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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2answers
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Linear Models - Regression Analysis

As a student learning Applied Regression Analysis, I come from a background with very little information about this topic. I understand that given $y = \beta_0 + \beta_1x_1 + \epsilon$ $E(y\mid x) = ...
2
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1answer
126 views

Are coordinate projections in the Skorokhod space continuous?

I was wondering whether coordinate projections in the Skorokhod space $D[0,1]$ are actually continuous (and, if so, how can this be proven)? many thanks for any comments/ideas. cheers!
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1answer
80 views

explanation of least squares

Can someone explain me how to start this problem: method of least squares fit, $y=Bx$ to the following $n=6$ points $(3,4),(1,2),(5,4),(6,8),(3,6),(4,5)$ I have calculated $x$ mean and $y$ mean, ...
0
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1answer
133 views

Can I use $P(x_1+x_2+…+x_n<k)$ to calculate $P(y_1+x_2+…+x_n<k)$?

Suppose that we have $n$ independent Bernoulli random variables, $x_1,\ldots,x_n$ such that each $x_i$ takes value 1 with success probability $p_i$ and value $0$ with failure probability $1-p_i$ ...
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1answer
665 views

What is the difference between a polynomial regression and a generalized linear model?

I have seen that a polynomial linear regression can have this form: $y = c_0 + c_1 x_1 + c_2 x_2 + \dots + c_k x_k $ but I have read that the general lineal model which is a form of the multiple ...
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1answer
379 views

Bounding the variance of an unbiased estimator for a uniform-distribution parameter

$X_1,\ldots,X_6$ is a sample from a uniform distribution $ \left[ 0, \theta \right] $, $\theta$ is $[1,2]$. Find an unbiased estimator for $\theta$ with variance less than $\dfrac{1}{10}$. I thought ...
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1answer
89 views

Help with understanding the notation?

I'm studying time series and in one of my resources I encountered the following: The linear model described by $\text{(2.1)}$ above can be conveniently written in a more general notation by ...
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3answers
51 views

Calculate the Variance

This is what the question that I'm having trouble with: I know how to calculate the variance in general but I'm not sure how they got this:
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0answers
124 views

Expected value of picking smallest k random variables out of n

The question is, I'm picking random variables from a normal distribution with mean $\mu$ and standard deviation $\sigma$. If I'm picking n random variables and sorting them in the ascending order, ...
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1answer
931 views

How would I go about finding the probability of one of these happening, but not the other two?

Say the probability of event A happening is 0.3, event B is 0.2, event C is 0.3, the probability of (A and B) is 0.15, (A and C) is 0.2 and (B and C) is 0.22, and (A and B and C) is 0.05. What's the ...
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1answer
48 views

Did I do this probability question right?

Say the probability of event A happening is 0.3, event B is 0.2, event C is 0.3, the probability of (A and B) is 0.15, (A and C) is 0.2 and (B and C) is 0.22, and (A and B and C) is 0.05. Is the ...
3
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1answer
431 views

Calculating Total Field Goals Using Poisson

With the Super Bowl around the corner, I've been attempting to price various prop bets. My question is, can I use the Poisson distribution to calculate the odds of the total number of field goals ...
2
votes
1answer
92 views

How do I use student's-t distribution without the sample size?

Here is my question (homework obviously): A sample from a normal population produced variance 4.0. Find the size of the sample if the sample mean deviates from the population mean by no more than 2.0 ...
3
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2answers
3k views

Poisson distribution, mean time and probability of waiting

We know that average number of planes landing at particular airport during an hour is 36. a) what is the mean time for waiting for the first landing during an hour? b) find probability of waiting ...
0
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1answer
72 views

Linear optimization of line under points.

I have a set of points in two-dimensional space. I'd like to find the line of best fit that minimizes the distance between the line and the points such that the line is below all points. How can I ...
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0answers
155 views

How to model mutual independence in Bayesian Networks?

It's well known that 3 random variables may be pairwise statistically independent but not mutually independent, for an illustration see: example pairwise vs. mutual relations. Can mutual ...
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1answer
465 views

What probability distribution formula to use?

I need help in solving the following problems. Homeland Security and missile defense technology make it paramount that we be able to detect incoming projectiles or missiles. To make the defense ...
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1answer
23 views

How would you model subjective opinions like “how fast time passes”?

