Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis.

learn more… | top users | synonyms

0
votes
0answers
4 views

Pearson correlation binary variables

In this paper, the Pearson product-moment correlation coefficient is stated to be used: Pearson product-moment correlation coefficient However, I don't understand how, from the Perason formula: $$\...
0
votes
1answer
13 views

What is the meaning of 'regression' in 'linear regression'?

I can see why linear regression is linear, i.e., because it is represented by a line, but what does regression have to do with the term as a whole? What is the meaning this word contributes to the ...
1
vote
1answer
11 views

If Mutual Information measures dependence, why is it symmetric?

From Wikipedia we can read: In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. ...
-1
votes
0answers
25 views

Newspaper statistical nightmare! [duplicate]

Right for a couple of days I have been suffering with this poorly worded question: The question! :) And so far I have handled the question like this but I am not sure if I am doing it properly.. i)...
2
votes
0answers
18 views

Find Probability that latency exceeds 10 ms given sample mean and variance

I am working on a statistics problem for my Engineering Statistics class. The problem goes like this: You are measuring the communications latency between two processors. You take 6 million data ...
-3
votes
0answers
48 views

Checking if a coin is fair [on hold]

I want to check if a coin is fair (lands 50% of the times on each side). I flip that coin multiple times and count the number of times it fell on heads and the number of times it fell on tails and ...
2
votes
2answers
67 views

Flipping coins- percentages of heads vs tails [on hold]

If I flip a coin multiple times and count the number of time it fell on heads and the number of times it fell on tails and keep a track of them. In how many flips on average will the delta between ...
-2
votes
1answer
28 views

The Bus Problem Related to Exponential Distribution [on hold]

There is a bus that departs from a bus stop every quarter hour from 6 am until midnight. You arrive at the bus stop between 7.10 am and 7.20 am, with the time in this interval being a uniform random ...
1
vote
1answer
23 views

Statistics - Chebychev's Inequalities

1) A box contains 100 tickets labeled with numbers. The average of the labels is -47.1 and the SD of the labels is 3.2. Sixteen tickets will be drawn independently at random with replacement from the ...
3
votes
1answer
30 views

Suggestions for Constructing a Random Variables from Correlated Observations

Let $\mathcal{X} \neq \phi $ be a finite set. Let $P_{XY_1Y_2}$ be a fixed joint distribution over $\mathcal{X}\times\mathcal{X}\times\mathcal{X}\ $ and that a random sample $(X,Y_1,Y_2 )$ is drawn ...
1
vote
0answers
12 views

Geometric mean, harmoinc mean and loss functions

Consider a sequence $(x_i)_{i \in I}$ of real numbers indexed on a set $I$. The mode of the series is the minimizing argument for the $L_0$ loss $$ \text{mode}[ \; (x_i)_{i \in I} \; ] = \arg\min_{u \...
0
votes
0answers
19 views

Method used to calculate average time spent

I need to calculate the average time spent on a single article. My data set ranges from a few seconds up to an hour. Dataset Summary .png file I'm currently using mean to calculate the average but ...
2
votes
2answers
66 views

If you roll two six-sided dice, what is the probability that the dice add to 10 or higher?

When answering these sort of questions people mostly resort to diagrams and I'm wondering if there is a way to calculate the probability without going through each outcome, just solely on the given ...
1
vote
2answers
54 views

Probability Conjecture

I think there is a flaw in my logic but I'm not sure where it would be. Let HHH denote the event of three coin flips. Let E(HHH) be the expected value of the number of coin flips until HHH. Let E(...
0
votes
1answer
51 views

An example where $E\left[\lim_{n \to \infty}X_n\right] \neq \lim_{n \to \infty}E\left[X_n\right]$

As in the title, what would be an example where $$E\left[\lim_{n \to \infty}X_n\right] \neq \lim_{n \to \infty}E\left[X_n\right]$$? with $E$ representing expectation and $X_n$ is random variable? (for ...
4
votes
1answer
33 views

probability/combinatorics question with marbles

An urn has 20 green out of 50 marbles. Draw all 50 marbles without replacement. Let X = # of green marble runs of any length. Example : GGGGBBBGGBBGBB. . . In the above example, there are 3 runs in ...
-2
votes
0answers
18 views

Confidence Interval for thirty people [on hold]

In a sample of 30 people, we have found that 12 them of have asthma. Find the 93.7 confidence interval of the population proportion.
0
votes
0answers
10 views

resource for derivation showing the computing of mutual information for normal random variables

If I have 2 correlated normal random variables, and they are not be jointly normally distributed, is there a closed form answer for their mutual information? I've seen that if two normal random ...
2
votes
1answer
37 views

Positive semidefinite ordering for covariance matrices

Suppose that X and Z are matrices with the same number of rows. Let $$ D = \left[\begin{array}{cc} X' X & X'Z \\ Z'X & Z'Z \end{array} \right]^{-1} - \left[\begin{array}{cc} (X' X)^{-1} & ...
1
vote
3answers
28 views

Jointly normal and correlated normal random variables

Is is true that if two normal random variables are correlated, then they are jointly normally distributed? I am not sure how to prove or disprove it.
3
votes
2answers
55 views

Is minimizing the squared errors optimal?

