Math Problem Statement
Solve this by using TI-84 plus calculator
Solution
To solve this problem using the TI-84 Plus calculator, follow these steps to find the cumulative distribution function for the given probability density function .
The given is:
\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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Probability
Cumulative Distribution Function (CDF)
Integration
Formulas
f(x) = 1/6 if 1 ≤ x ≤ 7, f(x) = 0 otherwise
F(x) = ∫(from -∞ to x) f(t) dt
F(x) = 0 for x < 1, F(x) = (x - 1)/6 for 1 ≤ x ≤ 7, F(x) = 1 for x > 7
Theorems
Fundamental Theorem of Calculus
Properties of Probability Distribution Functions
Suitable Grade Level
Undergraduate (Introduction to Probability or Statistics)
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