<Dd> M X (t) = E (e t X) = ∫ − ∞ ∞ e t x f X (x) d x . (\ displaystyle M_ (X) (t) = \ mathbb (E) \! \ left (e ^ (tX) \ right) = \ int _ (- \ infty) ^ (\ infty) e ^ (tx) f_ (X) (x) \, dx .) </Dd> <P> This is consistent with the characteristic function of X being a Wick rotation of M (t) when the moment generating function exists, as the characteristic function of a continuous random variable X is the Fourier transform of its probability density function f (x), and in general when a function f (x) is of exponential order, the Fourier transform of f is a Wick rotation of its two - sided Laplace transform in the region of convergence . See the relation of the Fourier and Laplace transforms for further information . </P> <P> Here are some examples of the moment generating function and the characteristic function for comparison . It can be seen that the characteristic function is a Wick rotation of the moment generating function M (t) when the latter exists . </P> <Table> <Tr> <Th> Distribution </Th> <Th> Moment - generating function M (t) </Th> <Th> Characteristic function φ (t) </Th> </Tr> <Tr> <Td> Bernoulli P (X = 1) = p (\ displaystyle \, P (X = 1) = p) </Td> <Td> 1 − p + p e t (\ displaystyle \, 1 - p + pe ^ (t)) </Td> <Td> 1 − p + p e i t (\ displaystyle \, 1 - p + pe ^ (it)) </Td> </Tr> <Tr> <Td> Geometric (1 − p) k − 1 p (\ displaystyle (1 - p) ^ (k - 1) \, p \!) </Td> <Td> p e t 1 − (1 − p) e t (\ displaystyle (\ frac (pe ^ (t)) (1 - (1 - p) e ^ (t))) \!) ∀ t <− ln ⁡ (1 − p) (\ displaystyle \ forall t <- \ ln (1 - p) \!) </Td> <Td> p e i t 1 − (1 − p) e i t (\ displaystyle (\ frac (pe ^ (it)) (1 - (1 - p) \, e ^ (it))) \!) </Td> </Tr> <Tr> <Td> Binomial B (n, p) </Td> <Td> (1 − p + p e t) n (\ displaystyle \, \ left (1 - p + pe ^ (t) \ right) ^ (n)) </Td> <Td> (1 − p + p e i t) n (\ displaystyle \, \ left (1 - p + pe ^ (it) \ right) ^ (n)) </Td> </Tr> <Tr> <Td> Poisson Pois (λ) </Td> <Td> e λ (e t − 1) (\ displaystyle \, e ^ (\ lambda (e ^ (t) - 1))) </Td> <Td> e λ (e i t − 1) (\ displaystyle \, e ^ (\ lambda (e ^ (it) - 1))) </Td> </Tr> <Tr> <Td> Uniform (continuous) U (a, b) </Td> <Td> e t b − e t a t (b − a) (\ displaystyle \, (\ frac (e ^ (tb) - e ^ (ta)) (t (b-a)))) </Td> <Td> e i t b − e i t a i t (b − a) (\ displaystyle \, (\ frac (e ^ (itb) - e ^ (ita)) (it (b-a)))) </Td> </Tr> <Tr> <Td> Uniform (discrete) U (a, b) </Td> <Td> e a t − e (b + 1) t (b − a + 1) (1 − e t) (\ displaystyle \, (\ frac (e ^ (at) - e ^ ((b + 1) t)) ((b - a + 1) (1 - e ^ (t))))) </Td> <Td> e a i t − e (b + 1) i t (b − a + 1) (1 − e i t) (\ displaystyle \, (\ frac (e ^ (ait) - e ^ ((b + 1) it)) ((b - a + 1) (1 - e ^ (it))))) </Td> </Tr> <Tr> <Td> Normal N (μ, σ) </Td> <Td> e t μ + 1 2 σ 2 t 2 (\ displaystyle \, e ^ (t \ mu + (\ frac (1) (2)) \ sigma ^ (2) t ^ (2))) </Td> <Td> e i t μ − 1 2 σ 2 t 2 (\ displaystyle \, e ^ (it \ mu - (\ frac (1) (2)) \ sigma ^ (2) t ^ (2))) </Td> </Tr> <Tr> <Td> Chi - squared χ </Td> <Td> (1 − 2 t) − k 2 (\ displaystyle \, (1 - 2t) ^ (- (\ frac (k) (2)))) </Td> <Td> (1 − 2 i t) − k 2 (\ displaystyle \, (1 - 2it) ^ (- (\ frac (k) (2)))) </Td> </Tr> <Tr> <Td> Gamma Γ (k, θ) </Td> <Td> (1 − t θ) − k (\ displaystyle \, (1 - t \ theta) ^ (- k)); ∀ t <1 θ (\ displaystyle \ forall t <(\ frac (1) (\ theta))) </Td> <Td> (1 − i t θ) − k (\ displaystyle \, (1 - it \ theta) ^ (- k)) </Td> </Tr> <Tr> <Td> Exponential Exp (λ) </Td> <Td> (1 − t λ − 1) − 1, (t <λ) (\ displaystyle \, \ left (1 - t \ lambda ^ (- 1) \ right) ^ (- 1), \, (t <\ lambda)) </Td> <Td> (1 − i t λ − 1) − 1 (\ displaystyle \, \ left (1 - it \ lambda ^ (- 1) \ right) ^ (- 1)) </Td> </Tr> <Tr> <Td> Multivariate normal N (μ, Σ) </Td> <Td> e t T (μ + 1 2 Σ t) (\ displaystyle \, e ^ (t ^ (\ mathrm (T)) \ left (\ mu + (\ frac (1) (2)) \ Sigma t \ right))) </Td> <Td> e t T (i μ − 1 2 Σ t) (\ displaystyle \, e ^ (t ^ (\ mathrm (T)) \ left (i \ mu - (\ frac (1) (2)) \ Sigma t \ right))) </Td> </Tr> <Tr> <Td> Degenerate δ </Td> <Td> e t a (\ displaystyle \, e ^ (ta)) </Td> <Td> e i t a (\ displaystyle \, e ^ (ita)) </Td> </Tr> <Tr> <Td> Laplace L (μ, b) </Td> <Td> e t μ 1 − b 2 t 2; for t <1 / b (\ displaystyle \, (\ frac (e ^ (t \ mu)) (1 - b ^ (2) t ^ (2))); (\ text (for)) t <1 / b \,) </Td> <Td> e i t μ 1 + b 2 t 2 (\ displaystyle \, (\ frac (e ^ (it \ mu)) (1 + b ^ (2) t ^ (2)))) </Td> </Tr> <Tr> <Td> Negative Binomial NB (r, p) </Td> <Td> (1 − p) r (1 − p e t) r (\ displaystyle \, (\ frac ((1 - p) ^ (r)) (\ left (1 - pe ^ (t) \ right) ^ (r)))) </Td> <Td> (1 − p) r (1 − p e i t) r (\ displaystyle \, (\ frac ((1 - p) ^ (r)) (\ left (1 - pe ^ (it) \ right) ^ (r)))) </Td> </Tr> <Tr> <Td> Cauchy Cauchy (μ, θ) </Td> <Td> Does not exist </Td> <Td> e i t μ − θ t (\ displaystyle \, e ^ (it \ mu - \ theta t)) </Td> </Tr> </Table>

Moment generating function for a continuous random variable