a. Prove that

\[\kappa_{5}=\mu_{5}^{\prime}-5 \mu_{4}^{\prime} \mu_{1}^{\prime}-10 \mu_{3}^{\prime} \mu_{2}^{\prime}+20 \mu_{3}^{\prime}\left(\mu_{1}^{\prime}ight)^{2}+30\left(\mu_{2}^{\prime}ight)^{2} \mu_{1}^{\prime}-60 \mu_{2}^{\prime}\left(\mu_{1}^{\prime}ight)^{3}+24\left(\mu_{1}^{\prime}ight)^{5} .\]

b. Prove that \(\kappa_{5}=\mu_{5}-10 \mu_{2} \mu_{3}\).

c. Suppose that \(X\) is an \(\operatorname{Exponential}(\theta)\) random variable. Compute the fifth cumulant of \(X\).