I was thinking about modelling mathematically (or finding the mathematical model) of the following experiemnt. Think about students $S_i$ saying how fast or slow they feel the lecture has passed - and ...
3
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5answers
370 views

Proving $e^\lambda$

I wish to show that $\lim_{n\to\infty}\left(1+\frac{\lambda}{n}\right)^n=e^\lambda$ as part of the proof of the Poisson limit theorem. My first avenue of attack was to take the log of both sides and ...
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1answer
1k views

Show that $\frac{X+1}{n+2}$ is a biased estimator of the binomial parameter $\theta$

The problem is: Show that $\frac{X+1}{n+2}$ is a biased estimator of the binomial parameter $\theta$. Is this estimator asymptotically unbiased? My way of doing this is following: Calculate ...
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1answer
90 views

Hypothesis testing, find out critical value $c$

I'm struggling with the following: "A certain offer is available for maximum 2 days. The sellsman claims, the offer will be over at a random date. You think, the probability is decreasing linear from ...
3
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2answers
1k views

What is a sampling density? Why is the sampling density proportional to $N^{\frac{1}{p}}$?

I'm reading a book named The Elements of Statistical Learning by Hastie, in section 2.5, Local Methods in High Dimensions, it says that the sampling density is proportional to $N^{\frac{1}{p}}$, where ...
5
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2answers
173 views

Improper integral of $\frac{x}{e^{x}+1}$

The improper integral of $\frac{x}{e^x-1}$ (along the positive real line) comes up in a lot of places, you can even invoke the Riemann-zeta and Gamma functions to solve it nicely. However, I just ...
0
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1answer
327 views

When to use likelihood ratio test? [closed]

I have a few questions regarding the use of likelihood ratio test in a logistic regression model. Suppose we have a logistic regression model like this: ...
0
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1answer
80 views

Expectation of independent normal distribution

I have a question attached below, and I really do't know how to solve it. Is there anyone can help me with this? $x,y\sim N(0,1)$, independent, what is $E(x\mid x+y=1)$, what about variance? What I ...
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1answer
4k views

Normal probability distribution with absolute value of X

Random variable X has a normal distribution N(30,5) find $P(|X| > 25)$ Having this I started to solve it normal way: $$P(|X| > 25) = 1 - P(|X| \le 25) $$ Now, normalize: $$1-P(|X| \le 25) = ...
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1answer
315 views

Find the joint distribution of two dependent, discrete random variables

A coin is tossed 4 times. Let X be the number of heads in the first two tosses and Z be the number of heads in the last three tosses. Describe the joint distribution of (X,Z) by means of a table. I ...
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2answers
424 views

Upper bound for the absolute value of an inner product

I am trying to prove the inequality $$ \left|\sum\limits_{i=1}^n a_{i}x_{i} \right| \leq \frac{1}{2}(x_{(n)} - x_{(1)}) \sum\limits_{i=1}^n \left| a_{i} \right| \>,$$ where $x_{(n)} = \max_i x_i$ ...
3
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2answers
162 views

Is this proof correct? Textbook has no solution.

Most of the problems in my textbook have numeric solutions in the back of the book except the proofs. Is this proof correct? Prove that if $A \cup B$ and $A \cap B$ are independent events, then ...
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2answers
260 views

How to break apart this sum?

I have a summation I need to break apart but I can't figure it out http://www.collectionscanada.gc.ca/obj/s4/f2/dsk1/tape10/PQDD_0027/MQ50799.pdf $p.15$, right after line $(3.8)$ Going from the ...
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1answer
2k views

How to calculate minimal member of set having mean and standard deviation?

Assume that we have given standard deviation and mean, and we know that our data follow normal distribution. ~ N(100, 16). What is the technique to calculate the minimal value that can appear in the ...
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1answer
40k views

What is a 'critical value' in statistics?

Here's where I encountered this word: The raw material needed for the manufacture of medicine has to be at least $97\%$ pure. A buyer analyzes the nullhypothesis, that the proportion is $\mu_0=97\%$, ...
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1answer
30 views

Have I tackled this hypothesis correctly?

