 Solution

ID: past-exam-of-the-probability-course-of-the-university-of-oxford/2015/1/b/i/solution

X

M_{X} (t) = E [e^{tX}]

for real

t

for which this expected value is finite. Because the integrand is nonnegative, the extended expected value is always defined, but it may equal

+ \infty

. We always have

M_{X} (0) = 1

; the usual uniqueness and convergence theorems require finiteness on an open interval containing zero. When this condition holds, derivatives at zero recover the moments,

M_{X}^{(r)} (0) = E [X^{r}]

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