1. **Problem:** Given the probability density function (pdf) $$f(x) = \begin{cases} 0.075x + 0.2 & 3 \leq x \leq 5 \\ 0 & \text{otherwise} \end{cases}$$ Find the constant and verify it is a valid pdf. 2. **Formula and rules:** - The total area under the pdf must equal 1: $$\int_{-\infty}^{\infty} f(x) \, dx = 1$$ - Since $f(x) = 0$ outside $[3,5]$, we only integrate over $[3,5]$. 3. **Calculate the integral:** $$\int_3^5 (0.075x + 0.2) \, dx = \int_3^5 0.075x \, dx + \int_3^5 0.2 \, dx$$ Calculate each part: $$\int_3^5 0.075x \, dx = 0.075 \cdot \frac{x^2}{2} \Big|_3^5 = 0.075 \cdot \frac{25 - 9}{2} = 0.075 \cdot 8 = 0.6$$ $$\int_3^5 0.2 \, dx = 0.2 \cdot (5 - 3) = 0.4$$ 4. **Sum the integrals:** $$0.6 + 0.4 = 1.0$$ 5. **Conclusion:** The total integral equals 1, so $f(x)$ is a valid pdf. **Final answer:** The pdf is valid as given.