Linked Questions

31
votes
4answers
9k views

Does mean=mode imply a symmetric distribution?

I know this question has been asked with the case mean=median, but I did not find anything related to mean=mode. If the mode equals the mean, can I always conclude this is a symmetric distribution? ...
40
votes
1answer
29k 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?
21
votes
4answers
62k views

Does mean = median imply that a unimodal distribution is symmetric?

For a unimodal distribution, if mean = median then is it sufficient to say that distribution is symmetric? Wikipedia says in relationship between mean and median: "If the distribution is symmetric ...
28
votes
3answers
18k views

Proof that moment generating functions uniquely determine probability distributions

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 $...
18
votes
5answers
14k views

How is the kurtosis of a distribution related to the geometry of the density function?

The kurtosis is to measure the peakedness and flatness of a distribution. The density function of the distribution, if it exists, can be viewed as a curve, and has geometric features (such as ...
14
votes
1answer
2k views

Identity of moment-generating functions

Are there any non-identical distributions which happen to have the same moment-generating function?
6
votes
2answers
244 views

What is the famous data set that looks totally different but has similar summary stats?

There is a famous example of a collection of datasets with similar summary statistics like mean, standard deviation etc., whose visual appearances are totally different. It is named after the famous ...
11
votes
1answer
986 views

How to fit an approximate PDF (i.e.: density estimation) using the first k (empirical) moments?

I have a situation where I am able to estimate (the first) $k$ moments of a data-set, and would like to use it to produce an estimation of the density function. I already came across the Pearson ...
4
votes
1answer
137 views

Estimate probability of event using moments of a distribution or a Taylor expansion involving the moments

Let's say we have four moments $(\mu_1, \mu_2, \mu_3, \mu_4)$ of a probabilty distribution of a random variable $X$ and the goal is to get the probability $\rm{P}(X \leq t)$ for a certain value of $t$....
3
votes
1answer
155 views

Finding mode using mean and skewness (and higher moments)?

I have a pdf that doesn't yield trivial derivatives, so I cannot differentiate it and find the root to determine where its max exactly occurs. However, I have a general formula to express all its ...
1
vote
1answer
136 views

Rebuilding a signal based on mean, std, length and more

For some given signals, I have these parameters: Mean Standard Deviation Skewness Kurtosis Length (number of samples) Now I would like to know if I can rebuild the signal (an estimation) based on ...
5
votes
0answers
222 views

median and mean of the sample mean of i.i.d. log-normal

Let $y:=\frac1n\sum_{i=1}^n x_i$, where $\{x_i\}_{i=1}^n$ is a set of i.i.d. random variables, and every $x_i$ has a lognormal distribution $x_i \sim\text{Lognormal}(\mu,\sigma^2)$. Let $\text{Med}[y]$...