and we want to find the number of individuals that are inactive. The probability of an individual being inactive in a period n is β. To find the expected value of Xn, we can use the formula µ = np, where µ is the mean, n is the number of trials, and p is the probability of success. In this case, success is being inactive, so p = β. Therefore, the expected number of individuals that are inactive in period n is µ = nβ.

Answer 2

The number of inactive individuals in each period can be obtained by multiplying the previous number of inactive individuals by the probability β. This analysis assumes that the probabilities α and β remain constant throughout the periods.

Step-by-step explanation:

In this scenario, we have a population of size N, where each individual is either active or inactive in each period. If an individual is active in one period, they have a probability of α to be active in the next period. Conversely, if an individual is inactive in one period, they have a probability of β to remain inactive in the next period.

To analyze this situation, let's define Xn as the number of individuals that are inactive in period n. Here's a step-by-step explanation:

1. In the first period (n = 0), we start with a certain number of individuals who are either active or inactive. Let's assume that there are N1 individuals who are active and N2 individuals who are inactive.

2. In period n = 1, the individuals who were active in period n = 0 will remain active with a probability of α. So, N1 * α individuals will be active in period n = 1.

3. Similarly, the individuals who were inactive in period n = 0 will remain inactive with a probability of β. So, N2 * β individuals will be inactive in period n = 1.

4. The total number of individuals in period n = 1 will be the sum of the active and inactive individuals: N1 * α + N2 * β.

5. In general, for any period n, the number of inactive individuals, Xn, can be calculated as follows:

• X0: Initial number of inactive individuals (given).
• Xn = Xn-1 * β: The number of inactive individuals in period n is equal to the number of inactive individuals in period n-1 multiplied by the probability β.

To summarize, the number of inactive individuals in each period can be obtained by multiplying the previous number of inactive individuals by the probability β. This analysis assumes that the probabilities α and β remain constant throughout the periods.