My Courses CPN 40338 E. Chapter 5 HW - Continuous Random Variables Curfent score: $$ 32.27 / 50 $$ pcirns "Al, Category: Homework The age of a first grader on February 4,2025 at Garden Elementary School is uniformly distributed from 4.5 to 7 yearg, We randomly select one first grader from the class. a. Define the random variable, $$ X= $$ $$ \qquad $$ A. The grade level of a randomly selected student on February 4,2025, 8 The weight of a randomly setected first grader on February 4, 2025, c. The age of a randomly selected first grader on February 4, 2025, D. The height of a randomly selected first grader on February 4,2025, - Your answer is correct. b. $$ K \sim $$ Enter your answer (2) 1 Enter your answer Enter your answer c. Graph the probabiity distribution. (Only plot the lower and upper bounds of this Uniform distribution on the number line below.)

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My Courses > CPN 40338 E. Chapter 5 HW - Continuous Random Variables Curfent score: $$ 32.27 / 50 $$ pcirns "Al, Category: Homework The age of a first grader on February 4,2025 at Garden Elementary School is uniformly distributed from 4.5 to 7 yearg, We randomly select one first grader from the class. a. Define the random variable, $$ X= $$ $$ \qquad $$ A. The grade level of a randomly selected student on February 4,2025, 8 The weight of a randomly setected first grader on February 4, 2025, c. The age of a randomly selected first grader on February 4, 2025, D. The height of a randomly selected first grader on February 4,2025, - Your answer is correct. b. $$ K \sim $$ Enter your answer (2) 1 Enter your answer Enter your answer c. Graph the probabiity distribution. (Only plot the lower and upper bounds of this Uniform distribution on the number line below.)

UpStudy AI · Accepted Answer

Great job on part **a**! Let's move on to parts **b** and **c** of your homework.

---

### **Part b: Define the Distribution of $$ X $$**

Given that the age $$ X $$ of a first grader on February 4, 2025, is uniformly distributed between **4.5** and **7** years, you can define the distribution of $$ X $$ as follows:

$$
X \sim \text{Uniform}(4.5, 7)
$$

**Explanation:**
- **Uniform Distribution** is appropriate here because every age between 4.5 and 7 years is equally likely.
- The notation $$ X \sim \text{Uniform}(a, b) $$ specifies that $$ X $$ is uniformly distributed between $$ a $$ and $$ b $$.

**So, your answer for part b is:**

$$
X \sim \text{Uniform}(4.5, 7)
$$
*or equivalently*
$$
X \sim U(4.5, 7)
$$

---

### **Part c: Graph the Probability Distribution**

To graph the **Uniform Distribution** of $$ X $$ between **4.5** and **7**, follow these steps:

1. **Draw a Number Line:**
   - Mark the lower bound at **4.5** and the upper bound at **7**.

2. **Plot the Probability Density Function (PDF):**
   - Since the distribution is uniform, the PDF is constant between 4.5 and 7.
   - The height of the PDF is calculated as:
     $$
     f(x) = \frac{1}{b - a} = \frac{1}{7 - 4.5} = \frac{1}{2.5} = 0.4
     $$
   - Draw a horizontal line at $$ y = 0.4 $$ between $$ x = 4.5 $$ and $$ x = 7 $$.

3. **Indicate the Bounds:**
   - Use solid lines or closed circles at $$ x = 4.5 $$ and $$ x = 7 $$ to show that these values are included in the distribution.

**Visual Representation:**

```
y
|
|         _________
|        |         |
|________|         |________ x
        4.5       7
```

- The horizontal line between **4.5** and **7** represents the constant probability density of **0.4**.
- The area of the rectangle equals **1**, which satisfies the total probability requirement.

**Key Points:**
- **Uniform Distribution** is represented by a rectangle on the graph.
- The **base** of the rectangle spans from the lower bound (**4.5**) to the upper bound (**7**).
- The **height** is $$ \frac{1}{b - a} = 0.4 $$.

---

**Summary of Your Answers:**

- **a.** *(Already Correct)*
  
  $$ X = $$ **C. The age of a randomly selected first grader on February 4, 2025.**

- **b.**
  
  $$
  X \sim \text{Uniform}(4.5, 7)
  $$
  
  *or*
  
  $$
  X \sim U(4.5, 7)
  $$

- **c.**
  
  *Graph a rectangle between 4.5 and 7 on the x-axis with a constant height of 0.4.*

---

If you have any further questions or need more clarification on any part, feel free to ask!
**Part b:** $$ X \sim \text{Uniform}(4.5, 7) $$ **Part c:** Graph a rectangle between 4.5 and 7 on the x-axis with a constant height of 0.4.

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