For questions about the moments of a random variable, that is, the expectation of $X^k$ where $X$ is a random variable and $k$ an integer (or a real number with $X$ non-negative).
8
votes
1answer
1k views
Existence of the moment generating function and variance
Can a distribution with finite mean and infinite variance have a moment generating function? What about a distribution with finite mean and finite variance but infinite higher moments?
5
votes
1answer
254 views
Proof that if higher moment exists then lower moment also exists
The $r$-th moment of a random variable $X$ is finite if
$$
\mathbb E(|X^r|)< \infty
$$
I am trying to show that for any positive integer $s<r$, then the
$s$-th moment $\mathbb E[|X^s|]$ is ...
1
vote
1answer
515 views
How can I calculate central moments of a joint pdf?
Let's say I have two signals $x_1$ and $x_2$, each having $N$ samples, i.e.:
$$ x_1 = \{ x_{11}, x_{12}, ..., x_{1N} \} $$
$$ x_2 = \{ x_{21}, x_{22}, ..., x_{2N} \} $$
The signals are both ...
19
votes
2answers
786 views
What's so 'moment' about 'moments' of a probability distribution?
I KNOW what moments are and how to calculate them and how to use the moment generating function for getting higher order moments. Yes, I know the math.
Now that I need to get my statistics knowledge ...
9
votes
4answers
1k views
A transform to change skew without affecting kurtosis?
I am curious if there is a transform which alters the skew of a random variable without affecting the kurtosis. This would be analogous to how an affine transform of a RV affects the mean and ...
7
votes
2answers
440 views
A proof involving properties of moment generating functions
Wackerly et al's text states this theorem "Let $m_x(t)$ and $m_y(t)$ denote the moment-generating functions of random variables X and Y, respectively. If both moment-generating functions exist and ...
6
votes
1answer
288 views
Central Moments of Symmetric Distributions
I am trying to show that the central moment of a symmetric distribution:
$${\bf f}_x{\bf (a+x)} = {\bf f}_x{\bf(a-x)}$$ is zero for odd numbers. So for instance the third central moment $${\bf ...
3
votes
2answers
130 views
Moments of the Kolmogorov distribution
Up to what order do the moments of the Kolmogorov distribution exist? References would be appreciated.
2
votes
1answer
225 views
Moment generating function of the inner product of two gaussian random vectors
Can anybody please suggest how I can compute the moment generating function of the inner product of two gaussian random vectors, each distributed as $\mathcal N(0,\sigma^2)$, independent of each ...
1
vote
2answers
240 views
Need an example of RV with a mean and no second moment
An example like the t-distribution with 2 degrees of freedom would not suffice as the second moment exists but equals inf.
