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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How to test whether a subset is representative of a population?

Given that I have an entire population's data, I'm looking to see if a subset that I take is representative of the population, on average. I'm not choosing randomly, as I plan to pick each item ...
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0answers
12 views

How can we measure the accuracy of prediction algorithm?

We have created a prediction algorithm, which predicts the chances of confirmation of ticket based on some parameters, and gives the prediction in percent. Now, how do I measure how close the ...
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16 views

Conditional Probability Dad, father

Can you help me please? The question is: if you were raised to call Packy dad, there is a 90% chance that he is your father. If you were raised to call someone else dad, there is still a 5% chance ...
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1answer
39 views

Expected value and variance of ratio of two sums of two sets of random variables

Let $X_1,X_2,\ldots,X_n$ be iid $\operatorname{Gamma}(\alpha,\beta)$ random variables. Suppose that, conditionally on $X_1,X_2,\ldots,X_n$, the random variables $Y_1,Y_2,\ldots,Y_n$ are independent ...
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1answer
36 views

poisson distribution probability problem

I am working on a Poisson distribution problem stated in the main question and got stuck and do not know how to proceed as I did not understand the next question on how to work it out The following ...
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1answer
26 views

If $w$ is a discrete random variable then is $P(w|x)$ a density or mass function? [on hold]

$w$ is a discrete random variable. $x$ is a continuous random variable. Then should I denote $P(w|x)$ as a probability density function or probability mass function, and why ?
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0answers
13 views

Nice applications of estimation theory and hypothesis testing

As a mathematics professor in an engineeer school, I want to write some lab work for students in Statistics. This work should last four hours and will be made in a language such as Matlab or Python. ...
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36 views

Did I work this out correctly? (Fitting a normal distribution)

Sorry for the long text. But please note that what I'm asking for is not a word by word correction, but I'm asking you to do a quick skim (or two) on my method of working, and I hope to get some ...
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3answers
52 views

distribution of one random over the sum of random variables

Suppose that $X_1,\ldots,X_n$ are independent random variables with $X_i\sim Gamma(\alpha_i,\beta)$. Define $U_i=\frac{X_i}{X_1+\cdots+X_n}$ for $i=1,2,\ldots,n$. Show that $U_i\sim ...
2
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1answer
32 views

How to compute the expected value of one random variable over sum of iid random variable

If $X_1,\ldots,X_n$ are independent identically distributed positive random variables, prove that $E(\frac{X_i}{X_1+\cdots+X_n})=\frac{1}{n}$, $i=1,\ldots,n$. Can someone give me a hint?
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1answer
43 views

What value to choose as representative of 100 votes?

I'm trying to use many votes to cast one vote. There are only 4 options, and I'm trying to use "consensus" to decide which to pick. 100 people vote. They can vote 1, 2, 3, or 4 stars. Distribution: 1 ...
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1answer
13 views

Basic hypothesis testing question: how to find the c.r. for a variance test?

Let's suppose that we have two normal distributions, and a sample is taken from each (of sizes $n_X$ and $n_Y$ respectively). $X \rightarrow N(\mu_X, \sigma_X ^2), Y \rightarrow N(\mu_Y, \sigma_Y ...
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1answer
16 views

Derivative of conditional expectation

I am completing exercises in Woolridge and am attempting the following: Define $E(y|x)$ = $\delta_0 + \delta_1(x-u) + \delta_2(x-u)^2$ where $u=E(x)$ Show that, when averaged across the ...
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1answer
25 views

probability distribution of the winning amount

Be A_n the event that a worker spends to process certain component with probabilities according to the table below: For each piece processed, the worker earns a fixed US 2.00, but if he processes ...
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2answers
31 views

Having an independent event with animals

In a building for 24 apartments. It is known that there is only one dog in 8 apartments and a single cat in 6 apartments. How many apartments must have cat and dog for events "have dog" and " have ...
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0answers
32 views

Normal and poissonian probability problems

I am working on a problem with a normal probability distribution but I am unsure of the results I calculated the probability asked for but still hesitate regarding the output and especially the first ...
4
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3answers
58 views

