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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Range of a marginal density function?

Given the joint density function: $$p(x,y) = \frac{4x^3}{y^3} \text{ when } 0 < x < 1\text{ and } y > x; \text{ otherwise } p(x,y) = 0$$ How does one calculate the marginal density for $Y$? ...
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17 views

Statistics question involving exponential distribution and (maybe) gamma function

This is from a past stat exam that I am studying for my final tomorrow (lol). I believe this might have to do with gamma function. Could someone guide me through step by step of how to do this? An ...
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0answers
32 views

Intuitive explanation of “unit root” sought

I am struggling to get my head around the concept of "unit root" in relation to time series. And it would be a great help if someone could give me a two or three sentence intuitive explanation of the ...
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1answer
18 views

Suppose $40\%$ of the population possess a given characteristic … What is the probability $44\%$ or fewer possess the characteristic?

I have the following question: Suppose $40\%$ of the population possess a given characteristic. If a random sample of size $300$ is drawn from the population, then the probability that $44\%$ or ...
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1answer
35 views

Binomial distribution giving me an answer above 1?

I am doing the following question. If i have a box of $20$ soccer balls and the independent chance of a soccer ball of being flat is $0.1$. What is the probability of having at least $4$ flat soccer ...
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0answers
7 views

Distribution invariant to orthogonal transformations

Let $X,Y$ be two real valued stochastic variables and define $$ \begin{pmatrix} \tilde{X}\\ \tilde{Y} \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} X\\ Y ...
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0answers
15 views

Grading system with multiple judges

This is a practical question, I'm not sure if it's on-topic here. So sorry if it's not. There is a competition where judges decide the score competitors by summing their performance mark in several ...
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2answers
31 views

Probability of getting a parking ticket

The city of Ithaca, New York, allows for two-hour parking in all downtown spaces. Methodical parking officials patrol the downtown area, passing the same point every two hours. When an official ...
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1answer
16 views

Statistics questions (normal distribution and possibly gamma function)

This is a question from a past stat exam that I am studying because my final is in two days (lol). It'd be great if someone could guide me through how do both parts of the problem. I know gamma ...
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0answers
32 views

Distinguishable balls in distinguishable boxes?

Suppose I have $n$ distinguishable balls and $N$ distinguishable boxes. A particular configuration of this 'system' is such that there are $k$ particles in a box, b, where $1\lt b \lt N$ (i.e. the ...
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1answer
13 views

Using chi-square test for statistics with multiple options in one variable

i think that this is more mathematical question, i am doing some statistics on survey. There is questions with multiple choices, so for example, if there are 4 choices, i can pick 1st ,2nd and 4th. ...
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1answer
22 views

Random Sample vs Simple Random Sample

I am reading, just for fun, the book Essentials of Statististics of Mario Triola. I am trying to see the differences between Random Sample and Simple Random Sample. In the book I found these ...
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1answer
13 views

Correlation Query

I can find $\mu_1, \mu_2, \operatorname{Var}(Y_1) \ \text{and } \operatorname{Var}(Y_2)$ but I am not sure how to get the co-variance of $Y_1$ and $Y_2$ in order to find the correlation of $Y_1$ ...
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1answer
39 views

Using the geometric distribution to find the probability that between 4 and 6 devices will be tested

Quality control tests spark plugs until they find one that doesn't work. If the probability of a spark plug working is 0.99, what is the probability that they will test between 4 and 6 (inclusive) ...
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2answers
21 views

Adding and Subtracting Normal Distributions

I think I know how to do this, but I'm not sure. I'm just hoping to check myself here before I do a bunch of work incorrectly. Suppose you have three independent normal distributions: Distribution A: ...
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1answer
17 views

Which function will fit this curve best?

I am trying to do a test of normality on this data set here. My QQ Plot looks like this . It looked like an arctan function to me. So my idea was to do a reverse "tan" function transformation on it. ...
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1answer
21 views

Analytic solution for matrix factorization using alternating least squares

The standard form for ridge regression aims to minimize the following cost function. $$ \min\ \ \sum_i(y_i-x_i^T\beta)^2 + \lambda\sum_j\beta^2_j $$ As described here, it's possible to differentiate ...
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1answer
12 views

Density function after a transformation.

