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Questions tagged [statistics]

Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory and other branches of mathematics such as linear algebra and analysis.

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Below average, average, and above average - miles driven per year

Assume a vehicle is driven, on average, 12,000 miles per year. What would be considered: Way below average? Below average? Above average? Way above average? Is there some sort of formula that could ...
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Scott and Jim play a game of chance, they obtain a “fair” coin and who ever flips a head first wins. using Geometric series

Scott and Jim play a game of chance, they obtain a “fair” coin and who ever flips a head first wins, is there an advantage to going 1st? Can you please explain it using the geometric series approach?
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Proving $V(aX+b)=a^2\sigma_x^2$

I am having trouble using what is given to prove the following. I think I have figured out a way to prove this using the variance shortcut formula, but that isn't what's being asked for I think. Does ...
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Statistics $E[V]$ Problems

I have 2 statistics problems I am stuck on. Can anyone help me? Use the definition in Expression 3.13 to prove that $V(aX+b)=a^2\sigma_x^2$. [Hint: With $h(x)=aX+b$, $E[h(X)=a\mu+b$ where $\mu=E(X)$....
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Find a bound $R(n,k,\tau)$

Let $\varepsilon\in ℝ^n$ be a Gaussian $N(0,I_n)$ vector and let $A\in ℝ^{n\times n}$ be a fixed matrix. Then, Gendre (1999) inequality states that for any $x>0$, one has $$P\left\{||A\...
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15 views

Using values 0<x<1 to scale standard deviation

Basically what I am trying to do is scale standard deviation using a value that is $0<x<1$ So for example, if standard deviation $(S)$= .0001, and $y$ = resulting scaled value using the value ...
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Question about Lundberg inequality

I have a task in a subject called "Risk Theory", with which I would need some help. Since I'm really stuck with it and don't know how to start. The task is as follows: Lundberg inequality) Let ...
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27 views

How much does the Top 1% have?

Please consider the following problem and my answer to it. The answer seems to low to me. Is my answer right? Thanks, Bob Problem: The median net worth of a certain population is $97K$. To be in the ...
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Does the theory of Gittins Indices solve the Multi-armed Bandit problem?

For example, both Wikipedia and Reinforcement Learning: An Introduction (page 33) seem to claim as much, which would suggest that the problem has been solved for over 40 years. However, doing as ...
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1answer
19 views

PDF of sums of independent random variables confusion

Suppose that $X$ and $Y$ are independent continuous RVs with PDFs $f_X$ and $f_Y$ respectively. I want to find the PDF of $Z = X + Y$. The CDF of $X + Y$ is $$F_{X+Y}(z) = P(X + Y \leq z)$$ $$=\...
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Minimum Variance Unbiased Estimator (MVUE) is unique proof - wrong [on hold]

Here is the proof of "MVUE is unique" that my lecturer gave: Now I understand the following: The first expansion is done using the formula for the sum of correlated random variables (https://en....
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26 views

Estimation of population minimum from sample

Given a sample of values $[x_1, x_2, ..., x_k]$ drawn from an unknown population $P$, how can we effectively estimate the minimum of $P$? I am interested in the general case where the distribution of ...
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What is the posterior distribution of a Pareto distribution?

I want to derive the posterior of the Pareto distribution, but I can't seem to make it work. I know that Pareto is given by: $p(\theta |\alpha,\beta) = \dfrac{\alpha\beta^\alpha}{\theta^{\alpha+1}} $ ...
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1answer
38 views

Probability that Month selected has $30$ days

In a Leap year a month is selected at random and a day is selected at random and found that its fifth Friday. What is the Probability that selected month has $30$ days. My try: Let $A$ be an event of ...
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1answer
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Markov Chains conditional probabilities [on hold]

Consider the Markov chain with one-step transition probability matrix: $$P= \begin{bmatrix} 1/2 &0 &1/2 &0 &0\\ 1/5 &1/5 &1/5 &1/5 &1/5\\ 1/2 &0 &1/2 &...
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Intrinsic Bayes factor

Here on the page 238 in the formula: the arithmetic intrinsic Bayes factor, $$B^A_{10}=\frac{1}{L}\sum_{x(\ell)}B_{10}^{(\ell)}=B_{10}(x)\frac{1}{L}\sum_{x(\ell)}B_{01}\left(x_{(\ell)}\...
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How can I prove Hat matrix*1=Hat matrix?? [on hold]

enter image description here How can I prove the equation is correct. Please help me
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1answer
19 views

