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.

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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. If this is not acceptable - administrators please delete, ...
4
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1answer
24 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 ...
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12 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 ...
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2answers
26 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 ...
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1answer
27 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. ...
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27 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, ...
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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 ...
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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?
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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 ...
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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 ...
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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 ...
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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 ...
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2answers
26 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 ...
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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 ...
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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?
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16 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 ...
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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 ...
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1answer
14 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}} \\ & = ...
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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 ...
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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 ...
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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 ...
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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 ...
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28 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 $X_1,\dots,X_n$ ...
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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, ...
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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?
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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 ...
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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 ...
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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 ...
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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% ...
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42 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. ...
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20 views

cluster sampling methods and stratifying sampling method

Briefly discuss one similarity and one difference between cluster sampling methods and stratifying sampling method.
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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 ...
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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 ...
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2answers
47 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 ...
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1answer
23 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 ...
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1answer
26 views

How can I calculate a campaign goal ongoing?

I work for a non-profit and we are continuing a campaign to serve 1000 children at $29 a month. As of today we have the funds to serve 398 children based on an average from the last 12 months. My ...
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6 views

Testing Relative Preference Fit (multi-variable)

I'm playing around with a data set that I have built which is meant to represent the subjective preferences of a person related to a particular product. I want to compare, then, the properties of a ...
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2answers
39 views

95% Confidence Interval Problem for a random sample

The sample mean of a random sample of $25$ observations is $9.6$ and the sample variance is $22.4$. Derive a $95$ confidence interval for the population mean. I calculated the following: Confidence ...
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Repeated Measures…(Analysis of Variance) [on hold]

What are within-subject variables and what are the difference scores between these variables referred to?
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9 views

Estimating Johnson Distribution Parameters by Quantile

In this paper: http://www.researchgate.net/publication/31291960_Quantile_Estimators_of_Johnson_Curve_Parameters the four parameters for the Johnson distribution ...
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11 views

the meaning of 4-wise hash function

If someone says: 4-wise independent sign (hash) functions $s_1,s_2, s_3 : [d] → \{+1, −1\}$, then what does it means? I cannot use k-wise Independence variables (the definition 1 ...
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Hypothesis Testing with given Pearson correlation.

There are two independent sets of samples , female and male. The problem is to calculate the 95 % C.I. of the mean of total( male+ female, two samples) population. -> Female : n=100, mean= 169.1, sd ...
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1answer
17 views

How to find the degrees of freedom for a chi-square variable

How does one find the degrees of freedom for a Chi-square random variable when trying to fit a distribution to a sample? I read an explanation regarding this in this source. I don't understand how to ...
3
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23 views

I want to know my hypothesis is right

Could you determine does the following formulation makes 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 Ratio ...
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1answer
11 views

Do the bounds change when multiplying a probability distribution by a constant?

Do the bounds change when multiplying a probability distribution by a constant? For example consider when an exponential distribution bounded by upper bound A, lower bound B and mean X, is multiplied ...
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50 views

Need simplified formula of probability equation

I have RV $x$ which is function of independent continuous RVs $x_1$ and $x_2$. After some manipulations, I came up with an expression for the outage probability of $x$ as $$P_\text{out}(y)=\Pr(x\leq ...
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34 views

Proving Convergence of a Derivative Free Method

Hopefully this question is on topic for the mathematics community (rather than the statistics) since the optimization method I am using relies on a statistical model. Well I am building an algorithm ...
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1answer
22 views

Univariate Normal Distribution Transformation

X is the time taken for a high school athlete to complete a 100m sprint, in seconds, which follows a normal distribution with a mean of 16. The athlete would complete the sprint under 19.495 seconds ...
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2answers
21 views

Cost per pound savings analysis between two data sets

Given 2 sets consisting of (type, cost, weight) and grouping the data into sub-sets based on type (for simplicity let's just say their are two types). How would you go about comparing these data sets ...
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32 views

black jack variance

Is it correct to compute variance in Blackjack using Bernoulli distribution? For example: $p = 41\%$ is the probability of winning $q = 9\%$ is the probability of tieing $t = 50\%$ is the ...