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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Difficult Survey Sampling question

Question: A Secretary of State wants to survey the primary owners of motorcycles registered in the state to estimate the proportion who want the license plates redesigned. (Primary owner means that ...
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Statistics: Converting logistic distribution to Uniform

Let X have a logistic distribution with pdf f(x) = e^-x/((1+e^-x)^2). Show that Y = 1/(1+e^-X) has a U(0,1) distribution. Hint: Find G(y) = P(Y <= y) = P(1/(1+e^-X)<=y) when 0 < y < 1
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1answer
22 views

Suppose $X\sim U[0,1]$ and $P(Y=1| X=x)= x = 1-P(Y=0| X=x)$. Find the expectation and variance of $Y$ [on hold]

Suppose $X\sim U[0,1]$ and $P(Y=1| X=x)= x = 1-P(Y=0| X=x)$. Find $E[Y]$ and $\operatorname{Var}[Y]$ I am trying to get $f_y(y)$ to be able to get $E[y]$ and $\operatorname{Var}[y]$. But I don´t ...
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1answer
19 views

Intuition behind Normal distribution forumula

In this formula $$ P(x) = \frac{1}{{\sigma \sqrt {2\pi } }}e^{ - \frac{ \left( {x - \mu } \right)^2 }{2\sigma^2}} $$ why do we divide by square root 2 pi and after that multiply everything by e in ...
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1answer
8 views

Proving something is strict stationary

Let $W$ be a uniform distribution on $(0,\pi)$. Let $Z_t=\cos(tw)$. I know that $Z_t$ is a strict stationary but I have no idea how to prove this. Can someone give me some methods?
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Stat problem help me! [on hold]

Hello! I got a problem when I was solving stat problem. I solved by(c), but after that, I found it hard to solve. Can you guys help me or give me a hint? Thank you anyway!
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2answers
19 views

Expectation of a function of pairs of random variables

For positive random variables $(X_1, Y_1)$ and $(X_2, Y_2)$, suppose that $(X_1, Y_1)$ and $(X_2, Y_2)$ have the same distribution and (the two pairs) are independent. Also suppose that $E[Y_1|X_1] = ...
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The number of ways to schedule six activities

I'm trying to review for Probabilities and Statistics and came upon this Question. If one needs to schedule a job interview for someone who wants to teach at a school. For the day of the interview, I ...
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1answer
28 views

Statistics and Probability. [on hold]

The weight of a certain type of brick has an expectation of 1.12 kilograms with a variance of 0.0009 kilograms2. How many bricks would need to be selected so that the average weight has a standard ...
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12 views

convergence of a weighted sequence of random variables

Suppose that we have an array of bounded random variables $\{ X_{j,n} \}$ with $j = 1,...,k(n)$. We cannot assume that $(X_{1,n},\dots,X_{k(n),n})$ are independent. It is known that $X_{j,n} ...
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1answer
16 views

Number of outcomes times probability of failure is number of failures?

I was hoping someone could spread some insight on why this is true? Suppose that in some situation, there are n! number of possible outcomes. Also, suppose that p is the probability of failure of ...
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39 views

about finding the probability when an event happens in competition

I am reading an interesting question about an ancient card game in China. I forgot the name but the rule is something like this. There are 1-50 participants playing the game each time. Each players ...
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0answers
4 views

Measure to compare quality of synthetic data generated?

What is a good measure to compare the quality of the synthetic data generated with respect to the original data? The synthetic data I have, is the scaled up version of the original data. I am confused ...
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0answers
14 views

Advanced Statistics Question [on hold]

I am currently struggling with this question big time, any help would be greatly appreciated.
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23 views

Probability of no person being present on a website [on hold]

Given that an average of 2 people are present on a website X at any given minute. What is the probability that no person is present on the website X in a 5 minute interval window (they need to take ...
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2answers
28 views

Ratio of Gamma random variables

If $X_i$, $i=1,2$ are independent gamma$(\alpha_i,1)$ random variables, find the distribution of $\frac{X_1}{X_1+X_2}$ and $\frac{X_2}{X_1+X_2}$. Attempt: Let $Y_1 = \frac{X_1}{X_1+X_2}$ and ...
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1answer
19 views