Heterozygous plants with red flowers (descendants of plants with red and white flowers) should have, according to Mendel's law, descendants in the proportion $3:1$ (plants with red flowers: plant with ...
5
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0answers
91 views

Does Multiplicative Version of Azuma's Inequality Hold?

We know that there are multiplicative version concentration inequalities for sums of independent random variables. For example, the following multiplicative version Chernoff bound. Chernoff bound: ...
2
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2answers
318 views

Statistics Probability Help

I just began to take this stats course in HS and I'm a little stuck on these 2 problems below. Can anybody please help me out with the solutions? Thank you. Anything is appreciated. Let $Y$ be a ...
2
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3answers
125 views

gambling probability problem

We are given a fair coin. We start out with 5 dollars. We keep tossing the coin. If the outcome is different than the previous one, we are awarded another 5 dollars. However, we do not get anything if ...
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1answer
283 views

Maximum Likelihood fitting of truncated, mixed, two population systems (Gaussian Examples)

TLDR: I am trying to do maximum likelihood fitting of a dataset having two mixed populations, observed over a subset of their parameter space, within it to two pdfs. I include working code with ...
0
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1answer
72 views

Optimal $p$ for biased coin?

You are given a biased coin with probability $p$ of getting $H$ and $1-p$ of getting tail. Each flip is independent of another. We keep flipping the coin until we get $4$ consecutive tails. For each ...
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2answers
88 views

Bernoulli Random Variable

Let $X$ be a Bernoulli random variable with probability of success $p$. Answer the following questions. (i) Derive the formulas for the mean, the variance, and the standard deviation of X. (ii) ...
1
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1answer
143 views

Probability that two people share the same Amino Acid sequence given these parameters.

This question will sound oddly specific like it's my homework but it's not. It's my research and I could use some help thinking about it. A person has 20 amino acids to choose from in constructing a ...
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2answers
102 views

Probability problem found in textbook

"Shown is the board for a simple dice game. You roll a die and move the same number of squares (for example if your first roll is a $3$, move to the $3$ square). If you land on an arrow's tail, you ...
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2answers
285 views

Working out minimum sample size

I have just started a course in statistics and have some general questions that have arisen trying to solve the following question: A survey organisation wants to take a simple random sample in order ...
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3answers
892 views

Statistics - expected number

Suppose the probability of success obtaining heads is $0.5$, number of trials $n=10$. In a class of $100$ students, each student tosses a coin (assumed fair) $10$ times. Write down the expected number ...
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2answers
118 views

Statistical Hypothesis Testing

Does stress have an influence on the time people need to finish a task? To answer this question 2 groups of test people are compared. One of the groups ($n_1=10)$ does a task under the influence of ...
3
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2answers
165 views

CDF of $\max(x_1,x_2)+\max(x_3,x_4)$ where all $x_i$s are iid from $U[a,b]$

I am looking for the cumulative density function of the sum of two variables, which are themselves the result of a rank order process. Thus, if $x_1, x_2, x_3$ and $x_4$ are all independent draws ...
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2answers
42 views

How do i apply multinomial laws in this question?

the Question is i assume i have 15 students in class A grade obtain probaiblity = 0.3 B grade obtain probability =0.4 C grade obtain probability = 0.3 and I have this question What is the ...
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1answer
243 views

Derive true value of a number within a sample of a larger population

I'm currently without direction in this problem and need guidance as to whether what I'm attempting is possible. I'm attempting to derive the length of a specific task from employee supplied data. ...
0
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1answer
127 views

I am getting a number below zero when caluclating out two standard deviations from the mean. Is this ok?

I am not a math wizard, so please keep your response simple enough. I need to complete a statistics screening exam for a methods course later on today and I am hung up on one topic that came up during ...
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1answer
82 views

Economics and Statistics

Let $Y$ be a random variable and write $\mu = E[Y ]$. Show that $E[Y −\mu]=0$. and Let $X$ and $Y$ be random variables. Prove that $\text{Cov}[X, Y ] = E[X(Y − E[Y ])]$.
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3answers
241 views

Stein's lemma condition

(Apologies if I break some conventions, this is my first time posting!) I am working on proving Stein's characterization of the Normal distribution: for Z $\sim N(0,1)$ and some differentiable ...