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The problem presented involves a discrete probability distribution where the number of daily reports of family violence in a community is described by a random variable \( X \), with the probability distribution given in the table. We will analyze the following parts:

### a) Describe the approximate probability model for the average number of reports over a 30-day month.
This involves finding the expected value \( E(X) \) of the daily reports and the variance \( \text{Var}(X) \), which will help describe the behavior of the daily average over a 30-day period.

We calculate the **expected value** \( E(X) \) using the formula:
\[
E(X) = \sum_{i} x_i \cdot p_i
\]
Where \( x_i \) represents the number of reports, and \( p_i \) is the corresponding probability from the table.

Similarly, the **variance** \( \text{Var}(X) \) is calculated as:
\[
\text{Var}(X) = \sum_{i} p_i \cdot (x_i - E(X))^2
\]

For the 30-day period, the **average number of reports** will approximately follow a normal distribution (according to the Central Limit Theorem), with mean \( E(X) \) and variance \( \frac{\text{Var}(X)}{30} \).

### b) What is the probability that the average number of reports over 30 days is between 2.6 and 3.3?
This involves calculating the probability \( P(2.6 \leq \overline{X}_{30} \leq 3.3) \), where \( \overline{X}_{30} \) is the average number of reports over 30 days. We approximate this using the normal distribution found in part (a), where the mean is \( E(X) \) and the standard deviation is \( \frac{\sqrt{\text{Var}(X)}}{\sqrt{30}} \).

Let me calculate the necessary values based on the probability distribution in the table.### Results:

- The **expected value** \( E(X) \) of the number of daily reports is \( 3.0 \).
- The **variance** \( \text{Var}(X) \) is \( 1.2 \).
- The **standard deviation** for the average number of reports over 30 days is \( 0.2 \).

#### a) Probability model description:
The approximate model for the average number of daily reports over a 30-day month can be described as following a **normal distribution** with a mean \( \mu = 3.0 \) and standard deviation \( \sigma = 0.2 \).

#### b) Probability that the average number of reports is between 2.6 and 3.3:
We can now calculate the probability \( P(2.6 \leq \overline{X}_{30} \leq 3.3) \) assuming the average follows a normal distribution with mean \( 3.0 \) and standard deviation \( 0.2 \).

Let me compute that next.The probability that the average number of reports over 30 days falls between 2.6 and 3.3 is approximately **91.04%**.

### Summary:
- **Expected daily average**: 3.0 reports
- **Variance**: 1.2
- **Standard deviation for 30 days**: 0.2
- **Probability** that the 30-day average lies between 2.6 and 3.3: 91.04%

Would you like more details or have any questions?

Here are 5 follow-up questions you might find useful:
1. How would the model change if the variance was larger?
2. Can we interpret this normal approximation for shorter periods, say 7 days?
3. How can we calculate confidence intervals for the average number of reports?
4. What if the probabilities were skewed toward lower or higher numbers?
5. How does the Central Limit Theorem apply to this problem?

**Tip**: Whenever working with averages over repeated trials, the Central Limit Theorem is your friend in approximating the distribution!

1) Un investigador social ha logrado establecer que el número de denuncias diarias por agresión familiar en una comunidad es una variable aleatoria X cuya distribución de probabilidades es descrita por la siguiente tabla:

| xi | 1 | 2 | 3 | 4 | 5 |
| pi | 0.1 | 0.2 | 0.4 | 0.2 | 0.1 |

 a) Describa el modelo de probabilidad aproximado del número promedio de denuncias durante un mes de 30 días.
 b) ¿Con qué probabilidad un promedio de denuncias de 30 días ocurre entre 2.6 y 3.3?

Learn how to calculate the expected value, variance, and normal distribution approximation for daily family violence reports. This problem involves analyzing the probability distribution of reports and determining the likelihood of a 30-day average between 2.6 and 3.3.

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