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

3
votes
0answers
304 views

Intuition for Fisher information metric

In statistical maniolds $S=\{p_\theta\}$,$\theta=(\theta_1,\dots,\theta_n)$, the Riemaanian metric usually defined is the Fisher information metric $$g_{ij}(\partial_i,\partial_j)=\int \partial_i(\log ...
2
votes
2answers
38 views

Would this still hold true if the probabilities of A and B followed a normal distribution?

Part a) Let A be the number that the left-hand die shows. Let B be the number that the right-hand die shows. Both dice are fair and therefore the probability that B is greater than A is equal to the ...
3
votes
1answer
237 views

Expectation of a compound random variable

Let $X_{1},X_{2},...,X_{n},...$ be independent random variables following the uniform distribution on $[0,1]$. Let $N$ follow the negative binomial distribution with the probability function ...
0
votes
1answer
440 views

Draw Random point Rectangle and Circle.

Plot $15$ points at random (a) inside a rectangle $3$ inches $\times$ $4$ inches. (b) inside a circle of $4$ inches. I think need to use Random Number table. It is a practical problem ...
4
votes
2answers
619 views

Two bags of colored marbles, different colors is each bag, how many combinations of complete pairings?

I've reduced a different problem I have to something that can be explained as below, but I'm not sure how to solve this last part. Assume there are two bags of colored marbles. Any color in the first ...
0
votes
1answer
381 views

Continuous random variables word problem

I'm having difficulty with a word problem on continuous random variables. Would you be able to help? Here it is: If you're n minutes early for an appointment, then you are charged n times 3 dollars, ...
1
vote
0answers
27 views

Confidence bounds on difference of sums

I am trying to get confidence bounds on the weighted sums of two buckets of iid normal random variables. I plotted the QQ plots using my variance estimate. The estimate works well for $N_A$, $N_B$ as ...
4
votes
1answer
901 views

Book on Measure Theoretic Statistics

I'm looking for a book, preferably a good one, on statistics from a rigorous, measure theoretic point of view. Ideally, this book should be introductory in nature and cover no more nor less than a ...
2
votes
1answer
482 views

Sufficient Statistic for a Parameter

Show that the sum of the observations of a random sample of size $n$ from a gamma distribution that has pdf $f(x,\theta)=(1/\theta)e^{-x/\theta}$, $0<x<\infty$, $0<\theta<\infty$, zero ...
1
vote
1answer
202 views

Obtaining the pdf of the sum of two iid random variables with pdf $f_{X}(x)= \sqrt{\frac{1}{2\pi x}e^{-x/2}}$

X1, and X2 are independent with the pdf: $f_{X}(x)= \sqrt{\frac{1}{2\pi x}e^{-x/2}}$ defined for x>0 Y=X1+X2 What is the pdf of Y? This is what I did so far: $$f_{Y}(y) = \int_{0}^{\infty } ...
1
vote
0answers
20 views

How is a statistic represented as a mapping with varying number of arguments?

Suppose that are $n$ sample points taking values in $A$. A statistic for sample size $n$ is a mapping from $A^n$ to another set $B$. But $n$ can vary in $\mathbb N$. There are also similarities ...
0
votes
1answer
794 views

Scaling a uniform distribution - Probability

I just have a simple question on scaling a uniform distribution. I know that uniform distribution has probability density of $1/(b-a)$ defined on the interval a to b. My textbook says that we can ...
-1
votes
1answer
261 views

Hypothesis Testing of a continuous distribution

Let $X_1, \dots, X_n$be a random sample of size $n$ from the continuous distribution with pdf given by $f_x(x|\theta) = \theta x^{\theta - 1} I(x)_{(0,1)} I(\theta)$. 1) Show that $T = ...
1
vote
0answers
62 views

Asymptotic behaviour of sums of covariances of RVs with LRD

Our assumptions are: $X_t$ is a stationary sequence of standard normal random variables such that $\gamma _X (k)\sim L_{\gamma}(k)k^{2d-1}$ with $d \in (0,1/2)$, where $L_\gamma (k)$ is a slowly ...
1
vote
0answers
311 views

Weighted linear least squares parameter covariance

I am currently trying to figure out the parameter covariance for a weighted linear least squares problem where $$y = X\beta$$ The parameters for which my objective function is lowest are given by ...
0
votes
1answer
42 views

Minimizing a function with vectors

This is a part of a problem that I'm having, and I'm unclear how to do this particular step. I'm dealing with a ridged regression and I need to minimize the equation $$\sum (Y_i - \beta^Tx_i)^2 + ...
1
vote
1answer
8k views

finding n value basic statistics

I have some practice homework questions. I did the first one I will go over the steps please tell me If I am doing it right. a) As mentioned earlier, it is claimed that 70% of households in Ontario ...
0
votes
1answer
84 views

