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

3
votes
1answer
54 views
+50

Is this alternative hypothesis valid?

Could anyone check that the alternative hypothesis is making sense? I wanted to prove that the "Mahalanobis distance ($\mathbf{(x_i - \bar{x})^T \Sigma^{-1}(x_i - \bar{x})}$)" is a Log Likelihood ...
3
votes
2answers
56 views

Proof of multivariate regression plane maximizes correlation in normals

I am doing a homework sheet as practice for an upcoming course in multivariate statistics and been stuck on the following problem: Let ...
0
votes
1answer
17 views

T distribution with n degrees degrees of freedom

I would like to prove that $\displaystyle \frac{\bar{X}\,\sqrt{n}\,}{\hat σ^2}\sim t_{n}$. Note that x~N(0,$σ^2$) and they are iid. Could someone explain why $\displaystyle ...
0
votes
1answer
302 views

How To Solve for Percentage When The Only Given Values Are Mean and Standard Deviation

How To Solve for Percentage When The Only Given Values Are Mean and Standard Deviation For Example: The scores of students in Mathematics examination is normally distributed with a mean of 60 and a ...
1
vote
1answer
348 views

Finding the regression line given the mean, correlation and std dev of x and y.

So we 100 observations for (x, y). The mean of x is 1.06, and for y it is 3. The std dev is 0.52 for x and for y it is 1.13. the correlation between x and y is 0.89 In the question we are told to: ...
2
votes
2answers
464 views

Definition of white noise vectors

As is shown in wikipedia: Click [here] (http://en.wikipedia.org/wiki/White_noise#Mathematical_definitions) A random vector (that is, a partially indeterminate process that produces vectors of real ...
0
votes
1answer
4k views

Calculating variance of Discrete Uniform distribution when its interval changes

I am not excited about grading exams. I would rather jam a dull stick into my leg. I will therefore randomly assign your grade by picking an integer uniformly from 77 to 100 (inclusively). ...
0
votes
1answer
12 views

inequality probability between order statistics of two independent distribution

Suppose we have two independent distributions $F_1$ and $F_2$ and from each distribution, we draw $k$ variables. Let us represent the $k$ i.i.d. variables from $F_1$ as $\{X_1, X_2, \ldots, X_k\}$. ...
1
vote
1answer
15 views

Standard deviation of the product of gaussians

What is the standard deviation of the product of two random variables that each have Gaussian Distributions? I don't even know where to begin on this problem.
1
vote
1answer
19 views

Wilcoxon signed-rank test

While reading Wikipedia, and my teacher's notes I found that Wilcoxon signed rank test for $n>10$ is given like below: Under null hypothesis, W follows a specific distribution with no simple ...
-7
votes
1answer
75 views

I'm taking a statistics class right now, and I get stuck on these problems. [on hold]

For a random variable $W$ where $P(W = 0) = 0.1$ and $P(W = 1) = 0.2$ and the density of $W$ for values between $0$ and $1$ is $f(w) = 1.4w$, draw a graph of the CDF. Is this a valid probability ...
2
votes
1answer
18 views

Statistical bias and the probability of an outcome.

A town referendum has occurred. The question posed to voters was YES or NO on a local law. There were 3 methods of voting: Electronic machine (voting booths), absentee ballot, and affidavit ballot. ...
0
votes
1answer
13 views

Derivation of t(n-1) distribution

While trying to prove that $\displaystyle \frac{\bar{X}\,-\,\mu}{S/\sqrt{n}}\sim t_{n-1}$ I came across a manipulation that I can not seem to understand the reasoning behind it. Why does ...
0
votes
2answers
24 views

Why is uncertainty in mean less

Any measurement, say length of any object, will have some errors. The random errors that are present in the measurement can be reduced if we take mean of a large number of samples. This is because the ...
-1
votes
1answer
23 views
-1
votes
3answers
43 views

Statistics (unsure how to do it)

A person's resting heart rate is the lowest number of heart beats per minute when fully relaxed and without distractions. Age, fitness, genetics, health status and gender affect the resting heart ...
1
vote
1answer
27 views

Calculating conditional probability of discrete uniform r.v.