Combining Weak Classifiers

In a binary classification problem, assume we have five classifiers with accuracies 0.4, 0.5, 0.6, 0.7, and 0.8. The errors of these individual classifiers are independent. What is a good decision ...
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1answer
18 views

basic purpose of characteristic functions [on hold]

I have studied for a long time, but i still can not understand why we need characteristic functions.I know the purpose of using them is to simplify but simplify what?
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0answers
34 views

Unbiased estimator of a square root of chi-squared distribution

Let $Y_1,Y_2,...,Y_n$ be a random sample from $N(\mu,\sigma^2)$. I need to show that $S$ is a biased estimator of $\sigma$. As from the definition, I see that $\frac{(n-1)S^2}{\sigma^2}\sim\chi^{2}_{(...
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Finding $P(XY \le a)$ when $x<y$ and the distribution of both is uniform

Let $X$ and $Y$ be random variables with joint pdf $$f_{XY}(x,y) = \begin{cases} 1, &0 < x <1,\ \ x < y < x+1 \\ 0, & \text{otherwise} \end{cases}$$ Find $P(XY \le a)$ ...
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1answer
21 views

Independence of random variables and their components

If we have that $X$ and $Y$ are independent, what do we know about there components $X^\pm, Y^\pm$. Are they independent as well? If so, which combinations are the independent ones and which ones are ...
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1answer
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How to use p value to determine statistical significance of change in student test scores.

I have around 100 students who have all sat 3 exams and are about to sit another. Since their last exam I have made changes to my instruction and I want to determine if any changes in student scores ...
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1answer
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Stuck on a term in $Var[\hat{\beta_o}]$ Proof

So I was trying to prove that $Var[\hat{\beta_o}]=\dfrac{\sigma^2n^{-1}\sum{(x_i)^2}}{\sum{(x_i-\bar{x}})^2}$ And I got stuck with the part $\dfrac{ -2\bar{x}}{\sum{(x_i-\bar{x})^2}}\sum[{(x_i - ...
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2answers
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Why is the degree of freedom not 5?

So we got out homework back and along with it the answers to the questions. I understood that the degree of freedom Can be obtained with N-1. So we have 6 sets of data then N =6. So df =5. But here it ...
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The distribution of a spatially correlated noise which always sums up to zero?

Consider a spatially discrete system: throwing $N$ balls independently into $n$ bins arranged linearly along the $x$-axis according to some discrete distribution $f(i),(1\le i\le n)$. Let the random ...
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2answers
17 views

Calculate expected end balance

There is a problem like this, Suppose you have a biased coin with a probability 0.1 of head. You win 100 dollars every time you get a head and lose 50 dollars if it is tail. You start with $0, it is ...
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45 views

Why CDF is not coming to 1? [duplicate]

Suppose, $F_X(x)=-\frac{x}{a^2}+\frac{2\sqrt{x}}{a}$ And, $f_X(x)=\frac{d}{dx}F_X(x)=\frac{1}{a\sqrt{x}}-\frac{1}{a^2}$, Here, $0\leq x \leq a^2$ Similar, $f_Y(y)=\frac{1}{a\sqrt{y}}-\frac{1}{a^2}$, ...
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1answer
46 views

Variance of X which is a sum of random m draws of different numbers among n numbers (1,2,3,…n)

You have a set of consecutive n natural numbers {1,2,3...,n}. m different numbers are drawn from these n numbers. Calculate the variance of the sum of these m numbers?. My try: Since each draw of m ...
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1answer
21 views

Function to calculate t-stat of similarity in survey answers

I have a survey of a large number of questions. Each question is multiple choice, and has three possible answers. Users get served random questions to answer. So they do not all answer the same ...
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1answer
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What are some statistical distributions with the irrational numbers e and pi in their functions? (apart from the most common - Normal, Poisson)

I've been researching on the application and origin of irrational numbers in probability theory and statistical distributions, so far having derived a unique proof of Stirling's approximation, and ...
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1answer
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Binomial Probability Problem -HARD!

In the final match a boxer is facing another boxer and is expected to win 48% of the time. In reality they win 80% of the time. What is the probability of that occurring in 10 matches? 2) In the ...
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1answer
18 views

Computing standard deviation of parameter

Given that I have a parameter $$\chi =\frac{\langle M^2\rangle}{TL^2},$$ how would I compute the standard deviation of $\chi$? I suspect it should be, $$\sqrt{\frac{\langle M^4\rangle-\langle{M^2}\...
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1answer
35 views

Is there a more efficient expected value estimator than the sample average?

