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

0
votes
1answer
62 views

Need help finding expected value and variance [closed]

Let $Y$ be a random variable with $E(Y) = \mu$ and $Var(Y) = \sigma ^2$ . Let another random variable $X = Y +c$, where $c$ is a known constant. Show that $E(X) = \mu +c$, and $Var(X) = \sigma ^2$ . ...
1
vote
1answer
61 views

Finding X with cdf $F(x)=1-\exp\left(-x^3\right)$

Suppose the following cdf $F(x)=1-\exp\left(-x^3\right), x \geq 0$ . How can I generate a stochastic variable $X$ with this cdf using the function runif() in R? Is ...
0
votes
1answer
37 views

Trouble solving these two equations

I'm doing a homework on uniform distribution and they gave me then mean and the standard deviation. I figured I had to solve for a and b to get min and max values. But when I try to solve for a and ...
0
votes
2answers
57 views

Probability question fair die roll

A fair die is rolled three times. What is the probability that the sequence of rolls is 1,2,3? I just don't understand how to go about this problem at all, it confuses me. Any help would be ...
0
votes
1answer
74 views

How can solve this integral

How ı can solve this integral, ı thınk that ı can seperate as above ı did but I dıd not do it, thanks for helping..
0
votes
1answer
125 views

Error Propagation - functions of the mean.

Given a number of measurements $\{x_i\}$ with values distributed according to a (known) probability distribution $\rho(x)$ with a theoretical mean $\langle x\rangle = \int dx x\rho(x) = f(y)$ and a ...
0
votes
1answer
104 views

what is the difference between maximum likelihood estimation and usual probability inference? [closed]

Can somebody tell me a clear difference between MLE from the usual probability inferences?
1
vote
1answer
75 views

What is the minimum Premium to be asked for a risk X?

Suppose that an insurer has an exponential utility function $u(x) =-2e^{-2x}.$ What is the minimum premium $P^{-}$ to be asked for a risk X? I got some hint for this, but I could not understand ...
3
votes
2answers
719 views

Variance of a MLE $\sigma^2$ estimator; how to calculate

Let $X_1, X_2,...,X_n$ be an i.i.d. random sample from $N(0, \sigma^{2})$. a. Find the variance of $\hat{\sigma}^{2}_{MLE}$ So I found $\hat{\sigma}^{2}_{MLE}$ by taking the derivative of the log ...
0
votes
1answer
615 views
0
votes
0answers
176 views

Checking to see if sample average weight is statistically larger than the standard weight?

For the fallowing question, we need to assume that the sample standard deviation is 16. We need to check to see if the sample average weight of 177 is statistically larger than the standard weight of ...
0
votes
1answer
80 views

statistics problem concerning probability, standard deviation, and mean

Keeneland is a popular horse racing track outside of Lexington that happens to have several drive-thru betting windows for people who would like to place wagers on races without entering the ...
1
vote
1answer
44 views

An equation related to covariance matrix, square root of the matrix, and Euclidean norm.

How can I prove this equation: $${ ({ x }^{ T }\Sigma x) }^{ 1/2 }={ \left\| { \Sigma }^{ 1/2 }x \right\| }_{ 2 }$$ In which $\Sigma $ is a covariance matrix. I tried some numerical examples in ...
1
vote
1answer
1k views

Relationship between Poisson and Exponential distribution, automobiles arrive per minute

I am struggling to understand the Poisson and Exponential distributions. The number of automobiles that arrive at a certain intersection per minute has a Poisson distribution with a mean of 5. ...
1
vote
0answers
89 views

If $X$ is a random variable, under which conditions is $g(X)$ also a r.v.?

In many instances, functions of random variables appear, and we usually treat them as random variables also. In the 3d edition, pp. 85-86, of this well-known book (now in its 4th edition), we find the ...
2
votes
4answers
5k views

Scaling the normal distribution?

I might just be slow (or too drunk), but I'm seeing a conflict in the equations for adding two normals and scaling a normal. According to page 2 of this, if $X_1 \sim N(\mu_1,\sigma_1^2)$ and $X_2 ...
1
vote
1answer
171 views

Stats - the Central Limit Theorem

Bottles filled by a certain machine are supposed to contain 12 oz of liquid. In fact the fill volume is random with mean 12.01 oz and standard deviation 0.2 oz. What is the probability that the mean ...
2
votes
0answers
74 views

Have averages, need variance

I have a large spreadsheet, generated by a colleague, that contains the results of $E$ experiments. For each experiment, he calculated the average of $M$ measurements. I need the calculate the ...
1
vote
1answer
165 views

Find the MLE of bivariate normal

Suppose that $X = (x_{ij})n*2$ follows a bivariate normal distribution $\mathcal{N}(\mu, \sigma^2I)$, where I is the $2\times 2$ identity matrix. How to find the maximum likelihood estimates of $\mu$ ...
2
votes
1answer
50 views

How I figure what (theoretical) dice are needed to achieve a certain curve?

I am posting here by suggestion of RPG.se I want to make character stats that fit within certain bell curves depending on choices during character creation (for example race, gender, class, sprokets ...
1
vote
4answers
92 views

Displacement law: $ \sum_{i=1}^{n} (x_i - \bar{x})^2 $ why is it squared?

$ \sum_{i=1}^{n} (x_i - \bar{x})^2 $ where $ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i $ (mean value) I would like to understand why the term inside the sum gets squared. Why isn't it enough to e.g. ...
3
votes
1answer
1k views

For the binomial distribution, why does no unbiased estimator exist for $1/p$?

Suppose that $X \sim \mathrm{Binomial}(n,p)$ for $0 < p < 1$ Why does no unbiased estimator exist for $1/p$? My approach: We try to find the structure of $E_p(U(x))$, where $U(x)$ is any ...
1
vote
2answers
163 views

Statistics-Math-Probability

An urn is filled with 10 green balls, 4 red balls, and 2 orange balls. Three balls are selected without replacement. Calculate the following probabilities: (a) P(at least one ball is red) I think ...
3
votes
0answers
154 views

Intuition behind (statistical) completeness

I was wondering if any of the members of the MSE community would like to share his/her intuition about completeness in statistics. For the sake of "completeness", here's the definition, taken from ...
1
vote
1answer
47 views

Regression towards the mean?

I am upset with what examples testing "regression to the mean" seem to allude to: People claiming to have "ESP" take a test, and A's score was 2 standard deviations above the mean, whereas B's score ...
1
vote
1answer
39 views

Statistics binomials

A recent medical survey reported that 45% of the respondents to a poll on patient care felt that doctors usually explain things well to their patients. Assuming that the poll reflects the feelings ...
1
vote
1answer
80 views

Meaning of off-diagonal multivariate covariance matrices

My terminology might be a bit sloppy. I apologize in advance. I'm reading on multivariate probabilistic distributions, particularly on Gaussian normal distribution (in the context of probabilistic ...
1
vote
1answer
128 views

Robust Standard Errors

For OLS, my professor said that you should always test for heteroscedasticity first, rather than going straight to the adoption of robust standard errors. I didn't quite follow this and no ...
2
votes
1answer
332 views

Correlation Coefficient Distribution Function: An Apparent Discrepancy?

I'd like to explain an apparent discrepancy between: (1) The sample correlation distribution function between sample vectors for a bivariate, correlated random variable (correlation coefficient = ...
2
votes
0answers
83 views

How many different ways can 10 octupuses touch legs?

There are 10 octopuses (octopi?). Each octopus has 8 legs. Legs on an octopus can only touch touch legs on other octupuses. Assuming each leg touches exactly 1 other leg, how many different ...
3
votes
0answers
102 views

Expected minimum of a finite random walk.

So I couldn't find any resource for how to calculate the expected minimum of a random walk. Since it is such the minimum of the random variables are actually not independent as they are cumulative ...
1
vote
1answer
45 views

Variance with correlated variables

A simple question that I don't manage to solve: I can use different methods to measure a magnitude $x$. The results of these methods are correlated and have some uncertainties. Combining the results ...
2
votes
0answers
45 views

What is Lagrange's Identity in terms of Covariance and Variance?

This is Lagrange's Identity What I've to put in the middle of this relation instead $\color{red}{?}$, which led to find Lagrange's Identity in terms of variance and covariance? $$\lvert ...
1
vote
1answer
328 views

Why isn't a t test used when comparing two proportions?

