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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6
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2answers
28 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
votes
1answer
28 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?
-1
votes
0answers
22 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 ...
0
votes
1answer
12 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 ...
0
votes
1answer
15 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 ...
0
votes
1answer
23 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 ...
0
votes
1answer
19 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 ...
1
vote
0answers
22 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
votes
3answers
53 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 ...
0
votes
1answer
14 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 ...
0
votes
1answer
32 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 ...
0
votes
0answers
13 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 ...
0
votes
1answer
30 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?
0
votes
1answer
18 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.
0
votes
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 ...
0
votes
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 ...
0
votes
1answer
24 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 ...
0
votes
2answers
21 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)? ...
-1
votes
2answers
31 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 ...
0
votes
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 ???
0
votes
1answer
22 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 ...
1
vote
0answers
26 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 $(1-\sigma^2)^{-\frac{n}{2}}$
1
vote
1answer
33 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 ...
-3
votes
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 ...
0
votes
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 ...
-4
votes
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: ...
1
vote
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
votes
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?
-4
votes
0answers
23 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. ...
0
votes
0answers
25 views

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 ...
-1
votes
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
0
votes
0answers
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?
0
votes
0answers
18 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 $\epsilon$ ...
0
votes
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 ...
0
votes
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 ?
0
votes
0answers
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
votes
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 ...
-1
votes
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.
0
votes
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 ...
0
votes
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 $\{X_1, X_2, \ldots, X_k\}$. ...
1
vote
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.
-7
votes
1answer
79 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 ...
2
votes
1answer
22 views

Statistical bias and the probability of an outcome.

A town referendum has occurred. The question posed to voters was YES or NO on a local law. There were 3 methods of voting: Electronic machine (voting booths), absentee ballot, and affidavit ballot. ...
0
votes
1answer
14 views

Derivation of t(n-1) distribution

While trying to prove that $\displaystyle \frac{\bar{X}\,-\,\mu}{S/\sqrt{n}}\sim t_{n-1}$ I came across a manipulation that I can not seem to understand the reasoning behind it. Why does ...
0
votes
2answers
25 views

Why is uncertainty in mean less

Any measurement, say length of any object, will have some errors. The random errors that are present in the measurement can be reduced if we take mean of a large number of samples. This is because the ...
-1
votes
1answer
23 views

calculate median for following data [on hold]

class interval F 45-49 14 ...
1
vote
1answer
28 views

Calculating conditional probability of discrete uniform r.v.

X is a discrete uniform random variable on $\{a, a+1, a+2, ... , b\}$ with mean 7 and variance 4. Find $Pr[X \leq 6| X > 4]$ I'm not familiar with the discrete uniform distribution. I was ...
-1
votes
3answers
54 views

Statistics (unsure how to do it)

A person's resting heart rate is the lowest number of heart beats per minute when fully relaxed and without distractions. Age, fitness, genetics, health status and gender affect the resting heart ...
-1
votes
0answers
24 views

Not sure what formula to use? (what to solve for?)

The question states, "The weight of people in a certain pacific island is normally distributed with a mean of 175 lb. and a standard deviation of 33 lb. They want to design a one-person canoe that ...
0
votes
1answer
18 views

Application of Law of Large numbers (1)

If we have an i.i.d random variable $X_i$ with mean variance $(\mu, \sigma^2)$. By Law of Large number, we have $\bar{X}\rightarrow^p \mu$. But can we use Law of large number as well and claim that ...