In least squares regression, we try to minimize the sum of the squares error terms. I was wondering if this would unfairly penalize a model for having terms that are too far away. For example, a term ...
-2
votes
0answers
21 views

Dependent or Independent [on hold]

In a random sample of 855 people, we have the following 4X4 tabular representation of their education level and racial background. African American Caucasian ...
2
votes
0answers
24 views

Impact of spurious regressors on out of sample prediction error

The true DGP is \begin{equation} y=\alpha_0 + \alpha_1 x_1 + \dots + \alpha_k x_k +\epsilon, \quad \epsilon\sim \mathcal{N}(0,1)\label{eq:1} \end{equation} but we instead estimate \begin{equation} y=...
0
votes
0answers
14 views

relative risk calculation when a cell is 0

the problem wants me to calculate the relative risk, but the dataset is like this 2 40 0 18 so the rr is $\frac{2}{42}/ \frac{0}{18} = \infty$. I thought ...
1
vote
1answer
30 views

How to calculate the median when each value from the data is increased by 10?

Given that a collection of scores on a quiz leads to a mean of 40, a median of 38, and a mode of 36. If we added 10 points to each score what would be the new median?
2
votes
4answers
67 views

Probability that the second throw of a fair die exceeds the first

A player throws an ordinary die and records the score $A$. The player then throws the die again and again records the score, $B$. if $B>A$ then we set a score for this player. What is the ...
3
votes
1answer
14 views

Data/Feature normalization

Let's say I have a set of random elements of the interval $(-1,1)$. $$S=\{0.03,-0.1,0.5,-0.45,...\}$$ I'm looking for a bijective function $f(x)$ which normalizes the elements of $S$ such that the ...
0
votes
0answers
49 views

Newspaper confusion and statistics.

As part of my new university course I am taking I need to do some statistics so I decided to do some practice. As I said I am new to statistics and I am currently very stuck on the following question: ...
1
vote
2answers
25 views

Probability to see all 6 numbers on a die after n throws

I am trying to work out the probability of seeing all 6 numbers on a fair die at least once after n throws, where n > 6. So I found a related question: Probability of rolling a dice 8 times before ...
4
votes
3answers
88 views

Continuous version of a Poisson R.V.

I am wondering if there is a continuous version of a Poisson random variable, that has the following two features: 1) Has a CDF that agrees with the discrete Poisson distribution on the integers, and ...
1
vote
0answers
19 views

statistical comparison, 3 groups, multiple columns

I am using R for some statistical analysis. I have a dataset listing number of deaths by eu regions. the dataset is annual and is for 2000-2008. I divided this data into 4 subgroups according to ...
0
votes
1answer
31 views

For $X,Y $ random variables, $h $ a function, show that $E (Xh(Y)|Y)=h (Y)E (X|Y) $ almost surely

Question in the title: For $X,Y $ random variables, $h $ a function, show that $E (Xh(Y)|Y)=h (Y)E (X|Y) $ almost surely My main problem is that I don't even understand what $E (Xh(Y)|Y)$ means.....
1
vote
1answer
21 views

Analogue of a “perfect” polynomial fit in $\mathbb{R}^p$, $p > 2$

I can't find a source for this right now - and it's been years since I've taken Linear Algebra - but my recollection is that in $\mathbb{R}^2$, if you have $n$ points with distinct $x$ values, there ...
0
votes
0answers
12 views

Comparing log functions of CDFs and PDFs (related to order statistics) with non-log functions of the same

Let $f$ and $F$ denote the respective pdf and cdf of a probability distribution on $\mathbb{R}$. Take any natural $n\geq3$ and any real $a$ and $c$ such that $a\leq c$, and $\rho\geq0$. We want to ...
0
votes
0answers
21 views

Predict the daily usage of Bandwidth of a Network

Context: I want to predict the daily usage of bandwidth of a network (consists a number of users) based on previous use . For example, I want to predict the amount of bandwidth during 8 pm to 9pm ...
8
votes
3answers
741 views

Is a data set really a set?