All the examples I've seen say to use a z test to compare two proportions. For example, n=13, x=0.22 versus n=10, x=0.44. Then all the examples warn that the z test doesn't work with low sample sizes. ...
0
votes
1answer
55 views

Differentiability of linear least squares

Show that least-squares $\|y-X\beta\|^2$ is twice differentiable and has minimizer. I understand that the second derivative is $X'X$. Also it is a composition of linear function which is ...
1
vote
1answer
170 views

Stat question about Markov Chain

An urn contains five red and three green balls. The balls are chosen at random, one by one, from the urn. If a red ball is chosen, it is removed. Any green ball that is chosen is returned to the urn. ...
0
votes
1answer
909 views

Standard error of Method of Moment estimator

Suppose that $X$ is a discrete random variable with $P(X=1)=θ$ and $P(X=2)=1−θ$. Three independent observations of $X$ are made: $x_1=1,x_2=2,x_3=2$. Find the MOM estimator of $θ$. What is ...
0
votes
1answer
50 views

Guessing the sex of my app user based on the overall app downloads statistics

I have got an app that has 72% male, 22% female, and 7% unknown users. In order to make more money from displaying ads, I want to pass the sex of the user to the ad server. How close would I be if I ...
0
votes
1answer
58 views

Question of Normal Distribution ??

I do some exercise, and there is an interesting question, can anyone can help me solve it. suppose that the heights in inches of the women in a certain population for a normal distribution with mean ...
0
votes
2answers
136 views

maximum likelihood estimation problem tutorial

The question is find the δ by The maximum likelihood estimation? My answer is δ=0 but I am not sure whether it is correct and how tho show its biasness?
2
votes
1answer
71 views

sampling distribution with no continuity correction

Assume that the distribution of the height of a plant is normally distributed with a mean height of $\mu = 83.4$ inches and standard deviation of $\sigma = 9.1$ inches. Find: if three such plants ...
1
vote
0answers
85 views

Alternate form of Cramer-von Mises type statistic

When I read a paper from Rothman and Woodroofe (1972) for testing symmetry, they used the following form: $$ R_n = \int_0^1 n \left[\hat{F}_n (x) + \hat{F}_n (-x) - 1 \right]^2 dF_n$$ where $ ...
0
votes
1answer
73 views

How to calculate two asset portfolio (problem finding the Population correlation coefficient)

I have two assets: A has an expected value of $12$ % and a standard deviation of $8$%. B has an expected value of $15$ % and a standard deviation of $12$ %. Suppose that we invest $75$ % in A and ...
0
votes
1answer
543 views

probability continuity correction

consider taking a random sample size of 25 from a population in which 42% of the people have type A blood. what is the probability that the sample proportion with type A blood will be greater than ...
0
votes
2answers
317 views

How leafy are my plants?

Suppose a botanist grows many individually potted eggplants, all treated identically and arranged in groups of four pots on the greenhouse bench. After 30 days of growth, she measures the total leaf ...
2
votes
1answer
47 views

Disagreement about rejection region for upper-tailed hypothesis test

If we look at the solutions for the first problem here and the first problem here, we see both problems are one-tailed tests for the upper-tail. However, in the first paper the rejection region is Z > ...
2
votes
1answer
60 views

How to calculate an expected value from a cluster of results

I have a vector of points r where each element has an x and y value. For every one of those points there is a z value, which corresponds to a measurement taken at position r. If I have say n ...
2
votes
2answers
2k views

X follows an exponential distribution, calculate Expected value of sqrt(X).

Problem: Let X follow an exponential distribution with expected value of 1. Define Y=sqrt(X). Calculate E(Y). This is my first course in probability theory (5 weeks ≈ about 5*40 hours of workload) so ...
1
vote
2answers
87 views

Variance of difference of two sample means from the same population

We take two samples from $Z$ of size $n_1$ and $n_2$ and take the difference of the mean of these samples. Both should have the same expected value, so the mean is zero. But what is the variance of ...
2
votes
1answer
438 views

Differentiation for least squares method?

Is there any reason that we use mathematical differentiation of least squares method for regression analysis? The theory say we use differentiation supposing the sum of errors is 0. I I don't really ...