MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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### Statistical Analysis and Probability Distributions: Understanding Binomial Distribution

---

#### Problem Context:
A survey was conducted in a town where each family had four children. The objective is to understand the distribution of the number of girls in these families.

#### Given Data:
- **Random Variable \(X\)**: Number of girls born to a randomly selected couple in a town with four children.
- The probability distribution for the random variable \(X\) is given, representing the number of girls (0, 1, 2, 3, or 4).

| \( X \) | P( \( X = x \) ) |
|--------|------------------|
|   0    |      0.0625      |
|   1    |      0.2500      |
|   2    |      0.3750      |
|   3    |      0.2500      |
|   4    |      0.0625      |

#### Tasks:

1. **Calculate the Mean**:
   - The mean number of girls in the family is \( 2.0 \).

2. **Calculate the Standard Deviation**:
   - The standard deviation for the number of girls is \( 1.00 \).

3. **Probability Histogram**:
   - You are provided with four histogram options to visually represent the probability distribution of the random variable \(X\).

---

#### Detailed Description of Histogram Options:

- **Option A**:
  - Probability values are represented on the vertical axis.
  - Number of girls is represented on the horizontal axis (from 0 to 4).
  - The heights of the bars correspond to the probabilities 0.0625, 0.2500, 0.3750, 0.2500, and 0.0625 respectively from 0 to 4.

- **Option B**:
  - Similar to Option A but with an incorrect probability representation.

- **Option C**:
  - Similar to Option A but with an incorrect probability representation.

- **Option D**:
  - Similar to Option A but with an incorrect probability representation.

---

#### Correct Histogram Selection:
- The correct histogram should reflect the probabilities 0.0625, 0.2500, 0.3750, 0.2500, and 0.0625 for the values 0, 1, 2, 3
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Transcribed Image Text:### Statistical Analysis and Probability Distributions: Understanding Binomial Distribution --- #### Problem Context: A survey was conducted in a town where each family had four children. The objective is to understand the distribution of the number of girls in these families. #### Given Data: - **Random Variable \(X\)**: Number of girls born to a randomly selected couple in a town with four children. - The probability distribution for the random variable \(X\) is given, representing the number of girls (0, 1, 2, 3, or 4). | \( X \) | P( \( X = x \) ) | |--------|------------------| | 0 | 0.0625 | | 1 | 0.2500 | | 2 | 0.3750 | | 3 | 0.2500 | | 4 | 0.0625 | #### Tasks: 1. **Calculate the Mean**: - The mean number of girls in the family is \( 2.0 \). 2. **Calculate the Standard Deviation**: - The standard deviation for the number of girls is \( 1.00 \). 3. **Probability Histogram**: - You are provided with four histogram options to visually represent the probability distribution of the random variable \(X\). --- #### Detailed Description of Histogram Options: - **Option A**: - Probability values are represented on the vertical axis. - Number of girls is represented on the horizontal axis (from 0 to 4). - The heights of the bars correspond to the probabilities 0.0625, 0.2500, 0.3750, 0.2500, and 0.0625 respectively from 0 to 4. - **Option B**: - Similar to Option A but with an incorrect probability representation. - **Option C**: - Similar to Option A but with an incorrect probability representation. - **Option D**: - Similar to Option A but with an incorrect probability representation. --- #### Correct Histogram Selection: - The correct histogram should reflect the probabilities 0.0625, 0.2500, 0.3750, 0.2500, and 0.0625 for the values 0, 1, 2, 3
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