Question: Let \(y_{X}(t)=\ln \left[M_{X}(t)\right]\), where \(M_{X}(t)\) is a mfg. The function \(y_{X}(t)\) is called the cumulant generating function of \(X\), and the value of the \(r\) th derivative evaluated at \(t=0\), \(\mathrm{K}_{r}=\psi_{X}^{(y)}(0)\), is called the \(\mathrm{r}\) th cumulant of \(\mathrm{X}\)

1. Show that \(\mu=\psi_{X}^{\prime}(0)\).
2. Show that \(\sigma^{2}=\psi_{X}^{\prime \prime}(0)\) "
3. Use \(y_{X}(t)\) to find \(\mathrm{m}\) and \(\mathrm{s}^{2}\) for the random variable with \(\mathrm{pdf}_{,} f_{X}(x)=e^{-(x+2)}\) if \(-2

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