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.

learn more… | top users | synonyms

1
vote
1answer
82 views

Find an unbiased estimator

Let $X$ be an r.v defined by $P(X=0)=p$ and $P(X=1)=1-p$. Find an unbiased estimator for $2p$. My solution: $E(X)=1-p$ so $2-2E(X)$ is unbiased. Is this correct?
2
votes
1answer
159 views

Expectation by conditioning

Problem: Independent trials, each of which is a success with probability $p$, are performed until there are $k$ consecutive successes. What is the mean number of necessary trials? Start on solution. ...
2
votes
1answer
105 views

Confidence interval for n=1, unknown standard deviation

Is there any way to calculate a confidence interval (or otherwise gauge the reliability of a sample) when you have a sample size of one and you don't know the population standard deviation? I work at ...
2
votes
2answers
1k views

Meaning of covariance

Can someone please give me an intuitive explanation for the meaning of covariance between two random variables? What does it measure?!
0
votes
1answer
2k views

Upper and Lower Bounds for Confidence Intervals on a one tail hypothesis test.

CI(99) = xbar ± Zcv( σ / √ n -1 ) My question is if the test is a one tail hypothesis test, do I still do two separate equations + and -. Here is the formula filled in (if it is the correct formula)...
0
votes
1answer
15 views

Why am I getting 84.25 instead of 97.68 for my Confidence Interval?

Why am I getting 84.25. I am doing 97.96*(6/sqrt[50-1]) = 97.96*(6/7) I am obviously not doing this correctly please help. It has been sometime since I have done any math.
0
votes
1answer
44 views

statistical significance in non normal distributed data

I have randomized strings undergoing two different tests. There's a pretty clear difference in their proportions, but it reminded me of apstats where you would often do 2prop Z-tests to compare to ...
2
votes
1answer
83 views
0
votes
1answer
40 views

Covariance Problem

I'm trying to find the solution to $cov(y_i, \frac{\sum{y_i}}{n})$, where $y_i = \beta_0 + \beta_1x_i + \epsilon_i$ (a regression model). Here's what I have so far... $cov(y_i, \frac{\sum{y_i}}{n}) ...
1
vote
1answer
65 views

Equality of joint probability distributions

Suppose I have $X$ and $Y$ as random variables and have their joint probability distributions. Is the following correct: $f_{X,Y}(x,y) = f_{Y,X}(y,x)$ ?
2
votes
1answer
96 views

Do I use the Standard Deviation of my sample or the population to find the standard error.

A professor is interested in determining if attending college influences the level at which an individual cooperates with the police. The professor is unsure if attending college will teach respect ...
0
votes
2answers
129 views

Understanding sampling from a normal distribution with zero mean

I'm studying probability. I came a cross "sampling from distributions". Given a probability density function $f_X(x)$, what I understood is that sampling means getting values of $x$ according to the ...
0
votes
1answer
318 views

How many times to roll a die before getting $n$ consecutive sixes given $m$ occurences?

Given a unbiased dice how to find the average number of rolls required to get $n$ consecutive sixes given already $m$ number of sixes occurred where $m\leq n$. I know how to solve using n consecutive ...
1
vote
0answers
26 views

Average of part of poisson distribution

Is there a way to find the average of part of poisson distribution? I tried it by finding the center of gravity of that part. However I faced the integration of factorial in the denominator of the pmf ...
1
vote
1answer
5k views

Standard deviation with exponential distribution

Let x denote the distance that an animal moves from its birth site to the first territorial vacancy it encounters. Suppose that x has an exponential distribution with parameter lambda = 0.01386. a. ...
1
vote
1answer
84 views

Find the probability generating function

I have an exercise of this type that I just can not solve "Are $x$ and $y$ be independent random variables, $X$-Poisson($a$), $Y$-Poisson($b$). Find the probability generating function of the random ...
1
vote
1answer
47 views

T statistic has t(n-1) distribution

I am trying to prove that $T_n=\frac{\bar{X}_n - \mu}{S/\sqrt{n}}\sim t_{n-1}$. One of the assumptions that seems to come up in proofs I saw of this is that $\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}\...
0
votes
1answer
16 views

Get overall sum having partial data

I have partial data of online game ingame sales. I know the data for N users. The overall game users count is M which is about 6 times greater than N. I can calculate overall revenue for N users. But ...
1
vote
2answers
23 views

Binomial probability on ports

This problem appears very simple, but I am almost positive that it should not be so simple. 10 ports. P1,P2,P3...P10 are connected to a computing device which polls them in order to check which ...
0
votes
1answer
308 views

Probability statements in a true or false format

The following questions are to be answered as preparation to my exam next Friday. I feel I understand the terms, such as "complement", "union", and "intersection", but when confronted with questions ...
1
vote
1answer
98 views

Normal Distribution - Statistics

True or false statements The central limit theorem implies that any sample of any size can be approximated using normal distribution. Let X be a normal random variable with mean 3 and variance 16. ...
1
vote
1answer
60 views

How do we square a random variable?