Assume $Y$ has the density function $q(y) = 3y^2$ when $y \in (0,1)$, overwhise zero. Let $Z = -log(Y)$. Then; $$P(Z \le z) = P(-log(Y) \le z) = 1 - P(Y \le e^{-z}) = 1 - F_Y (e^{-z})$$ Then I ...
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0answers
19 views

How to do sampling for the following problem. [on hold]

There are 200 students with a mathematics exam marks. According to marks students are divided into five categories 0-20,20-40,40-60,... and I want to choose two random sample with 25 for a group. ...
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1answer
23 views

Assumptions of a probability distribution

Let $X$ be a continuous real-valued random variable indicating the fragility of a firm. Suppose that the firm defaults if $X$ takes a value above a threshold $u>0$. Hence $$ Prob(X>u) $$ is the ...
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1answer
14 views

Moment generating function of a uniform random variable.

My first attempt to this question was to find the first few moments about the mean and try to rearranging the those moments to obtain the general function as desired. However, when I tried to ...
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0answers
15 views

Finding variance with method of moments

In part (a), I know Var(Ni)=p(p-1) but how do I find the variance of the estimator with this result? And for part (b) of this question, I have to clue on how to tackle this question. I don't get ...
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1answer
26 views

Sufficient conditions for monotonicity with probability distributions

Let $X_i$ be a continuous non-negative real-valued random variable and $i=1,...,n$. $X_i$ are not necessarily independent over $i$. Let $b>0$, $\delta>0$. Consider $$ ...
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0answers
34 views

What distribution would describe this?

I start with 100 eggs, 10 of them being broken. I randomly select eggs without replacement until they are all split into baskets of 10 eggs each. Here's what I know: Best case scenario all 10 bad ...
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1answer
19 views

Statistics; Normal distribution question

I'm not sure if I am solving this question correctly A used-car dealership has found the length of time before a major repair is required on car it sells is normally distributed. Witha mean = 10 ...
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0answers
17 views

How do I find percentiles of data sets (Even vs odd)?

Given the following data set with an even number of values: $100, 100, 105, 113, 129, 132, 146, 152, 176, 200$ The value representing the 30th percentile, using the formula n(p/100) where n = sample ...
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1answer
25 views

Does Binomial variables independence implies Bernoulli variables independence

$X$, $Y$ are independent variables with Binomial distribution. $X={\sum_{i=1}^nX_i}$, $Y={\sum_{i=1}^nY_i}$. $X_i$, ($1\le i\le n$) are independent Bernoulli variables. Same applies for $Y_i$ Is the ...
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1answer
36 views

Probability of a point from one normal distribution being higher than a point from another independent normal distribution

Given two independent normal distributions: Distribution 1: Mean $= 23.95$, SD $= 7.44$ Distribution 2: Mean $= 16.29$, SD $= 7.79$ How often on average will a point from Distribution 2 be greater ...
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1answer
19 views

Finding method of moments

Question: In order to solve find the method of moments estimator, I know I need to first find the expected value of Y. But after finding $E(Y)= \frac{\theta}{2}$, what should I do next?
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0answers
12 views

Confidence level using unknown median

This is a general question. Lets say we are giving a sample space n. The sample space has an unknown median The order statistics of each person, $**X_{i}**$ in sample space is measured (lets say ...
3
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1answer
26 views

Calculating the probability of winning roulette after x bets

I'm going through all of my homeworks to study for my final and I'm getting hung up on this one problem I never figured out... A roulette wheel has 38 slots, numbered 0, 00, and 1 through 36. If you ...
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0answers
22 views

standard deviation and adjusted R-squared for simultaneous regressions

I am conducting a study that requires two steps of statistical estimation. First, I run a regular OLS regression, from which I gather three outputs that I need: coefficient values standard ...
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0answers
27 views

Statistical research [on hold]

I'm preparing for an exam and I came across this question in a book. Below, I understand that we have to assumed a split plot in time analysis with 3 way factorial with 3 drug $\times$ 2 sex $\times$ ...
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0answers
46 views

Statistical research question [on hold]

I'm preparing for an exam and I came across this question in a book. An experiment is done to examine ways to detect phlebitis during the intravenous administration of a particular drug. Phlebitis ...
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1answer
36 views

When using the Central Limit Theorem, how to scale the mean and variance depending on the number of samples?

