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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Derive minimum length confidence bounds for a F distribution variance …

Derive minimum length confidence bounds for a F distribution variance $\sigma^2$ and the ratio of two F distribution population variances $\frac{\sigma_1^2}{\sigma_2^2}$. What I got so far is $$ P[...
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54 views

sum of two normal distributed random variables

Consider $Z = \frac{1}{2}(X+Y)$ where X and Y are normal distributed variables. It is easy to show that the variance of $Z$ is $1/2$ and the mean is zero. I'm stuck because I want to show this using ...
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107 views

Given $X$ and $Y$ are independent N(0,1) random variables and $Z = \sqrt{X^2+Y^2}$ from the marginal pdf of $Z$

Let $X$ and $Y$ be independent $N(0; 1)$ random variables. