Questions tagged [statistics]
Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory and other branches of mathematics such as linear algebra and analysis.
2,334
questions
7
votes
3
answers
3k
views
Ways of getting a number with $n$ dice, each with $k$ sides
Assume the dice are numbered from $1$ to $k$. My hunch is that this will form a normal distribution with a median at $n\cdot\frac{k}{2}$. However, I have no idea as to
turn this fact into an answer (...
- 1,776
7
votes
1
answer
6k
views
Joint distribution of range $(R=X_n-X_1)$ and mid-range $(V=\frac{1}{2}(X_1+X_n))$ order statistics
Let $X_1,X_2, \ldots , X_n$ be independent and identically distributed Uniform
random variables on the interval (0, a) for a > 0, each having a density function
$f(x) = \frac{1}{a}$, $0<x<a$. ...
- 929
5
votes
2
answers
9k
views
Finding the maximum likelihood estimators for this shifted exponential PDF?
Consider a random sample $X_1, X_2, \dots, X_n$ from the shifted exponential PDF
$$f(x; \lambda, \theta) = \begin{cases}\lambda e^{-\lambda(x-\theta)} ;& x \geq \theta\\
\theta ...
- 697
3
votes
2
answers
831
views
Differential Entropy
I'm a little temporarily confused about the concept of differential entropy. It says on wikipedia that the differential entropy of a Gaussian is $\log(\sigma\sqrt{2\pi e})$. However I was thinking as $...
- 26.1k
64
votes
17
answers
16k
views
Is the Law of Large Numbers empirically proven?
Does this reflect the real world and what is the empirical evidence behind this?
Layman here so please avoid abstract math in your response.
The Law of Large Numbers states that the average of the ...
- 767
25
votes
3
answers
87k
views
How do I combine standard deviations of two groups?
I have 2 groups of people. I'm working with the data about their age. I know the means, the standard deviations and the number of people. I don't know the data of each person in the groups.
Group 1 :
...
- 253
23
votes
3
answers
57k
views
Distribution of $-\log X$ if $X$ is uniform.
For $X$ and $Y$ random variables; $X$ follows the uniform distribution.
(1): if $Y=-\log X$
(2): then it can be shown that $-\log X$ is distributed as $\exp(1)$ {i.e. exponential with mean 1}.
Why ...
- 567
17
votes
6
answers
7k
views
Arithmetic mean. Why does it work?
I've been using the formula for the arithmetic mean all my life, but I'm not sure why it works.
My current intuition is this one:
The arithmetic mean is a number that when multiplied by the number ...
- 1,760
16
votes
2
answers
22k
views
Derivation of Mode of grouped data
A formula to calculate the mode for grouped data's is given in my text book:
Mode = $l + \dfrac{(f_1 - f_0)h}{2f_1 - f_0 - f_2} $
Where, $l = $ lower limit of the modal class,
$h = $ size of the class ...
- 163
15
votes
2
answers
16k
views
Distribution of Sum of Discrete Uniform Random Variables
I just had a quick question that I hope someone can answer.
Does anyone know what the distribution of the sum of discrete uniform random variables is?
Is it a normal distribution?
Thanks!
- 321
14
votes
2
answers
17k
views
Prove that the sample median is an unbiased estimator
My book says that sample median of a normal distribution is an unbiased estimator of its mean, by virtue of the symmetry of normal distribution. Please advice how can this be proved.
- 1,327
12
votes
2
answers
11k
views
Joint density of order statistics
I need some help to understand the following proposition (mainly to understand how it is proven):
Let $Y_1,Y_2...,Y_n$ be $n$ random variables which are independent, identically distributed random ...
- 1,567
12
votes
1
answer
10k
views
Are squares of independent random variables independent?
If X and Y are independent random variables both with the same mean (0) and variance, how about $X^2$ and $Y^2$? I tried calculating E($X^2Y^2$)-E($X^2$)E($Y^2$) but haven't been able to get anywhere.
- 123
11
votes
1
answer
34k
views
Finding mode in Binomial distribution
Suppose that $X$ has the Binomial distribution with parameters $n,p$ . How can I show that if $(n+1)p$ is integer then
$X$ has two mode that is $(n+1)p$ or $(n+1)p-1?$
- 999
8
votes
2
answers
12k
views
Probability density function of a product of uniform random variables
Let $z = xy$ be a product of two uniform random variables, with $x$ having the range $[a, b)$ and $y$ the range $[c, d)$.
What is the probability density function of $z$, and how is it calculated?
- 191