1. **State the problem:** Find the constant $k$ for the probability density function (pdf) of the random variable $Z$ given by $$f(z) = \begin{cases} kze^{-z^2} & z > 0 \\ 0 & z \leq 0 \end{cases}$$ and draw its graph. 2. **Recall the rule for pdf:** The total area under the pdf curve must equal 1: $$\int_{-\infty}^{\infty} f(z) \, dz = 1$$ Since $f(z) = 0$ for $z \leq 0$, we only integrate from 0 to $\infty$: $$\int_0^{\infty} kze^{-z^2} \, dz = 1$$ 3. **Evaluate the integral:** Use substitution $u = z^2$, so $du = 2z \, dz$ or $z \, dz = \frac{du}{2}$. Rewrite the integral: $$\int_0^{\infty} kze^{-z^2} \, dz = k \int_0^{\infty} ze^{-z^2} \, dz = k \int_0^{\infty} e^{-u} \frac{du}{2} = \frac{k}{2} \int_0^{\infty} e^{-u} \, du$$ 4. **Integrate:** $$\int_0^{\infty} e^{-u} \, du = \left[-e^{-u}\right]_0^{\infty} = 1$$ 5. **Solve for $k$:** $$\frac{k}{2} \times 1 = 1 \implies k = 2$$ 6. **Final pdf:** $$f(z) = \begin{cases} 2ze^{-z^2} & z > 0 \\ 0 & z \leq 0 \end{cases}$$ 7. **Graph description:** - The graph starts at 0 when $z=0$. - It rises to a peak and then decays exponentially as $z$ increases. - The shape is similar to a right-skewed bell curve starting at zero.