Gaussian random variable(GRV)

$X$ is a Gaussian random variable $N(2,2)$. Also given are values x1=1 and $x=3. i. Write a program to calculate the probability Pr(x1 ≤ X ≤ x2). ii. Write a program to calculate the probability Pr(|X ...
0
votes
1answer
82 views

Statistics - Interval Estimation - Proof

Please tell me how to start on this proof or give me some kind of hint. Please click on this link to see the question Show that if $X_1,X_2,\dots,X_n$ denotes an iid sample from $N(\mu,\sigma^2)$ ...
0
votes
1answer
2k views

Find distribution, calculate expected value and variance of two random variables

I am given two random variables $X$ and $Y$, both are independent: and X distributioN: -2 -1 3 0.2 0.3 0.5 and Y distribution: -1 0 2 0.2 0.4 0.4 I have to create distribution and then I ...
2
votes
0answers
78 views

Likelihod ratio test: $f_0(x)=2x$ vs $f_1(x)=3x^2$ : $2n$ degree of freedom?

Let $\left\{ X_i \right \}$ be an $n$-sample with pdf $f$. Show that the likelihood ratio test statistic for \begin{align} H_0 &: f(x) = 2x \\ H_1 &: f(x)= 3x^2 \end{align} has a $\chi^2$ ...
1
vote
2answers
2k views

Expected value of the inner product of two random vectors

$X=[x_1, x_2,...x_n] Y = [y_1, y_2,...y_n]$ If $x_i, y_i$ are both random variables with $P(x=1) = .5$ $ P(x=2) = .5 $ $P(y=1)=.5$ $P(y=2)=.5$ How would I find the expected value of the inner ...
0
votes
1answer
96 views

Prove minimal sample size

A point estimator for a Bernoulli probability $\theta$ is $\hat{\theta}$ is $\hat{\theta}=\frac{X}{n}$ where $X$ is the number of successes in $n$ Bernoulli trials. Using the normal approximation ...
1
vote
1answer
30 views

How to show $(N-1)\sigma^2_{N-1}=N\sigma^2_N-\frac{N}{N-1}(Y_N-\mu_N)^2$?

Show that $$(N-1)\sigma^2_{N-1}=N\sigma^2_N-\frac{N}{N-1}(Y_N-\mu_N)^2$$ where $\sigma^2_N=\frac{1}{N}\sum_{i=1}^N(Y_i-\mu_N)^2$ and $\mu_N=\frac{1}{N}\sum_{i=1}^NY_i.$ Trial: \begin{align} ...
1
vote
1answer
66 views

Equations of significance probabilities

Consider a population of independent light bulbs with an exponential lifetime distribution with mean $\mu$. It is claimed that their expected lifetime is 1000 hours. A definition of a 100(1−$\alpha$)% ...
1
vote
1answer
69 views

Transformation of confidence intervals

I'm using Matlab to perform a linear regression. In order to prevent the prediction of negative values I used a box-cox-transformation of the dependent variable ($=y_t$) with $\lambda = 0.5$. ...
0
votes
1answer
110 views

mutual information problem

Consider the following problem: What is $I(X;Y)$ where $X$ is the outcome of a roll of a fair 6-sided die and $Y$ is whether the outcome of THAT SAME ROLL was even or odd? Intuitively, I thought ...
1
vote
1answer
66 views

$X_i \sim N(\theta,1), \theta \in \Bbb Z$: $T=\left\lfloor \bar X_n \right \rfloor$ not consistent for $\theta$

Let $\{X_i\}$ be a sample from a normal $N(\theta,1)$, where $\theta \in \Bbb Z$. Show the estimator $$ T=\left\lfloor \bar X _ n \right \rfloor, $$ is not consistent for $\theta$, with $\bar X_n$ ...
1
vote
1answer
169 views

Significance levels, mean testing

Slabs of toffee are specified by a manufacturer to have a mean weight of 200 g and the standard deviation is known to be 5g. Weights are assumed Normally distributed. It is found that two samples, ...
0
votes
1answer
3k views

Matlab Time Series (AR, MA, ARIMA)

Is there a function which calculates an AR(p), MA(q), ARIMA(p,q) process in MATLAB which is free. I know of Econometrics toolbox, but I have to pay for that. Is there a way to get around that?
0
votes
1answer
7k views

How to calculate standard deviation for a series containing both positive and negative numbers?

I got a stream of numbers in one of my apps to represent an electrical signal. I've observed that the signal ranges from -100 to +100. Other than that, the signal is fairly random and crosses 0 in ...
-1
votes
1answer
73 views

How to calculate the following variance?