How do we square a random variable? For example, Let $Y=X^2$. $$f_X(x)={\frac{1}{\sqrt{2\pi}}} \cdot e^{\tfrac{-x^2}{2}}$$ How do we derive $f_Y(y)$? Thanks in advance.
0
votes
1answer
25 views

Test for Validity of Artificial Samples

I have a model that actually is learned from some observed samples. Then I use the model to generate several artificial data. My question is: Which test should I use to test if the data is of the ...
2
votes
0answers
40 views

Quantitatively comparing event trains of different lengths for Poissonness

I have a parameterized, effectively black box process that generates a series of events (simulated action potentials). Different parameter values often lead to different numbers of events. How can I ...
0
votes
1answer
64 views

Confidential interval for ratio of mathematical expectations

I've got 2 sequences of Bernoulli random variables: ${X_1, ... , X_n}$ and ${Y_1, ... , Y_n}$ For each mathematical expectations ($EX$ and $EY$) of these sequences i can find confidence intervals. But ...
0
votes
1answer
26 views

Find alpha when we know valuse of student distribution

Assume, we have $t_{1-\alpha/2,39} = 2.252401$. I don't know how can find $\alpha$. Please help me and how find it in R.
0
votes
1answer
69 views

What is the corrected sample standard deviation when there is only one sample?

What do we say about the corrected sample standard deviation when the sample size is one? Do we just assume it is 0? Is it even meaningful? The usual equation seems to fall apart here because N - 1 ...
1
vote
1answer
38 views

Solving for a ridge penalty given a fitted model

This is kind of embarrassing; I once knew this stuff, and I've forgotten it. I've got a fitted ridge regression: $$ \hat\beta = \left(X'X+\lambda\right)^{-1}X'y $$ X is n by k y is n by 1 $\lambda$...
1
vote
1answer
43 views

Help with the birthday probability problem

Say there are lottery tickets with three numbers. Each number has $100$ possibilities. There are $N$ tickets sold where $N > 1000$. I've tried $$\frac{1 − 100!}{(100 − N)! 100^N}$$ ...
0
votes
2answers
110 views

Returns probability

I have following investment data: mean-8 standard deviation 15 Assume that if it is normal distribution, what is the probability that returns will exceed 23%. How to calculate this?
1
vote
1answer
581 views

Method of Moments and Maximum Likelihood question

Suppose that $X_1,X_2,…,X_n$ are an i.i.d. random sample from a Rayleigh distribution with parameter $\theta > 0, f(x|\theta) = \frac{x}{\theta^2}e^{-\frac{x^2}{2\theta^2}}, x>=0$ Find the ...
0
votes
1answer
164 views

questions on bias of estimator

a) Let $X_{1},...,X_{n}$ be i.i.d Uniform$[0,\theta]$. Show that estimator $\beta(X)=max(X_{1},..,X_{n})$ is a biased estimator for $\theta$.Find an unbiased estimator, based on $\theta$. My attempt: $...
2
votes
1answer
223 views

What is the pivotal quantity

I had a question of two parts. I solved the first part but I am stuck on the second. Any hints or partial solutions would be greatly appreciated. a)$ X_1,....X_n$ are uniform iid on the interval $(0,\...
1
vote
1answer
102 views

what value for $c$ yields the estimator for $σ^2$ with the smallest mean square error among all estimators of …

If $S'^2 = \dfrac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n}$ and $S^2 = \dfrac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n-1}$ then $S^{'2}$ is a biased estimator of $σ^2$, but $S^2$ is an unbiased estimator of the ...
1
vote
4answers
9k views

