Let $X$ be a real random variable with cumulative distribution function $F_X(x) := {\bf P}(X \leq x)$. Show that $$\displaystyle {\bf E} e^{tX} = \int_{\bf R} (1-F_X(x)) t e^{tx}\ dx$$ for all $t > 0$. If $X$ is nonnegative, show that $$\displaystyle {\bf E} X^p = \int_0^\infty (1-F_X(x)) p x^{p-1}\ dx$$ for all $p>0$.

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Exercise 1.38 (Moment-generating function)

Let $X$ be a real random variable with cumulative distribution function $F_{X} (x) : = P (X \leq x)$ . Show that

E e^{tX} = \int_{R} (1 - F_{X} (x)) t e^{tx} dx

for all $t > 0$ . If $X$ is nonnegative, show that

E X^{p} = \int_{0}^{\infty} (1 - F_{X} (x)) p x^{p - 1} dx

for all $p > 0$ .