1. **State the problem:** We are given a probability density function (pdf) defined as $$f(x) = \begin{cases} 0 & x \leq 0 \\ 1 & 0 < x < 1 \\ 0 & x \geq 1 \end{cases}$$ We need to show that this is a valid probability density function. 2. **Recall the properties of a pdf:** - The function must be non-negative for all $x$. - The total area under the pdf over the entire real line must be 1, i.e., $$\int_{-\infty}^{\infty} f(x) \, dx = 1$$ 3. **Check non-negativity:** From the definition, $f(x)$ is either 0 or 1, so $f(x) \geq 0$ for all $x$. 4. **Calculate the integral over the entire real line:** Since $f(x) = 0$ outside $(0,1)$, the integral reduces to $$\int_{-\infty}^{\infty} f(x) \, dx = \int_0^1 1 \, dx = [x]_0^1 = 1 - 0 = 1$$ 5. **Conclusion:** Since $f(x)$ is non-negative everywhere and integrates to 1 over the real line, it satisfies the conditions of a probability density function. **Final answer:** The given function $f(x)$ is a valid probability density function.