I want to calculate the expectation and variance in the following scenario: $w$ is my initial wealth With probability $0< q_i <1$ with$ i \in\{a,b,c\}$ I lose $a,b$ or $c$ repectively. $U()$ ...
2
votes
1answer
201 views

How do I calculate the cdf over this range? I'm very confused.

I don't want the answer, I just want an explanation. Say $F(x) = 0.25 x^2$ for $0 \leq x < 2$. If $F(x) = 1$ for $x \geq 2$ and $F(x) = 0$ everywhere else, how would I calculate $P(X \leq 1.5)$?
2
votes
3answers
5k views

Deriving the mean of the Geometric Distribution

I am missing something that might be trivial in deriving the mean of the geometric distribution function by using expected value identity $$ \sum_x x \theta (1-\theta)^{x-1}. $$
0
votes
2answers
108 views

Central limit theorem using it with formulas

1.The average amount of money spent at lunch in the wissakion cafeteria is 3.00 dollars with a standard deviation of 75 cents. Assume the distribution of money spent is normal. a. What is the ...
0
votes
1answer
64 views

Probability Coupled with Payout

My brother and I made a bet which I took. He had two coffee cups which were roll up the rim to win. The odds of a cup winning were 1 in 6. He was feeling very confident and said "if I don't win ...
2
votes
3answers
97 views

Poisson probability differs from combinatorial probability, why?

I am trying to figure out why the Poisson probability differs from the combinatorial probability. For example, assume that 10% of people are left handed. What is the probability that a classroom of ...
0
votes
1answer
4k views

Likelihood function of a gamma distributed sample

I missed the day of class where we went over likelihood functions, and I had a quick question. If $X_1,...X_n$ are i.i.d. $ {\Gamma}(\alpha,\beta)$ r.v.s, I'm trying to find the likelihood function ...
0
votes
1answer
266 views

Better alternative to run tests for testing randomness

Is there a better alternative to using run tables for testing for randomness in a sample? Currently, for statistical process control charting, I am testing the number of runs identified in a sample ...
1
vote
1answer
49 views

Homework help with statistics

I have just answered a question and i am not to sure whether i have got it correct or if i have missed something. Question: Suppose that X and Y are independent random variables, with N(-2,2) and ...
0
votes
1answer
28 views

Statistical variable transformation

i'm preparing my statistics exam and I dont know how to solve the following exercice: Let X be a Statistical variable having the following values: $x_1$,$x_1$,...$x_r$. It is known that ...
2
votes
1answer
998 views

Negative Binomial Question Without Exact Values

The question I am working on is: Three brothers and their wives decide to have children until each family has two female children. What is the pmf $X=$ the total number of of male children born to ...
0
votes
1answer
319 views

How can I determine number of votes cast based on percentages before and after voting?

I have a poll with two options, it does not display the number of votes cast. prior to casting a vote, the ratio was 91.5% to 8.5%. After a single vote cast, the percentages moved to 91.7% and 8.3%. ...
1
vote
1answer
225 views

Probabilities using Z scores help

How to calculate the probability of a z-transformed value does not appear in the table? e. g What is the probability of randomly drawing a Z-transformed value of 2.465 or greater? P(Z ≥ 2.46) = 0.0069 ...
4
votes
0answers
150 views

Using Bernoulli distribution approximate the $q$-th moment

Let $x$ be vector in $R^n$. Let $\pi(⋅)$ be a permutation on the set $\{1,\ldots,n\}$ with a uniform distribution. Let $|m|\leq n, m \in Z$. Using Bernoulli (or maybe some other) distribution ...
4
votes
3answers
2k views

Definition of an “Experiment” in Probability

One can define the fundamental concepts of probability theory (such as a probability measure, random variable, etc) in a purely axiomatic manner. However, when we teach probability, we start off with ...
1
vote
1answer
174 views

A Problem about Hypothesis Testing and Decision Making

I have a problem about the hypothesis testing and decision making as follows: A botanist wishes to test the null hypothesis that the average diameter of the flowers of a particular plant is ...
3
votes
1answer
2k views

Taking the derivative of definite integral?

I'm having trouble understanding the derivative of definite integral. For example, why is the following true? $\frac{d}{dx}\displaystyle\int_{0}^{x}F_{1}(x-v)f_{1}(v)\, \mathrm{d}v = ...
0
votes
1answer
590 views

Hypergeometric Distribution Problem With Rocks!

The question I am working on is: A geologist has collected 10 specimens of basaltic rock and 10 specimens of granite. The geologist instructs a laboratory assistant to randomly select 15 of the ...
1
vote
0answers
78 views

How to determine the degrees of freedom of an inner-product matrix of two random matrices?

I have two random matrices A and B, with N columns each. The columns of A and B are independent but not necessarily identically distributed. A and B may be considered as two instances of an underlying ...