X is a discrete uniform random variable on $\{a, a+1, a+2, ... , b\}$ with mean 7 and variance 4. Find $Pr[X \leq 6| X > 4]$ I'm not familiar with the discrete uniform distribution. I was ...
0
votes
1answer
18 views

Application of Law of Large numbers (1)

If we have an i.i.d random variable $X_i$ with mean variance $(\mu, \sigma^2)$. By Law of Large number, we have $\bar{X}\rightarrow^p \mu$. But can we use Law of large number as well and claim that ...
-1
votes
0answers
20 views

Not sure what formula to use? (what to solve for?)

The question states, "The weight of people in a certain pacific island is normally distributed with a mean of 175 lb. and a standard deviation of 33 lb. They want to design a one-person canoe that ...
-2
votes
0answers
10 views

Cumulative Distribution Function Proof with Discrete Random Variables [on hold]

Let $Y$ be a discrete random variable on the natural numbers $0,\, 1,\, 2,\, 3,\, 4,\ldots$ Let $F_y$ its CDF. Prove that $E[Y] = \sum_{k=0}^\infty(1-F_y(k))$
1
vote
0answers
26 views

Sampling substrings of a beaded necklace to determine the necklace composition

I have a necklace composed of 100 beads, where each bead is one of 13 colors. If I am only able to look at one 4 bead sub-sequence at a time (connected, as they would be on the necklace) , how many ...
0
votes
0answers
32 views

Difference between the “Hazard Rate” and the “Killing Function” of a diffusion model?

I posted this question on Cross Validated - but I think it applies here too. Also, it increases the chances of the question being answered. Link here If this is not acceptable - administrators ...
0
votes
2answers
41 views

Partition-based entropy of a sequence

The entropy $H$ of a discrete random variable $X$ is defined by $$H(X)=E[I(X)]=\sum_xP(x)I(x)=\sum_xP(x)\log P(x)^{-1}$$ where $x$ are the possible values of $X$, $P(x)$ is the probability of $x$, ...
5
votes
1answer
41 views

Why do polynomial regressions have larger variance at the end?

In reading the book "An Introduction to Statistical Learning with Applications in R", I came across this graph: It shows that the point-wise variance is larger at the ends of the regression curve. ...
0
votes
0answers
8 views

Repeated Measures ANOVA [on hold]

What is the model equation for one-way repeated measures ANOVA? Is that for the 2-way rANOVA similar to the 2-factor experiment with interaction term present in the model equation as follows: Yijk = ...
0
votes
0answers
13 views

Why are the Real and Imaginary parts of a field in k-space uncorrelated?

I am in the process of generating a (real) Gaussian random field $\delta(\vec{x})$ from a given power spectrum $P(k)$. The way I define the power spectrum is, in Fourier space, $\left\langle ...
1
vote
1answer
30 views

Determining bounds for change sum of continuous r.v.'s

I'm trying to understand how to determine the bounds when computing the sum of continuous random variables. Here is a sample question: X and Y have the following joint pdf: $f_{X,Y}(x,y) = 4xy, 0 ...
0
votes
0answers
15 views

Divergence based robust inference

I have learnt that the inference based on minimizing the following divergence is robust to outlying observations for some specific range of $\alpha\in\mathbb{R}$. $$D_{\alpha}(g,f) = ...
3
votes
1answer
501 views

Maximum likelihood estimators, hypergeometric and binomial

I'm trying to solve a two part problem. The set up is as follows: consider a bag with $\theta$ red marbles and $7-\theta$ blue marbles, with $\theta$ being unknown. Let $x$ denote the number of red ...
1
vote
1answer
33 views

3-sigma approximation

I am making a system involving a sensor who has to be really precise. I found on their datasheet a diagram that shows the typical performance of the sensor. There's the mean value, the +3 sigma, ...
0
votes
1answer
343 views

Purpose of curly braces in Statistics? (Programming in R)

I'm fairly new to Statistics and have just started Programming in R. I'm trying to write the following expression in R ready to ...
4
votes
1answer
28 views

sufficient statistics of a sequence of normal random variable

If $X_1, X_2\ldots,X_n$ are independent variables with $X_i \sim \mathcal N(i\theta,1)$, $\theta$ is an unknown parameter. What is a one dimensional sufficient statistic $T$ of this sample? I have a ...
1
vote
1answer
320 views

Numerical calculation of fisher information

I am trying to obtain numerically the fisher information. Given a likelihood function $$ f(X,\theta),$$ with $X \in [0,1]$. The fisher information is given by $$ ...
-2
votes
0answers
13 views

What course would you recomment to take? Elementary Statistical Methods or Introduction to Biostatistics? [on hold]

I am required to take either or for my major (Nursing). What course would you recommend? Which one is easier? What are the pros and cons about choosing one over the other. What exactly are each course ...
0
votes
2answers
31 views

If $x_i$ is from a random sample is $Var(\bar x \mid x_i)=0$?