Algebraic and combinatorial proof of an identity

For any two integers $2 \le k \le n-2$, there is the identity $$\dbinom{n}{2} = \dbinom{k}{2} + k(n-k) + \dbinom{n-k}{2}.$$ a) Give an algebraic proof of this identity, writing the binomial ...
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1answer
15 views

Goodness-of-Fit tests for Multinomial and Binomial Data

A box has 4000 red, 5000 blue and 1000 orange balls. A selection of 70 balls is made, with 25 reds, 35 blues, and 10 oranges being observed. Can one essentially prove that the selection was NOT a ...
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1answer
33 views

Probability and balls

An urn contains four blue balls and three white balls. A second urn contains five blue and four white balls. Pass up a ball from the first to the second urn and then extracted a ball second urn. How ...
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0answers
14 views

What's the difference between MCMC and particle MCMC?

Markov chain Monte Carlo (MCMC) methods are a class of algorithms for sampling from a probability distribution based on constructing a Markov chain that has the desired distribution as its equilibrium ...
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1answer
31 views

Normal Distributed( Probability of X)

The random variable $X$ can take negative and positive values. $X$ is distributed normally with mean 3 and variance 4. How can i find the probability that $X$ has a negative value?
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1answer
19 views

How to find the mean and standard deviation [on hold]

How do I find the mean and the standard deviation both are unknown? The given information is the area of each tail.
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0answers
9 views

Completely Randomised Design data

A biologist produced the following data set that represents the expressions of 9 genes before (control) and after a treatment has been administered. The biologist is interested in determining whether ...
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1answer
22 views

Experimental design

An experimental is interested in studying the effects of consuming chocolates on cardiovascular health. She decides to use three different types of chocolates: 100g of dark chocolates, 100g of dark ...
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1answer
26 views

Coin flipping and probability

One in each two people launches three equilibrated coins. How likely is it that take the same number of heads??? guy $1$, just head $1/2 \cdot 1/2\cdot 1/2= 1/8$ guy $2$, just head $= 1/8$ But how ...
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2answers
22 views

multinomial coefficient

Twenty-five girls are bored. a) In how many ways may five girls be chosen to go to a party and five different girls be chosen to volunteer at a soup kitchen (and the remaining 15 girls stay home)? ...
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2answers
33 views

lamps and statistic [on hold]

I tried so hard this question, but I was not be able to answer it.... Could you help me to understand it? In a supermarket 2,000 lamps from three different factories A, B and C. The A produced 500 ...
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1answer
21 views

Say we have a double-decker Lazy Susan with two levels that can be turned independently. If we have n + k dishes in total, how many ways

Say we have a double-decker Lazy Susan with two levels that can be turned independently. If we have n + k dishes in total, how many ways is that solution is correct ???
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1answer
24 views

Probability problem - Electrical circuit.

A guided missile has five distinct sections through which a signal must pass if the missile is to operate properly. Each of the individual sections has two circuits through which the signal ...
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0answers
27 views

UMVUE of parameter $(1-\sigma^2)^{-\frac{n}{2}}$

suppose $X_1,X_2,\ldots,X_n$ be random sample of $N(0,\sigma^2)$. how can I calculate UMVUE of parameter $(1-\sigma^2)^{-\frac{n}{2}}$
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1answer
34 views

Is my work correct? (Easy problem, confidence intervals)

The r.v. $X$ represents the time taken by a computer in company $1$ in order to perform a certain job, and $Y$ represents the same thing but for company $2$. A sample of $n_X = 12$ computers are taken ...
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1answer
23 views

Probability question - multiple choice experiment. [on hold]

Suppose a student who is about to take a multiple choice test has only learned $60\%$ of the material covered by the exam. Thus, there is a $60\%$ chance she will know the answer to the ...
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1answer
21 views

Multinomial Coefficients

Coach Cramer has 15 basketball players. 4 centers, 5 forwards, and 6 guards. She starts one center, two guards, and two forwards. How many different groups of bench-warmers are possible? The answer ...
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0answers
17 views

Cumulative failure rate for hard drives [on hold]

Google have reported on the average failure rates of population of hard drives over time. They report the following statistics (approximated from their graph) for average failure rate: ...
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0answers
26 views

