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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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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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 ...
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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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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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91 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 ...
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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 ...
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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 ...
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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: ...
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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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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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318 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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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 $x_{(n)} = \max_i x_i$ ...
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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
261 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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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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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 ...
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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: ...
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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 ...
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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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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 ...
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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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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) ...
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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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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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895 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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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 ...
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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
246 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. ...
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128 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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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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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 ...
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1answer
200 views

Finding the standard error of mean

Suppose you have a process $x_{t} = \mu +w_{t} -0.8*w_{t-1}$ where $w_{t}$ ~ $wn(0,\sigma_{w}^2)$. How do I calculate the standard error of $var(\bar{x})$ for estimating the mean. I know: ...
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1answer
186 views

autocovariance function for nonlinear time series

I want to find the autocovariance function of $y_{t} = \exp(x_{t})$ which is a nonlinear time series. I am assuming $x_{t}$ is stationary with mean $\mu$ and covariance $\gamma(h)$. When I try to ...
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2answers
426 views

Transformation of beta distribution into gamma distribution

How can I convert a Beta Distribution to a Gamma Distribution? Strictly speaking, I want to transform parameters of a Beta Distribution to parameters of the corresponding Gamma Distribution. I have ...
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1answer
1k views

Can I use ANOVA when I have negative values in my data?

I was trying to analyze a data with negative values. Is it possible to use ANOVA in this case?
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116 views

Texas Hold 'Em textbook question

"Texas Hold-em is a poker game in which players are each dealt two cards face down (called your hole or pocket cards), from a standard deck of 52 cards, followed by a round of betting, and then five ...
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2answers
212 views

Probability distribution function of the length of an interval taken from a uniform probabilty distribution.

This is a consequence of my suggested solution to this question. Consider the probability distribution function that is uniform over the interval $[-a,a]$: $$F(x)=\begin{cases} 0 & x \leq -a\\ ...
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1answer
165 views

Textbook question about a weird lottery

The 10,000 tickets for a lottery are numbered 0000 to 9999. A four-digit winning number is drawn and a prize is paid on each ticket whose four-digit number is any arrangement of the number drawn. For ...
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4answers
628 views

Another textbook problem on probability

Here's a problem in my textbook: "From a set of 2n + 1 consecutively numbered tickets, three are selected at random without replacement. Find the probability that the numbers of the tickets form an ...
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1answer
690 views

How do I find the number of bit strings with 3 consecutive 0s in a bit string of length n?

Say n is 8. How would I ever solve this problem? I've Googled around and searched this site but I haven't come up with much. I'm not even looking for the answer necessarily, just the process by ...
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1answer
88 views

Stationary time series

Suppose $V(t)= \frac{1}{2q+1}\sum_{j=-q}^q X(t-j)$ where $X_t = b_o + b_{1}*t + w_t$ where $w_t$ is ~$N(0,1)$. I know the mean is simply bo + b1t but when I calculate the autocovariance function I get ...
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50 views

Time Series: existence of moments $\Rightarrow$ existence of distribution?

This might come to you as a bit silly, because normally we are used to the vice-versa question. But here is what I have: a nonlinear time-series model, for which I can derive by infinite backwards ...
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251 views

Challenging combinatorial problem with balls and boxes

There are $N$ boxes. I want to distribute $(j\cdot D)$ balls into $N$ boxes, where $j$ and $D$ are arbitrary natural numbers. However, more than $(D-1)$ balls is not allowed to be put in each box. ...
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7k views

“Normalize” values to sum 1 but keeping their weights

I am not really sure what this operation might be called, but I have some numbers, for example: 40 10 I need to format these numbers so that they form the sum 1, but they should keep their ...
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45 views

how to check system is linear or nonlinear by simulation?

I am trying to understand whether the given equation is linear or not? or which part of the equation is linear? and how can I search which kind of approximation is suitable for this by simulation? ...
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1answer
160 views

Is Lottery probability really the same for all combos?

http://justwebware.com/uklotto/uklotto.html Test run quickpick Test run 1,2,3,4,5,6 Test run (single digit,teens,twenties,twenties,thirties,forties) 1000 times or more each cycle for as many ...
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1answer
102 views

Quantile in normal distribution

Let's think that we have a random variable $X \sim N(0,1)$. And is it true that quantile(let's call it z) z($\alpha$/2) = - z($1- \alpha/2$)?