Linked Questions

8
votes
1answer
4k views

if 2 random variables have exactly same mean and variance [duplicate]

If two continuous random variables have exactly the same expected value and variance, do they always have the same distribution?
12
votes
5answers
10k 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 ...
16
votes
1answer
4k views

Whether distributions with the same moments are identical

Following are similar to but different from previous posts here and here Given two distributions which admit moments of all orders, if all the moments of two distributions are the same, then are they ...
12
votes
2answers
3k views

How would you explain Moment Generating Function(MGF) in layman's terms?

What is a Moment Generating Function (MGF)? Can you explain it in layman's terms and along with a simple & easy example? Please, limit using formal math notations as far as possible.
11
votes
1answer
797 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 ...
5
votes
2answers
185 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 ...
4
votes
2answers
950 views

Moment Generating Function for Lognormal Random Variable

I'm working through the proof of a lognormal random variable and am having some difficulty in moving through it. I understand the following: Our CDF is $\Phi(\frac{logx - \mu}{\sigma})$, and thus our ...
4
votes
1answer
118 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
0answers
164 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 $\text{Med}[y]$...
3
votes
1answer
89 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 ...
2
votes
0answers
133 views

Sampling from distribution known only by its moments

Is there an efficient, numerically stable algorithm to sample from a distribution for which we know only a certain (large) number of moments?