So I'm reviewing my notes for the central limit theorem for my final and I'm getting hung up on one detail. The two questions below both utilize the central limit theorem, but they use it in ...
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0answers
26 views

case deletion formula restricted least square estimator [on hold]

hi any one please help me to find out a case deletion formula for restricted least square estimator? $$ \hat\beta = (X' X)^{-1} X'y-(X' X)^{-1} R' [R' (X' X)^{-1} R]^{-1} R(X' X)^{-1} X'y $$ i need a ...
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2answers
23 views

How to show U1 and U2 are independent (Stats)

Prompt: Assume that $Y_1, Y_2, Y_3$ and $Y_4$ are independently and identically distributed $N(\mu,\sigma^2)$ random variables. Show that $Y_1 + Y_2 – Y_3 – Y_4$ and $Y_1 – Y_2 + Y_3 – Y_4$ are ...
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2answers
27 views

explain how this confidence interval is correct [on hold]

An srs of $16$ households is selected in Houston and the number of remote controls is counted. we are interested in a $99\%$ confidence interval for the population mean number. In the sample, the mean ...
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2answers
38 views

Markov chain steady state existence

Is it possible for a Markov chain to have no steady state solution ? What is an example of such system ?
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2answers
14 views

Stats probability addition rule, multination rule

The directions are to calculate the following probability based on drawing cards without replacement from a standard deck of 52. What is the probability of drawing a 2 or a king on the first draw and ...
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2answers
28 views

Defining median for discrete distribution

In probability theory, a median of a probability distribution is a number $M$ such that the CDF of this distribution $F_\xi(x)$ satisfies $F_\xi(M)=\frac{1}{2} \tag1$ This works for continuous ...
2
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1answer
35 views

Number of samples needed to get a given expected distance

Suppose I have a surface in $\mathbb{R}^3$ with surface area $A$. How many points do I need to (uniformly at random) sample so that the expected distance from each point to its nearest neighbor is ...
2
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2answers
33 views

Standard deviation of $Z = 9 - 3Y^3$

If Y is a random variable with probability mass function: Y | Pr(Y = y) -1 | 0.4 0 | 0.5 1 | 0.1 I need to find the standard deviation of $Z = 9 - 3Y^3$. ...
1
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1answer
40 views

Approximate using the central limit theorem

This is a homework question in a class I'm TAing, and I just wanted to make sure I'm not doing anything wrong, before I conclude for sure that the book has an error. Suppose $S_n$ is binomially ...
0
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1answer
24 views

Poisson Distribution and sample mean

I am currently working on some error analysis homework and I am having trouble understanding some basic concepts. In particular I don't understand if there is any difference between the sample mean, ...
1
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1answer
19 views

Normal approximation of Poisson Distribution

Hi currently studying for a final exam and I just want to confirm my approach/answers to this problem are correct: Suppose that $X \sim \mathrm{Poisson}$. We wish to test $H_0: \lambda = 50$ vs ...
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1answer
15 views

inequality for real-valued Gaussian sums

I saw the following Lemma in an article: Let $\mathbf{b}\in \mathbb{R}^N$ be fixed, and let $\mathbf{\epsilon}\in \mathbb{R}^N$ be a random vector whose N entries are i.i.d. random variables drawn ...
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1answer
21 views

Mean and Variance of probabilities [on hold]

For a certain commodity which you buy, you can make either a $500$ profit with probability $0.5$ when you sell it, or $200$ with probability $0.3$ or lose $100$ with probability $0.2$. a. Find the ...
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0answers
9 views

Confusion with QQ plots

I've got some difficulty understanding QQ plots and the specific slope and intercept of the line which the data will approximate if it really is generated from the distribution we're comparing with. ...
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0answers
35 views

The efficiency property of estimator for second moment. [on hold]

Please help to improve the efficiency property of estimation for second moment. Statistical population is normally distributed. Sorry for my bad english, if something is wrong. Thanks in advance.