Finding distribution of random variable if X is exponential $(1)$

Let X be an exponential (1) random variable, and define Y to be the integer part of X+1, that is $\hspace{15mm}Y=i+1$ if and only if $\hspace{5mm}i \leq X \leq i+1, i = 0,1,2,...$. Find the ...
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21 views

Correlation and First Order Stochastic Dominance

Suppose we have a random variable $X \sim [0,1]$ with a continuous distribution $F_X(x)$. Suppose $I \in \left\{0,1\right\}$ is a discrete random variable with $\text{Prob}(I=1 \ | \ X=x)$ strictly ...
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1answer
11 views

Percentages in Normal Distribution

A statistics problem involves: Lengths of a certain type of carrot have a normal distribution with mean 14.2 cm and standard deviation 3.6 cm. (i) 8% of carrots are shorter than c cm. Find the value ...
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1answer
27 views

Suggest any Comic based books for Learning calculus and Statistics? [on hold]

I seen some manga comics for learning statistics and calculus. Suggest other books.
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2answers
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Is there any compact notation for the count [on hold]

Can any one suggest what is the best compact notation that I can use for the following pseudo problem. I think it is simple counting with some constraints but don't know if is there any notation for ...
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IQR of the sampling distribution of S^2

Suppose $X_1,X_2,...,X_5$ is a random sample from a $N(33,9)$ population and let $S^2$ be the sample variance. [a] Find the IQR of the sampling distribution of $S^2$. [b] Find the IQR of the ...
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19 views

weights go to infinity in logistic regression with linearly separable data

I have the loss function of logistic regression $L(W)$ = - $\sum_{i=1}^n {y_i}.log[\sigma(w^Tx)] + {(1-y_i)}.log[1- \sigma(w^Tx)]$ I have derived the Hessian and proven it's positive semi-definite ...
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1answer
18 views

(Statistics)Probability of given sum in dice tossing [on hold]

I need some help with this problem: By tossing two dice, what is the probability of: i) Total sum of 7 ii) Difference of 5 iii) Total sum multiple of 7 Thanks everyone ~Chris
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1answer
20 views

Showing something converges, in distribution, to a normal distribution

I'm not sure how relevant the first few parts are, but I will post it just in case... $(X_i,Y_i), i=1,\dots,n$ are independent where $X_i$ has an exponential distribution $\mathcal{E}(\lambda_i)$ ...
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1answer
15 views

Showing the multivariate normal is log-concave?

I'm trying to show that $\log p(x) = -\frac{1}{2}(x-\mu)^{T} \Sigma^{-1}(x-\mu)$ is concave. How would I go about this in $\mathbb{R}^n$? I've tried taking derivatives but I'm getting stuck once I get ...
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1answer
26 views

Statistics - Show that $\hat{\theta}$ hat is a biased estimator of $\theta$

I'm asked to solve this exercise, but I can't manage to find something satisfying. Any help/hint would be much appreciated. Let $Y_1, Y_2,\dots, Y_n$ denote a random variable sample of size n from a ...
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7 views

Get SNR from several noisy signal

I want to get the SNR of a noisy signal(signal length 3000). But the problem is I don't have any ground truth for that. I can repeat the experiment for almost any time as I want (get many noisy ...
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1answer
50 views

How to count importance of bought rate?

We are presenting lists of products to our users. Users can buy products. We have about 100 000 products. Users are watching only two or three pages of products. It's important to show best products ...
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1answer
29 views

Question about the Central Limit Theorem

The version of the CLT in my book states that if $X_1,...,X_n$ is a random sample, with mean $\mu$ and standard deviation $\sigma^2$, then $W=\frac{\bar X-\mu}{\sigma/\sqrt n}$~$N(0,1)$ as ...
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25 views

simple random sampling without replacement proof

For simple random sampling without replacement, starting with the expectation of $\sum_1^n(y_i-\bar Y)^2$, show that $V(\bar y)= (1 − f )S^2/n$ this looks very hard i tried to simplify the right ...
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How to conceptualize and calculate the variance in the waiting time?

