# Linked Questions

11 questions linked to/from Identity of moment-generating functions
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?
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 ...
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 ...
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.
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 ...
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 ...
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 ...
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 $t$....
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 $\text{Med}[y]$...
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 ...
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?