Originally I thought that in statistics, a data set is just a set of real numbers, and that was it. But in the case of a set, there can only be one instance of any given entry, e.g. in set theory $$\...
0
votes
0answers
26 views

norma distribution and log-normal distribution

I often see when people analyzing data, they assume data has either normal or log-normal distribution, and trying to fit data into a distribution for the convenience of data analysis (e.g. by ...
1
vote
0answers
30 views

How to Calculate the “Drop Off” of a Set

So I have never taken a formal class of statistics and this is likely just a case of me not knowing the right name for what I am looking for. Nonetheless, say I have a set of numbers in descending ...
-2
votes
1answer
22 views

Suppose X and Y have joint density f (x, y) = 2 for 0 < y < x < 1. Find P (X − Y > z). [on hold]

Suppose X and Y have joint density f (x, y) = 2 for 0 < y < x < 1. Find P (X − Y > z). Solution is (1-z)^2
0
votes
0answers
18 views

Matrix Calculation Significance and Multivariate Bayesian Methods

Suppose I have the matrix given by: $$X = \begin{bmatrix}1 & 0 & 0\\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{bmatrix}$$ This matrix actually represents whether a user interacted with a ...
0
votes
2answers
40 views

Show that $(\bar{X})^2$ is not an unbiased estimator for $\mu^2$

If $X_1, ... , X_n$ are $n$ identical distributed independent random variables each with mean $\mu$ and variance $1$. A little confused by this question. Is it asking for if $(\bar{X})^2$ != $\mu^2$....
1
vote
2answers
25 views

Differentiating $\int\cdots \int f(X_1,X_2,\ldots,X_n)\varphi_1(x_1,\theta)\cdots\varphi_n(x_n,\theta)~dx_1\cdots dx_n$

Differentiating:$$\int_{-\infty}^\infty \cdots \int_{-\infty}^\infty f(X_1,X_2,\ldots,X_n)\varphi_1(x_1,\theta)\cdots\varphi_n(x_n,\theta)\,dx_1 \cdots dx_n$$ with respect to $\theta$. The result is ...
0
votes
0answers
46 views

expectation and variance of an implicit estimator

Suppose the following equation holds \begin{align*} p_2=\int\limits_{-\infty}^{\Phi^{-1}(p)}\int\limits_{-\infty}^{\Phi^{-1}(p)} \frac{1}{2\pi\sqrt{1-\rho^2}}\exp\bigg({-\frac{1}{2}\frac{x^2-\rho xy+...
0
votes
1answer
22 views

How many time the standard deviation, do I need to travel from mean in both directions such that I cover a given percentage of data?

I do not have much experience in Statistics. However, I read this rule on a page and followed it up on Wikipedia: https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule I wanted to know ...
2
votes
1answer
59 views

probability of rank of a number

Suppose I have 10 sample means. I want to find the probability of rank of the population means using sample means. Therefore, I want to perform two experiments. First experiment: I pick one of the ...
0
votes
0answers
40 views

Expectation and Variance of an Estimator

Imagene following equation holds \begin{align*} p_2=\int\limits_{-\infty}^{\Phi^{-1}(p)}\int\limits_{-\infty}^{\Phi^{-1}(p)} \frac{1}{2\pi\sqrt{1-\rho^2}}\exp\bigg({-\frac{1}{2}\frac{x^2-\rho xy+y^2}{...
2
votes
0answers
53 views
+50

Help with conditional expectation of a convolution of exponential random variables

I'm working through this paper, with lots of help from all the great people on this site. Obviously my statistics/probability is a lacking to follow all the mathematical steps. Currently, I'm trying ...
0
votes
0answers
15 views

Show $G^2=2\sum o \log \frac{o}{e}$ is approximately $X^2=\sum \frac {(o-e)^2}{e}$

Show $G^2=2\sum o \log \frac{o}{e}$ is approximately $X^2=\sum \frac {(o-e)^2}{e}$ $o_i$ = observed $e_i$=expected (I removed $i$'s for ease) The solution is: $$G^2=2\sum o \log \frac{o}{e}$$ $$=2\...
2
votes
2answers
74 views

Gradient and Hessian of function on matrix domain

Let $A \in R^{k \times p}$. Define $f(X) : R^{p \times k} \rightarrow R$ to be $f(X) = \log \det(XA + I_{p})$, where $I_{p}$ is a $p \times p$ identity matrix. I want to know what is the gradient and ...
0
votes
0answers
20 views

Convolution of multiple exponential distributions

I'm trying to figure out the derivation presented on page 442 of this paper. Given a probability distribution $$f_n(t) \frac{\binom{n+1}{2}}{2N}\exp{\left(-\frac{t\binom{n+1}{2}}{2N}\right)}$$ ...