I'm trying to calculate the mean and variance in the waiting time for the following situation: You are at the grocery store waiting to check out. There are 100 people in the line, and 10 cashiers. ...
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1answer
30 views

Sum of Binomial Coefficient products

I am trying to prove that $$\sum\limits_{y=0}^d \frac{{2x \choose y} {2d-2x \choose d-y} }{2d \choose d} = x $$ So far, I have tried using induction on $d$ but I am having trouble using the ...
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1answer
27 views

Standard Normal Distribution and CDF

I have a data set which consists of measured time in seconds. Secs= ${3000, 3857, 2400, 3323}.$ Mean $\mu =3145$. Standard deviation $\sigma=609.556$. I calculated the Standard Normal variable for ...
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2answers
25 views

weird null hypothesis

I've done loads of hypothesis tests for the mean where the null hypothesis is of the form $H_{0}=a$. Now I've come across one where the hypotheses are like $H_{0}{\leq}a$ and $H_{1}>a$. How does ...
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1answer
45 views

What is the bound on $E\|Y_n\|^4$ in terms of $n$?

Let $X_n,n\in\mathbb{N}$ be i.i.d. zero-mean random variables in some separable Hilbert space with $E\|X_n\|^8<\infty$ and $Y_n=\frac{1}{n}\sum_{i=1}^nX_n$. I need to find bounds on $E\|Y_n\|^4$. ...
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1/1000 chance of a reaction. If you do the action 1000 times, whats the new chance the reaction occurs?

A hypothetical example: You have a 1/1000 chance of being hit by a bus when crossing the street. However, if you perform the action of crossing the street 1000 times, then your chance of being ...
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1answer
45 views
+50

Monte Carlo p-test and early stopping

Say you have a coin with some probability $p$ of falling on heads. You would like to determine if this probability is less than or equal to $0.05$ with some reasonable degree of confidence and stop ...
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1answer
33 views

Expected Value on code

I'm trying to figure out the expected number of times this algorithm will print. I'm stuck on how to go about doing so. I used an indicator variable to keep track of the number of print statements ...
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1answer
32 views

Convergence to the normal distribution [on hold]

I am looking at the following exercise: Find a function $g( \cdot )$, such that: $$ \sqrt{n} (g( \hat{\lambda})-g(\lambda)) \xrightarrow[n \to +\infty]{d} N(0,1)$$ Could you give me a hint, how to ...
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1answer
33 views

Using the inverse Gaussian integral to find percentiles

I need some help with the following: Let $$R=\mu+\sigma*\epsilon \hspace{1cm} \epsilon \sim N(0,1)$$ I want to argue that $$ \mu + \sigma*\Phi^{-1}(u)$$ are the percentiles of the model when ...
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1answer
68 views

H Test Question

Really have no idea where to start :( In an assessment comparing two plans (to determine muscle lost) C and D 20 test subjects were matched into pairs so that within each pair the subjects were as ...
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1answer
19 views

Cumulative probability of Chi-squared distribution

If $X$ is distributed $\frac{\chi_{10}^2}{10}$ , find the probability that $X > 1.83$ The formula for the Chi-squared CDF I'm using is the following, which is the integral of the PDF formula: ...
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Finding a sufficient statistic for an iid sample of the Gumbel distribution

$G(x;\alpha, \beta) = \exp\{-\beta e^{-\alpha x}\}$ for $x \in \mathbb{R}$ is a distribution (Gumbel family). Side question: is $G(x;\alpha, \beta)$ a member of the exponential family? I do not think ...
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1answer
16 views

Finding probability given mean and standard deviation

I don't know how to approach this problem: X is normally distributed with a mean of 200 and a standard deviation of 10. Find P(X ≥ 203)
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1answer
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Generating random variables with complicated probability distribution functions

I have an interesting question I need to solve, and as much as I try, I cannot wrap my head around it. Given a postive random variable X with p.d.f. ...
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1answer
13 views

confidence Interval of multivariate Gaussian Distribution

Suppose I have an single variable which is gaussian distributed (with mean value 0 and standard deviation sigma). Then I certainly know that there is 68.2% of the chance that this variable should lie ...
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1answer
27 views

How to estimate the mutual information

I have two discrete non-negative random variables $X$ and $Y$. I know $X$ is the number of heads you get by tossing $n$ unbiased coins and I know $Y$ is in the range $0,\dots n$. I can sample from ...
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Stat problem! Why is this? [duplicate]

This is a statistics problem. although this is not a problem which needs an answer, I want to know the reason Why this is right. Can you guys help me ? Thanks in advance!
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Bayesian estimation (bernoulli)

If i'm given a prior uniform distribution for theta from (0,1), where theta is the probability of successful outcome. How do I find the bayes estimate for theta for squared error loss if the outcome ...