If $x_i$ is from a random sample, is the conditional variance of the mean (or the sum of squares, really any statistic based on $x$) just treated as a constant? I saw this in a OLS variance of a ...
3
votes
2answers
2k views

Same mean, different standard deviation in data sets

How would a data set containing the values of a variable with a mean of 50 and a standard deviation of 3 compare with another data set containing the same variable, but a mean of 50 and a standard ...
0
votes
0answers
33 views

Normal distribution of $t$.

I have a population with $10000$ individuals, I took $10$ to my samples. The values that I have inside $\bar{X}$ are: $$\begin{matrix} 70.22502193 & 70.26042017 \\ 70.28977621 & 70.2905717 ...
0
votes
0answers
19 views

what is the intuition behind the SRSWOR formula?

I earlier asked about Slovin's Formula, and learned shortly thereafter that it was derived from this formula. $n=\dfrac{n_0}{1+\dfrac{n_0}{N}}$, Where $n_0=\dfrac{z^2p(1-p)}{e^2}$. So, breaking it ...
5
votes
3answers
60 views

Doubt about a probability excercise

I'm a statistics teacher at a college. One day a student came with a doubt about an exercise about probability. The text goes like this: A person has two boxes $A$ and $B$. In the first one has ...
1
vote
1answer
22 views

Double integral proving that a function is a probability density

If $$g(x,y)=f(x+y)/(x+y)$$ for $x,y>0$ and $$\int_0^{\infty} f(z) \, dz = 1$$ How do you show that $$\int_0^{\infty} \int_0^{\infty} \frac{f(x+y)}{x+y} dx \, dy = 1$$ as well?
0
votes
2answers
28 views

Games and statistics

Three individuals A, B and C alternate in contention of a game according to the following rules: A plays with B and the winner plays with C. The game continues until one of the individuals to win two ...
0
votes
3answers
52 views

Intuitive meaning of the probability density function at a point

I understand how to integrate probability density functions to find probability within a certain range. However, what I don't understand is what it would mean to set the variable (say x or y) to a ...
2
votes
2answers
3k 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 ...
0
votes
0answers
11 views

SSE distribution in simple linear regression

I'm looking at the typical simple linear regression model $Y_i = \beta_0 + \beta_1X_i + \epsilon_i$, where there $\epsilon_i$s are iid $N(0, \sigma^2)$ random variables. We have unbiased estimates ...
0
votes
2answers
27 views

defective component and probability

An electronics industry uses three plants from A, B and C in the ratio of $3$ to $2$ to $1$. However $1\%$ of components manufactured by A, $2\%$ of components manufactured by B and $3\%$ of ...
4
votes
3answers
1k views

What does relaxing the iid assumptions mean? Intuitive and technical perspectives.

I believe the most restrictive assumption we can place on a series of observations is that they are iid. It is possible to relax these assumptions. For example relaxing the independent distribution ...
0
votes
0answers
5 views

Concentration bounds on Pearson correlation matrix

I am interested in (rather sharp if not the finest) tail/concentration bounds for the Pearson correlation matrix: let $X_1,\ldots,X_N \sim \mathcal{N}(0,1)$ be correlated random variables; let ...
1
vote
1answer
9 views

Binomial Distribution formula

If $X\sim B(n,p)$, show that $P(X=r+1)=P(X=r) \cdot \frac{p(n-r)}{q(r+1)}$ for $r=0,1,...,n-1$ My attempt, $P(X=r+1)={_n}C_{r+1}(p)^{r+1}(1-p)^{n-(r+1)}$ How to proceed then?
3
votes
2answers
907 views

Expressing “Probability that #successes is an even number” mathematically

Needing a little help with my probability concept. Here's the question: An urn contains $10$ red balls, $20$ green balls and $30$ blue balls. Each trial consists of drawing a ball from the urn ...
0
votes
0answers
17 views

Calculate pdf and cdf of exponential random variable

I am not looking for an answer as this is for a homework exercise i just want to be pointed in the right direction so i can learn how to do it. The question is: Let X ∼ Exp(λ). Calculate the cdf and ...