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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Is there a value in keeping 'history of statistics' (how statistics over a set of data changed over time)?

I currently have statistics calculated (mean, average, mode, range, variance) for my dataset (timestamped data), but it's overwritten every time - so I can access only latest measurements. My ...
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75 views

Man walking on the bridge. Stopping time and probability. [closed]

A man is standing on a bridge, the length of the bridge is 100 meters, and the position of the man at time $t = 0$ is at 29 meters to the East. The man walks randomly in each second to the direction ...
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47 views

Probability of N unrelated events, each with different probabilities, what is the chance X number of outcomes occur

Given the probability of N unrelated events, each with different probabilities, what is the chance X number of outcomes occur? Said specifically there are 8 unrelated contracts, what is the chance a ...
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52 views

Proofs related to chi-squared distribution for k degrees of freedom

I was reading a proof related to chi-squared distribution for k degrees of freedom from wiki. https://en.wikipedia.org/wiki/Proofs_related_to_chi-squared_distribution I think I might understand the ...
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96 views

Difficulty formulating $H_0$ and $H_1$ in hypothesis testing.

For example in this example I don't know $H_o$ is whether $\eta>28 $ or $\eta<28 $ ,and in general how should we choose $H_o$? I'm really confused with this,any help would be appreciated.
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30 views

Knowing that the probability to drill success (produced) exploratory hydrocarbon well is $0.2$ by certain company

Knowing that the probability to drill success (produced) exploratory hydrocarbon well is $0.2$ by certain company, this company has drilled four success exploratory wells in different areas on the ...
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1answer
19 views

MLE in introductory probabilistic information theory

Consider sending a bit that is either $\{0,1\}$ through a noisy symmetric channel, such that for a given input $x$ and a given (potentially noisy) output $y$, $\forall i,j \in \{0,1\}. P(y = i | x = j)...
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16 views

Fisher Information Matrix for a certain multivariate distribution

Consider a multinomial distribution with $n$ trials and $k$ outcomes per trial. I want to find the Fisher Information matrix of this distribution. I am guessing that the likelihood of seeing this ...
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121 views

What is the probability that a person wearing a blue t-shirt will sit next to one wearing red?

9 people sit in a row linearly. 2 dressed in Red, 3 blue and 4 in yellow. What is the probability that a person in blue will sit next to a person in red? Why? RRBBBYYYY this sequence from what I ...
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11 views

Converting standard deviation of a score to standard deviation of a percent?

I have the standard deviation for scores on a test that is out of 105; it is 1.9. If I want to convert scores to percents, I do: score * 100/105. But how do I do the same with standard deviation? Can ...
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19 views

Cumulative bivariate normal

How do I calculate the cumulative probability distribution function for a bivariate normal distribution with conditions $P( x>a , y>b)$? Is there any method to solve $$P(x>a,y>b)\\\int_{...
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22 views

How do I calculate the degrees of freedom for this odds ratio?

I don't have a math background so I'm kinda improvising right now. I have a feeling the regular n-1 or n1+n2-2 rule doesn't apply here. These are the conditions. 522 people with diabetes 346 ...
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24 views

Error of a mean of data

I am taking the mean of a set of data, each with their own error. I would like to know what the error is for that value. Not the uncertainty of the mean, but the uncertainty of the measurement, if ...
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2answers
47 views

Is it correct to say $\text{max}(x_1,x_2,\dots,x_n)=\lim_{p \to \infty} \frac{x_1^p+x_2^p+\dots+x_n^p}{x_1^{p-1}+x_2^{p-1}+\dots+x_n^{p-1}}$

There are many possible way to represent the maximum function, I came up with this one: $$\text{max}(x_1,x_2,\dots,x_n)=\lim_{p \to \infty} \frac{x_1^p+x_2^p+\dots+x_n^p}{x_1^{p-1}+x_2^{p-1}+\dots+...
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13 views

Combining standard deviations - given mean, sample size, and SD of two sets?

I have two groups A and B. I know A and B's sample sizes, mean performances on a test (not their individual scores however), and the standard deviations of A and B. I know how to combine them to find ...
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1answer
21 views

Maximum-Likelihood Estimator of $p_1,…,p_k$ the probabilities of $Y=y_i$ respectively?

$y_1,....,y_k$ are the only possible outcomes of Y, and we can assume all of them appear at least once in the sample. $L(p_1,...,p_k)=\prod_x p_x=\prod_i^k p_i^{\#y_i}$ Normally now I would derive ...
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1answer
23 views

Joint Probability Distribution function

I have two random variables $\alpha(x)$ and $\beta(x)$, and they are correlated. $\alpha(x)$ obeys log-normal distribution while $\beta(x)$ obeys normal distribution. How do I construct a joint ...
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26 views

Taking into account a predictor's accuracy when calculating probability.

