Math Problem Statement
Solution
The problem presents the cumulative distribution function (CDF) of a continuous random variable . To find the expected value (mathematical expectation) of , we first determine the probability density function (PDF) by differentiating with respect to .
Given:
\begin{cases} 0, & x \leq 0, \\ \frac{x^2}{16}, & 0 < x \leq 4, \\ 1, & x > 4. \end{cases}$$ ### Steps: 1. **Differentiate $$F(X)$$:** The PDF $$f(x)$$ is the derivative of $$F(X)$$ with respect to $$x$$: $$f(x) = \frac{d}{dx} F(X).$$ Differentiating piecewise: $$f(x) = \begin{cases} 0, & x \leq 0, \\ \frac{d}{dx} \left( \frac{x^2}{16} \right) = \frac{x}{8}, & 0 < x \leq 4, \\ 0, & x > 4. \end{cases}$$ So, the PDF is: $$f(x) = \begin{cases} \frac{x}{8}, & 0 < x \leq 4, \\ 0, & \text{otherwise}. \end{cases}$$ 2. **Expected Value ($$\mathbb{E}[X]$$):** The expected value is given by: $$\mathbb{E}[X] = \int_{-\infty}^\infty x f(x) \, dx.$$ Since $$f(x)$$ is nonzero only in the range $$0 < x \leq 4$$, we compute: $$\mathbb{E}[X] = \int_0^4 x \cdot \frac{x}{8} \, dx.$$ 3. **Simplify and Compute the Integral:** $$\mathbb{E}[X] = \frac{1}{8} \int_0^4 x^2 \, dx.$$ The integral of $$x^2$$ is: $$\int x^2 \, dx = \frac{x^3}{3}.$$ Evaluate the definite integral: $$\int_0^4 x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^4 = \frac{4^3}{3} - \frac{0^3}{3} = \frac{64}{3}.$$ Substitute back: $$\mathbb{E}[X] = \frac{1}{8} \cdot \frac{64}{3} = \frac{8}{3}.$$ ### Final Answer: $$\mathbb{E}[X] = \frac{8}{3}.$$ --- Would you like further explanation or a breakdown of any step? Here are five related questions to explore: 1. What is the variance of $$X$$ for this distribution? 2. How is the cumulative distribution function related to the PDF? 3. Can you derive $$F(X)$$ if the PDF $$f(x)$$ is given? 4. How does the range of $$X$$ affect the expected value calculation? 5. What happens to $$\mathbb{E}[X]$$ if $$f(x)$$ is scaled by a constant? **Tip:** Always verify the validity of the PDF by ensuring the total probability integrates to 1 over its domain.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Cumulative Distribution Function (CDF)
Probability Density Function (PDF)
Expected Value
Integration
Formulas
f(x) = d/dx F(x)
E[X] = ∫(x * f(x)) dx
Theorems
Differentiation of CDF to find PDF
Expected Value Formula for continuous random variables
Suitable Grade Level
College-level (undergraduate)
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