# Characteristic function vs Moment generating function

Mathematics Asked by nessy on November 9, 2020

Normal distribution $$N(mu, sigma^2$$) has the moment generating function $$m_X(t) = exp (mu t+frac{sigma^2t^2}{2})$$ and the characteristic function $$phi_X(t) = exp (i mu t-frac{sigma^2t^2}{2})$$ which looks almost the same. In fact, it satisfies the equation $$m_X(it) = phi_X(t)$$ for all $$tin mathbb{R}$$.

My question : Is there a criterion for a distribution to satisfy $$m_X(it) = phi_X(t)$$ ? I’m especially interested in continuous distributions.

I had a course on measure theory, and I’m new to probability theory. I know that moment generating function can be failed to be defined for all $$tin mathbb{R}$$. I saw some examples of moment generating functions and characteristic functions, and all of them satisfy the equation above.

## Related Questions

### How to check if two functions only touch in one point?

3  Asked on December 13, 2021

### $ker(T)^{bot} = overline{im(T^*)}$ if $T$ is a linear operator between Hilbert spaces

1  Asked on December 13, 2021 by rino

### Partial Fraction Decomposition of $frac{1}{x^2(x^2+25)}$

2  Asked on December 13, 2021 by mjoseph

### How to project (draw) a rectangle on an incline plane

1  Asked on December 13, 2021 by alexander-cska

### How to find a basis of complementary subspace of a subspace not in $mathbb R^n$?

1  Asked on December 13, 2021 by jon-g

### Equality of ideals in $mathbb{Z}[sqrt{-7}]$

1  Asked on December 13, 2021

### An orthonormal basis in a subspace.

1  Asked on December 13, 2021 by ryuta-osawa

### Is there a simple-ish function for modeling seasonal changes to day/night duration and height of the sun?

3  Asked on December 13, 2021 by saganritual

### Parallel transport on the Stiefel manifold

1  Asked on December 10, 2021 by gcc

### Remainder when $^{40}C_{12}$ is divided by $7$.

2  Asked on December 10, 2021 by yash-bhatia

### Showing that for some group it is abelian iff $x • (y • x ^{−1} ) = y$

2  Asked on December 10, 2021 by cheese12345

### Summation involving Gamma function

1  Asked on December 10, 2021 by paranoid

### Probability one random variable is less than another random variable but higher than the same other random variable with a factor

1  Asked on December 10, 2021 by sottoj

### Find x in a multiple arithmetic sequence

1  Asked on December 10, 2021 by billy-senders

### Proof that this is a Primitive Recursive function

2  Asked on December 10, 2021 by user6767509

### Convergence and absolute convergence of an infinite product of terms in $(0,1]$.

1  Asked on December 10, 2021 by lce

### Describing the set ${nin Bbb Nlvert (n>1)wedge (forall x,yin Bbb N)[(xy=n)implies (x=1lor y=1)]}$

2  Asked on December 10, 2021

### What is the permutation of choosing the just 3 balls in a pool of 16 balls?

1  Asked on December 10, 2021 by mahesh87

### Determine number of divisors of $x^n -1$ of degree $k$. Why are they such a number?

1  Asked on December 10, 2021

### why does a set of parametric equations need to be smooth and not cross itself in order to evaluate its arc length?

0  Asked on December 10, 2021 by lightbulb

Get help from others!