Lets assume that you are blind. And you are also given the task of sorting a line of randomly ordered humans into a group of males and females. Let us also assume that the actual population ...
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1answer
27 views

Mean of samples from Exponential distribution

I please ask someone to check if my calculations are right. I have $X_1, ..., X_n$ from a $\mathcal{E}(\lambda): f(x, \lambda) = \lambda e^{-\lambda x}$, $\mathcal n = 50, {\lambda} = 3$. I have to ...
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I need help finding the linear equation for an exponential regression

I've done nearly everything but at the end when I need to $\alpha$ it doesn't work and I'm not sure what's wrong. This is the function: $y = \alpha\beta^x$ I transformed it into a linear equation ...
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1answer
18 views

Sufficient statistic for Uniform distribution.

We are given that $ X_i \sim U( 0 , \theta )$ where $\theta$ is unknown. We need to find a sufficient statistic. First we write the conditional distribution : $ f( x_1 ,........ ,x_n | \theta ) = \...
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12 views

goodness of fit doubt where the next data value is twice

I have a list of data, and the following problem: A particular mobile phone company , determined that by launching its totally innovative new product and never before seen in the market, each day ...
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1answer
97 views

showing SSE of simple regression model is larger than or equal to SSE of multiple regression model

Lets say we have 2 linear regression models: $y_i = B_0 + B_1x_{i1} + \epsilon_i,$ where $\epsilon_i$ follows $N(0,σ_1^2)$ $y_i = B_0 + B_1x_{i1} + B_2x_{i2} + \lambda_i,$ where $ \lambda_i$ follows ...
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11 views

ANOVA standard error for treatment difference

can anyone please help me with question b) (I link the image below). It says $\sigma_\Delta=2.241528$ But I got $\sigma_\Delta=2.00488$ I used the equation $\sqrt {\frac {\sigma_i^2}{n_i}+\frac {\...
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25 views

Linear Regression And Coding

I'm revising for my A-level Statistics exam and need to figure out where / when coding has an effect. What I'm currently stuck on is the effect coding has on values for $Sxy$ where $x= y \;$or$\;x $. ...
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12 views

Meaning of inverse mills ratio

Can someone, please, explain in layman terms what is the meaning of using inverse mills ratio for incidental truncation? I am non-mathematician self-studying this topic and have problems explaining to ...
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41 views

Picking cards sequentially vs consecutively

We have a pack of 6 cards over the table. Cards are: {A, A}, {B, B}, {C}, {D}. There are 3 players (Papa, Pepe, Popo) sat around the table. Cards are all upside down, so the players cannot see which ...
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1answer
36 views

What kind of process is this? [closed]

I have encountered some difficulties in the following exercise: Let $\{X_{t}, t=0,1,2, \dots\}$ be a process defined by $X_{0}=0$ and $$X_t=\mu+X_{t-1}+u_{t}$$ where $u_{t}\sim IID(0,\sigma^{2}_{u})$....
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27 views

A computer test of a very fast primarily test

Fermat's little theorem states that if $p$ is a prime and $a$ is any integer not divisible by $p$, then $a^{p-1} - 1$ is divisible by $p$. $$a^{p-1}\equiv 1\pmod p$$ This can be used to test if a ...
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28 views

What do attendance figures tell me about regularity? What does the average tell me about individual attendance?

Suppose I have a group of $N$ people, attending a series of $M$ events, and (for simplicity) let's assume the overall attendance happens to be the same at each event, say $A$ people (ranging between 1 ...
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1answer
34 views

How to combine correlated estimates to test variable is > 0?

Let X1 and X2 be two unbiased but correlated Gaussian estimators of a true value x. 1. What is the proper way to combine two observations of X1 and X2 to test whether x > 0? 2. How does the answer ...
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1answer
31 views

Logistic regression MLE example. What is this “logistic function”?

This is a basic question (I think). I am trying to grasp the idea behind this example, where we define a "logistic function" and use that to work towards the maximum likelihood estimate (MLE). We ...
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32 views

Understanding a Substep of the Proof for the Law of Total Variance

In the proof for the Law of Total Variance, the following lemma seems to be appealed to (when going from the 2nd to the 3rd step of the proof): $$ E[E[Y^2 \mid X]] = E[\text{Var}[Y \mid X] + [E[Y \...
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System of stochastic equations

I want to know if this system of SDE: $$dX_{t}=b(X_{t})dt+\sigma( X_{t}) dB_{t}$$ $$dY_{t}=b_{0}(Y_{t})dt+\sigma( Y_{t}) dB_{t}$$...
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1answer
181 views

Why is the classical Secretary Problem about Ranks?

