To construct the probability distribution, we'll calculate the probability \( P(X = x) \) for each value of \( X \), where \( X \) is the number of days and \( P(X = x) \) is the probability that a randomly chosen person would wait exactly \( x \) days.

### Probability Calculation:
The probability for each value of \( X \) is given by:
\[
P(X = x) = \frac{\text{Frequency of } X = x}{\text{Total number of observations (N)}}
\]

### Probability Distribution Table:

| **Number of Days (X)** | **Frequency (f)** | **Probability \( P(X = x) \)** |
|------------------------|-------------------|--------------------------------|
| 0                      | 27                | \( \frac{27}{823} \)           |
| 1                      | 443               | \( \frac{443}{823} \)          |
| 2                      | 257               | \( \frac{257}{823} \)          |
| 3                      | 70                | \( \frac{70}{823} \)           |
| 4                      | 12                | \( \frac{12}{823} \)           |
| 5                      | 14                | \( \frac{14}{823} \)           |

Let’s calculate the exact probabilities.### Probability Distribution:

| **Number of Days (X)** | **Frequency (f)** | **Probability \( P(X = x) \)** |
|------------------------|-------------------|--------------------------------|
| 0                      | 27                | 0.0328                         |
| 1                      | 443               | 0.5383                         |
| 2                      | 257               | 0.3123                         |
| 3                      | 70                | 0.0851                         |
| 4                      | 12                | 0.0146                         |
| 5                      | 14                | 0.0170                         |

This table shows the probability distribution for the number of days people wait before seeking medical treatment.

Would you like further analysis on this distribution or have any additional questions?

### Follow-up Questions:
1. How do these probabilities compare to a uniform distribution?
2. What is the cumulative distribution function (CDF) for this data?
3. How would adding more days affect the probabilities?
4. What is the expected value using this probability distribution?
5. Could this distribution be modeled with a known probability distribution (e.g., Poisson)?
6. What is the median number of days based on this distribution?
7. How would the mode be determined from this distribution?
8. What does this distribution tell us about people's behavior regarding medical treatment?

**Tip:** The sum of all probabilities in a probability distribution must equal 1. This is a good way to check if the distribution is correct.

Learn how to construct a probability distribution using frequency data. Includes step-by-step calculation and analysis of probabilities for different values. Suitable for undergraduate students and anyone interested in probability theory.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation