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

1
vote
0answers
7 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
11 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
18 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
20 views

calculate median for following data [on hold]

class interval F 45-49 14 ...
1
vote
1answer
22 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
3answers
29 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
votes
0answers
15 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 ...
0
votes
1answer
17 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 ...
-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
21 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
7 views

Repeated Measures ANOVA

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
15 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 ...
0
votes
0answers
11 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) = ...
0
votes
0answers
11 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
29 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
28 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 ...
4
votes
1answer
27 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 ...
-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
30 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 ...
5
votes
1answer
39 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. ...
1
vote
1answer
31 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, ...
5
votes
3answers
59 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
21 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
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 ...
0
votes
3answers
50 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 ...
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 ...
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
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?
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 ...
0
votes
0answers
23 views

How many poker hands until statistically significant winner

How many poker hands do I have to play to determine a statistically significant winner? What is the best approach to get a 95% confidence interval? To give some more context: I have been building a ...
0
votes
1answer
16 views

Summatory problem | Ordinary least square estimator

How I can transform the first expression in the second? \begin{align} \hat{\beta}_{1} & =\frac{n\sum X_{i}Y_{i}-\sum X_{i}\sum Y_{i}}{n\sum X_{i}^{2}-\left(\sum X_{i}\right)^{2}} \\ & = ...
0
votes
1answer
42 views

Rumor and probability

31 people in a community, a person has a rumor to a second which, in turn, repeats to the third, etc. At each step the person receiving the rumor is randomly chosen among 30 people available. a) Find ...
0
votes
2answers
39 views

Two urns, A and B, each with two drawers.

If you have two urns, A and B, each with two drawers. The urn A has a gold coin in a drawer and a silver coin in the other drawer while the urn B has a gold coin in each drawer. An urn is chosen at ...
1
vote
0answers
20 views

Wick's theorem: Classical Version. Derivation question.

I am trying to prove the classical version of Wick's theorem: For a set of random variables ${a_i}$, with covariance matrix $M$ and $\rho(\vec a)$ a Gaussian probabilitiy density: $$\langle a_j a_k ...
-1
votes
0answers
4 views

assumptions of t-test of correlated terms [on hold]

Good Day ,i just want to ask if what are the assumptions of t-test of correlated means.I've been browsing the internet since yesterday because im just so curious if what does this statisticall tool ...
1
vote
0answers
30 views

Uniformly boundedness of convolutions

Assume $X$ is an absolutely continuous random variable with pdf $f:\mathbb{R}\to[0,\infty)$. Assume further there exists $M>0$ s.t. $|f(t)|\leq M \quad\forall t\in\mathbb{R}$. Let $X_1,\dots,X_n$ ...
0
votes
1answer
18 views

Probabilty of Drawing Specific Cards

If I have a deck of randomized cards and I draw 5 cards, what is the probability that I will draw at least one 2 and at least one 3. In other words, I am looking for any hands of the form x2xx3, ...
1
vote
0answers
30 views

What is the intuition behind slovin's formula?

Here is the formula. $n=\dfrac{N}{1+Ne^2}$ I don't understand why this equation works. What value does $1+Ne^2$ represent?
0
votes
0answers
5 views

Fitting power law to existing integral

I have empirical data - people from cities - a certain number of people for a certain number of cities. I know the exact number of cities, as well as the exact number of total people - e.g. the ...
1
vote
1answer
35 views

“Well known properties” of Poisson distribution

I'm working with Bradley Efron (2010): Large Scale Inference and my question concerns the proof of Lemma 2.3. Here we have $z_i \sim F_0$ with probability $\pi_0$, $z_i \sim F_1$ with probability ...
0
votes
1answer
35 views

Finding Probability of picking one ball out of N balls.

presented with n identical balls, one with a prize in it. Picks each ball out idependently one at a time till gets prize. I need to find the mean and variance of the number of balls needed to pick ...
0
votes
1answer
6 views

Success ratio improvement

I got the following question from a friend: Suppose the success ratio improved from 98% to 99.5%. What is the ratio of the improvement? This is the answer I gave him which he deemed correct: 99.5% ...
0
votes
0answers
44 views

Product of +1 and -1 with all combinations

I am looking for an algorithm or a smart way to do this in excel. I have this table. ...
0
votes
0answers
24 views

cluster sampling methods and stratifying sampling method

Briefly discuss one similarity and one difference between cluster sampling methods and stratifying sampling method.
3
votes
1answer
28 views

Expected value of trials to obtain a red ball in a box of white balls.

I have a problem that involves a box containg N balls, one of which is red and the rest of which (N-1) are white. The question involves finding the expected value and variance for the number of trials ...
0
votes
0answers
37 views

Where can I find this definition of “expected value”?

I need bibliography or some text about this definition: "Define the expected value of a function by: $E_{t}(x(t))=(\frac{1}{t})\int_{0}^{t} x(s)ds$. " I think that it's statistics or functional ...
1
vote
2answers
49 views

Solve $\frac{1}{2^\theta}\sum_{k=0}^{\theta} {\theta\choose k} \delta(k)=\theta$ for $\delta$

The following arises in unbiased estimation of a parameter for the binomial distribution, but that information is not needed for solving the question. I tried solving this by manipulating the sum to ...
0
votes
1answer
24 views

Biased sample from biased sample

A webpage has users, where each user has a number of projects uniquely assigned to him or her. I want a random sample of users by randomly sampling projects and then taking the users connected to this ...