The Secretary Problem goes like this: To an observer there are presented objects in a row. He may either choose the object one it is presented or move on - there is no turning back. Objective: stop at ...
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Estimation of suitable significance level for high numbers of experiments

I am a scientist and work with high numbers of experiments. For example: I have tested $ 1000 $ different parameters between $2$ groups. I found $10 $ parameters, which are significantly different ...
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how to Obtain standard error of estimates Sy.x and Sx.y. given summated values. please show steps

The data for 10 years on profit earned (Y) by a company and dividend paid (X) by it yielded the following summated values in Kwacha: ΣX =15, ΣY =110, ΣXY = 400, ΣX2 = 250, and ΣY2 = 3200. Obtain i) ...
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1answer
91 views

How to find the expected value of this game?

How can I create a function that figures out the expected value for a value that can change. Example: I can flip a (fair) coin $n$ number of times. The pot starts at $\$1$. If I lose, $50$ cents is ...
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1answer
34 views

let f(x)=(2x)/9 where 0<x<3 and Find p(x=2)? [closed]

$f(x)=\frac{2x}{9}$ where $0<x<3$ $F(x)=\frac{x^2}{9}$ where $0<x<3$ Find $p(x=2)$? Please help me how can I find the probability when it equal?
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32 views

A few questions about a Probability problem involving airline flights and using the phrase “or fewer”.

The question goes as follows: An airline reports that $77.5\%$ of its flights arrive on time. A check of $200$ randomly selected airline flights shows that $140$ of them arrived on time. Estimate ...
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63 views

Measuring the degree of convergence of a stochastic process

Consider a set of random variables $(X_1,X_2,X_3,...X_k)$ that are i.i.d. $Bernoulli(p)$ While I do not know $p$, I can estimate it using $$ Y(k)=\frac{1}{k}\sum_{i=1}^k X_i $$ Notice that $Y(k)$ is ...
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Proof of Random Treatment's Effect on the Causal Regression Function

Consider $C(x)$ to be the outcome a subject would have if they recieve dose $x$ on some real interval. the observed response is given by the random variable relationship $Y = C(X)$. We treat ...
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VC Dimension of star-shaped sets on $\mathbb{R}^2$

Consider a set $U \subset \mathbb{R}^2$ to be star-shaped with respect to a point $\epsilon \in \mathbb{R}^2$ if for every $x \in U$, the segment joining $\epsilon$ and $x$ is contained in $U$. ...
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Statistics. How to find the median of two other medians?

I got median of 226 and another median of 304. Now the question is how to find the overall median? Averaging them is not the correct answer.
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15 views

Determining sample size knowing change effect

I want to figure out the sample size to consider the experiment statistically significant. I did some research and found the following formula: Now, I have $\alpha$, $\beta$, $p_1$, $\Delta$, $k$ ...
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2answers
28 views

Calculate probability from a Negative Binomial Distribution with multiple success scenarios - 7th child born is either 3rd or 4th boy

So the problem I am trying to solve states that, if a newborn is equally likely to be a boy or girl, what is the probability that 1) the third child is the first boy 2) the eighth child is the second ...
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36 views

Estimate counts with different sample sizes

Given an arbitrary time period, lets say one week, but it could be five days, one month etc.., I have a sample from a population. My sample consists of shoppers at a store. For week one my sample is ...
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1answer
16 views

Maximum Likelihood Estimate for a uniparametric family

Say we have a simple random sample of size $n$ distributed according to a function of the family: $$ f(x|\theta)=\frac{\theta}{(1+x)^{1+\theta}} $$ where $\theta >0$. We want to compute the ...
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1answer
32 views

What is $cov(Y_1,Y_2)$ if $Y_1 = \prod_{i=1}^{n}X_i$,$Y_2 = \prod_{i=2}^{n+1}X_i$ , and $X_{1..n+1}$ are $n+1$ bernoulli-distributed random variables?

$X_1,X_2,...,X_{n+1}$ are $n+1$ random variables that follow a Bernoulli distribution with parameter $p$. We define $Y_1 = \prod_{i=1}^{n}X_i$ and $Y_2 = \prod_{i=2}^{n+1}X_i$. How can I find out ...
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Probability that n-digit number is divisible by some number(s)?

I have came across a number of problems in our probability course that deal with this kind of question. And for two digit numbers I have always "brute-forced" the solution by writing them all out and ...