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Solve this by using TI-84 plus calculator

To solve this problem using the TI-84 Plus calculator, follow these steps to find the cumulative distribution function \( F(x) \) for the given probability density function \( f(x) \).

The given \( f(x) \) is:
\[
f(x) = 
\begin{cases}
\frac{1}{6}, & \text{if } 1 \leq x \leq 7 \\
0, & \text{otherwise}
\end{cases}
\]

### Step-by-Step Solution:

1. **Understanding the cumulative distribution function (CDF):**
   The CDF, \( F(x) \), is defined as:
   \[
   F(x) = \int_{-\infty}^{x} f(t) \, dt
   \]
   This means we need to integrate \( f(x) \) from the lower bound to \( x \).

2. **Set up the intervals for \( F(x) \):**
   - **For \( x < 1 \):** Since \( f(x) = 0 \), the integral will also be 0.
     \[
     F(x) = 0, \quad \text{if } x < 1
     \]

   - **For \( 1 \leq x \leq 7 \):** Here, \( f(x) = \frac{1}{6} \). Integrating this from 1 to \( x \) gives:
     \[
     F(x) = \int_{1}^{x} \frac{1}{6} \, dt = \frac{1}{6} (x - 1)
     \]
     Thus,
     \[
     F(x) = \frac{x-1}{6}, \quad \text{if } 1 \leq x \leq 7
     \]

   - **For \( x > 7 \):** Since \( f(x) = 0 \) beyond 7, the CDF will remain constant at its maximum value:
     \[
     F(x) = 1, \quad \text{if } x > 7
     \]

3. **Summary of the solution:**
   \[
   F(x) = 
   \begin{cases}
   0, & \text{if } x < 1 \\
   \frac{x-1}{6}, & \text{if } 1 \leq x \leq 7 \\
   1, & \text{if } x > 7
   \end{cases}
   \]

### Using the TI-84 Plus Calculator:

1. **Integrating \( f(x) \):**
   - Press the `MATH` button and select `fnInt(` for integration.
   - Input the function as `fnInt(1/6, X, 1, X)` for \( 1 \leq X \leq 7 \).

2. **Graphing \( F(x) \):**
   - Define \( Y1 = 0 \) for \( x < 1 \), \( Y2 = (X-1)/6 \) for \( 1 \leq x \leq 7 \), and \( Y3 = 1 \) for \( x > 7 \).
   - Use the `piecewise` function or manually set up different ranges for graphing.

Would you like more details or have any questions on this? Here are some related questions to consider:

1. What is the interpretation of the CDF in probability theory?
2. How would the CDF change if the interval for \( f(x) \) was different?
3. How can we verify that \( F(x) \) is indeed a valid CDF?
4. What would be the effect of scaling \( f(x) \) by a factor?
5. Can the TI-84 Plus handle cases where \( f(x) \) is not piecewise constant?

**Tip:** When working with piecewise functions, clearly separate each region to avoid confusion in calculations.

Suppose Melique has the probability density function f(x) = 1/6 if 1 ≤ x ≤ 7 and 0 otherwise. Find the cumulative distribution function of X.

Learn how to find the cumulative distribution function (CDF) for a probability density function defined as f(x) = 1/6 for 1 ≤ x ≤ 7, and f(x) = 0 otherwise. This example covers concepts such as the CDF, integration, and properties of probability distributions.

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