I'm wondering if there is a known estimator for expected value that is more efficient than the sample average. If that is not the case for an arbitrary random variable, then maybe there are examples ...
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Failure Rate calculation based on manufacturing defects (electronics-capacitor)

Electronics failure rates typically follow the exponential distribution and models like Prism, Mil-Hdbk-217 or 217Plus (newer) are used to predict failure rates. I'm using 217Plus model for ...
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1answer
24 views

Unbiased estimator having a deviation less than $0.05$

In a mass production of items produced indepedently of eachother the probability of an item being defect is $p$. An unbiased estimator for $p$ is $\hat p = \frac{X}{n}$ where $X$ = amount of items ...
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log transformation in Regression and correlation coefficient [on hold]

Let μX and μY be respective means of X and Y , let σ2 be the common variance of X and Y , and let ρ be the correlation coefficient of X and Y . Also, let β be the slope of regressing log Y on log X ...
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1answer
31 views

A and B be two events of the same probability space. Equivalent

I have this probation calculation demo exercise and I do not know very well Let $A$ and $B$ be two events of the same probability space. They are equivalent $A$ and $B$ are independent $A$ and $B^\...
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1answer
18 views

Calculate recurring probability with break and change in probability

I'm somewhat uncertain on how to calculate recurring probability with two timeframes. Assume an event occurs on a daily basis with the probability of 1/600. If I wanted to calculate how many days it ...
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1answer
19 views

Finding $a$ and $b$ such that $\hat u$ is an unbiased estimator

A chemist wants to decide the amount of a certain substance $\mu$ in a specific type of food. In the lab, the chemist has two measuring intruments $A$ and $B$. The results from the instruments can be ...
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2answers
31 views

How do you infer the success probability of a Bernoulli random variable from independent samples

Let's say we have a coin (not necessarily fair) and we flip it 100 times and all of the outcomes were tails. We can immediately conclude that the probability of getting tails is not 0 and we ...
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2answers
27 views

Probability of average times called being inside an interval

Assume that when a man calls a restaurant to order some food, his call has a probability $0.2$ of reaching through. Every attempt is independent and we can assume that the restaurant is open every day....
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1answer
51 views

What is the probability that a red ball will be selected?

What is the probability that a red ball will be selected? Suppose there are two jars, $A,B$ $A$ has $2$ red, $4$ green $B$ has $3$ red, $5$ green An urn is selected at random, giving each of the ...
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Total Possible Combinations when exhausting a group

I have 3 green basketballs, 2 red, 6 yellow and 4 green. How many combinations are possible if I shoot one basketball at a time with the stipulation that once a color is chosen, I have to shoot all of ...
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2answers
37 views

Sample Distribution of $S^2$

Let $X_1,...,X_n$ are iid sample from $N(\mu,\sigma^2)$. Then $\bar X$ and $S^2$ are independent. I was stuck on proving above statement. The joint PDF of $(X_1, ... ,X_n)$ is given by $$f(x_1,...,...
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1answer
17 views

Structural Equation Modeling Reference Text

I'm looking for a good introductory book on the theory behind structural equation modeling in the social sciences. I have a solid background in multivariate statistics and in pure mathematics, so ...
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2answers
116 views

Is inequality between two distributions' tail ends guaranteed?

If this question is true it is profound with countless predictions and implications (some of which are unforunate, but there are some good ones for doing better science and correcting the bad ...
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1answer
38 views

Convolution without known pdfs in terms of z

I have looked at many sites with almost enough information for this to make sense... just not quite there yet. So I have this convolution formula (w instead of z and $dx$ instead of $dy$): $$f(w) = \...
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2answers
24 views

Draw two cards at random without replacement. What is the chance of not getting all kings? [on hold]

I am at a loss for how to solve this problem. Can someone please help and link the steps below? Thank you so much!
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1answer
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Show equivalence of minimization techniques using Lagrangian multiplier.

Define a cost function : $$L(x,w) = \frac{1}{2} \sum_{i=1}^N\left(y^{(i)} - w^T\phi\left(x^{(i)}\right)\right)^2 + \frac{\lambda}{2}\sum_{j=1}^M w_j^2$$ Show that minimizing $L$ is equivalent to ...