a. Prove that (kappa_{4}=mu_{4}^{prime}-4 mu_{3}^{prime} mu_{1}^{prime}-3left(mu_{2}^{prime}ight)^{2}+12 mu_{2}^{prime}left(mu_{1}^{prime}ight)^{2}-6left(mu_{1}^{prime}ight)^{4}). b. Prove that (kappa_{4}=mu_{4}-3 mu_{2}^{2}), which is often called the
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a. Prove that \(\kappa_{4}=\mu_{4}^{\prime}-4 \mu_{3}^{\prime} \mu_{1}^{\prime}-3\left(\mu_{2}^{\prime}ight)^{2}+12 \mu_{2}^{\prime}\left(\mu_{1}^{\prime}ight)^{2}-6\left(\mu_{1}^{\prime}ight)^{4}\).
b. Prove that \(\kappa_{4}=\mu_{4}-3 \mu_{2}^{2}\), which is often called the kurtosis of a random variable.
c. Suppose that \(X\) is an \(\operatorname{Exponential}(\theta)\) random variable. Compute the fourth cumulant of \(X\).
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