<P> Note that for the case where X has a continuous probability density function ƒ (x), M (− t) is the two - sided Laplace transform of ƒ (x). </P> <Dl> <Dd> M X (t) = ∫ − ∞ ∞ e t x f (x) d x = ∫ − ∞ ∞ (1 + t x + t 2 x 2 2! + ⋯ + t n x n n! + ⋯) f (x) d x = 1 + t m 1 + t 2 m 2 2! + ⋯ + t n m n n! + ⋯, (\ displaystyle (\ begin (aligned) M_ (X) (t) & = \ int _ (- \ infty) ^ (\ infty) e ^ (tx) f (x) \, dx \ \ & = \ int _ (- \ infty) ^ (\ infty) \ left (1 + tx+ (\ frac (t ^ (2) x ^ (2)) (2!)) + \ cdots + (\ frac (t ^ (n) x ^ (n)) (n!)) + \ cdots \ right) f (x) \, dx \ \ & = 1 + tm_ (1) + (\ frac (t ^ (2) m_ (2)) (2!)) + \ cdots + (\ frac (t ^ (n) m_ (n)) (n!)) + \ cdots, \ end (aligned))) </Dd> </Dl> <Dd> M X (t) = ∫ − ∞ ∞ e t x f (x) d x = ∫ − ∞ ∞ (1 + t x + t 2 x 2 2! + ⋯ + t n x n n! + ⋯) f (x) d x = 1 + t m 1 + t 2 m 2 2! + ⋯ + t n m n n! + ⋯, (\ displaystyle (\ begin (aligned) M_ (X) (t) & = \ int _ (- \ infty) ^ (\ infty) e ^ (tx) f (x) \, dx \ \ & = \ int _ (- \ infty) ^ (\ infty) \ left (1 + tx+ (\ frac (t ^ (2) x ^ (2)) (2!)) + \ cdots + (\ frac (t ^ (n) x ^ (n)) (n!)) + \ cdots \ right) f (x) \, dx \ \ & = 1 + tm_ (1) + (\ frac (t ^ (2) m_ (2)) (2!)) + \ cdots + (\ frac (t ^ (n) m_ (n)) (n!)) + \ cdots, \ end (aligned))) </Dd> <P> where m is the nth moment . </P>

When does the moment generating function not exist