Establishing the upper and lower bounds of normal using standard deviation

I understand the concept of standard deviations and z-values, but I'm trying to figure out if standard deviations alone are good for establishing the upper and lower bounds for normal. For example, if ...
0
votes
0answers
37 views

standard deviation estimation after changing part of data to be constant

If we have a sample of known average and standard deviation, for example, 5 and 1. Suppose that we have a constant for which the CDF is 10%. The values below the constant are set to equal the constant....
0
votes
1answer
32 views

some misunderstand about student distribution and R

I see on book : $t_{0.025,59}$ = 2.009 But when call qt in R. We have this result: qt(0.025,59) [1] -2.000995 I don't know why have different in here. As I think, $t_{0.025,59}$ must is ...
0
votes
1answer
261 views

Prove that the usual (1-$\alpha$)% confidence interval for $\sigma^2$ is NOT the shortest interval.

Prove that the usual (1-$\alpha$)% confidence interval for $\sigma^2$ is NOT the shortest interval. In particular, show that the minimum length interval satisfies $f_{(n+3)}(a) = f_{(n+3)}(b)$, where ...
1
vote
0answers
185 views

If $S'^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n}$ and $S^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n-1}$, find $V(S'^2)$.

If $S'^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n}$ and $S^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n-1}$ then $S'^2$ is a biased estimator of $σ^2$, but $S^2$ is an unbiased estimator of the same ...
1
vote
2answers
957 views

show that MSE$(\hat{\theta}) = E[(\hat{\theta} − θ)^2] = V(\hat{\theta}) + (B(\hat{\theta}))^2$.

Using the identity $(\hat{\theta} − θ) = [\hat{\theta} − E(\hat{\theta})] + [E(\hat{\theta}) − θ] = [\hat{\theta} − E(\hat{\theta})] + B(\hat{\theta})$, I need to show that MSE$(\hat{\theta}) = E[(\...
1
vote
1answer
40 views

Marginal Density Question

I am faced with the following question, which I think is quite simple, but I can't put together for some reason. Given that $f(x,y)=(6/5)(x+y^2)$ for $0<x,y<1$, ($f(x,y)=0$ everywhere else), I ...
2
votes
1answer
226 views

If $ X = \sqrt{Y_{1} Y_{2}} $, then find a multiple of $ X $ that is an unbiased estimator for $ \theta $.

Problem: Suppose that $ (Y_{1},Y_{2},Y_{3},Y_{4}) $ denotes a random sample of size $ 4 $ from a population with an exponential distribution whose probability density function $ f $ is given by $$ f(...
0
votes
1answer
20 views

Help with Poisson Random Variables.

The problem is if $\lambda=1/2$. Find $E[X]$, $E[2-X]$, $E[X^2]$ and $Var[2X]$. I know that $E[X]$ is simply $1/2$. But as for finding the other ones, I am lost. I'm assuming I'll have to create ...
1
vote
2answers
197 views

Find the distribution of $W$

Let $X \sim N(0,1)$ and $Y \sim N(0,1)$, independent. $$W = Y \;\;\text{if } Y-X >0, \;\;\;\;\;\;\;\;\; W = -Y \;\;\;\; \text{if } Y-X <0 $$ Then, find the distribution of $W$ Here is a my ...
0
votes
1answer
27k views

Percentage with only standard deviation and mean given. [closed]

I have some questions that I really need help with. The mean mark for an IQ test in the population is 100, with a standard deviation of 16.5. The IQ is normally distributed. Your IQ is 113. a. ...
0
votes
1answer
22 views

Did I correctly calculate the specificity and the false negative rate?

So I filled out the summary table of the data, but I'm not quite sure if I calculated the specificity and the false negative rate correctly from the table. Can someone please check that I'm doing it ...
0
votes
1answer
246 views

Take -log of a Beta distributed R.V.

X1.....Xn~Beta(a,1) Y = -log(X) Use the transformation formula to calculate the pdf of Y. What named distribution does it have? I am confused what method to use here. A beta does not converge to a ...
-1
votes
1answer
73 views

Expectation formula proof [closed]

Let $X$ have a normal distribution with mean $\mu$ and variance $\sigma^2$. Prove that $E(X-\mu)^2$=$\sigma^2$
1
vote
1answer
139 views

Statistics and Probability, finding unbiased estimates of mean and variance given sigma x and sigma (x^2)

The random variable $X$ is normally distributed with unknown mean $\mu$ and unknown variance $\sigma^2$. A random sample of $20$ observations on $X$ gave the following results $\sum_i X_i = 280, \...