Problem with statistics notation for a density function

I'm reading a paper about partitioning of driving data and producing synthetic driiving profiles and I'm uncapable of understanding some of its equations. Just to give an example, if we consider the ...
4
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1answer
24 views

Geometric mean of 2 sets

If $2$ finite sets of positive integers have different cardinality but the same arithmetic mean, does the set with the greater number of elements always have a lower geometric mean?
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24 views

Standard of deviation [on hold]

If the number of trials be $n$ and the probability of occurrence be $p$ then the standard deviation with respect to $np$, a. $\sqrt{np}$ b. $\sqrt{np(1-p)}$ c. $\sqrt[4]{np}$ d. ...
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0answers
31 views
+50

Consequences of exchangeability of random variables

Consider two random variables $X_i$ and $U_i$ respectively distributed as $F_{X_i}(\cdot)$ and $F_{U_i}(\cdot)$ for $i=1,...,N$. Let $X:=(X_1,...,X_N)$ and $U:=(U_1,...,U_N)$ be respectively ...
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2answers
21 views

Sample variance equation [on hold]

I'm studying statictics, I don't understand why the equation like this i attached. x bar is sample variance https://www.youtube.com/watch?v=D1hgiAla3KI : 5:08
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10 views

How to show sufficient statistics are complete

By writing out the likelyhood function, I can show that $(X_{(1)}, X_{(n)})$ is sufficient statistics, but how to show they are complete?
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19 views

Confidence interval for an overfitted model may be wider.

(a) Consider the multiple linear regression model: $Y = X_1β_1 + X_2β_2 + \epsilon$, where Y is n × 1, $X_1$ is $n × p_1$, $β_1$ is $p_1 × 1$, $X_2$ is $n × p_2, β_2$ is $p_2 × 1$, and $\epsilon$ ...
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0answers
9 views

Multiple imputation for fuzzy logic

In my data set I have missing values for all variables and one particular variable has 12 out of 25 data points missing. So I used multiple imputation method to handle the missing values. What I want ...
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1answer
11 views

Modulus of Z (Normal distribution)

The random variable $Z$ is distributed such that $Z \sim N(0,1)$ find the probability of $P(\left|Z\right| >2.4)$. How to solve this modulus type of question ?
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18 views

Random Variables kith order statistic

If a random variable has density f(x) = αc^(α)x^(−α−1), x > c, where c and α are positive constants, we say that it has a P(c,α) distribution. Consider a random sample of n observations on a P (1, 3) ...
-3
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1answer
37 views

Cats and Dogs = Idenpedent events [on hold]

I did not get this question. Could you explain it to me? In a building for 24 apartments. It is known that there is only one dog in 8 apartments and a single cat in 6 apartments. How many apartments ...
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0answers
15 views

Probability Data Management [on hold]

A bag contains 54 black marbles and 63 white marbles. Use Pascal’s Triangle to determine how many combinations and how many permutations are possible if 7 marbles are drawn out of the bag.
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2answers
43 views

T distribution with n degrees degrees of freedom

I would like to prove that $\displaystyle \frac{\bar{X}\,\sqrt{n}\,}{\hat σ^2}\sim t_{n}$. Note that x~N(0,$σ^2$) and they are iid. Could someone explain why $\displaystyle ...
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1answer
15 views

inequality probability between order statistics of two independent distribution

Suppose we have two independent distributions $F_1$ and $F_2$ and from each distribution, we draw $k$ variables. Let us represent the $k$ i.i.d. variables from $F_1$ as $\{X_1, X_2, \ldots, X_k\}$. ...
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1answer
22 views

Standard deviation of the product of gaussians

What is the standard deviation of the product of two random variables that each have Gaussian Distributions? I don't even know where to begin on this problem.
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1answer
82 views

I'm taking a statistics class right now, and I get stuck on these problems. [on hold]

For a random variable $W$ where $P(W = 0) = 0.1$ and $P(W = 1) = 0.2$ and the density of $W$ for values between $0$ and $1$ is $f(w) = 1.4w$, draw a graph of the CDF. Is this a valid probability ...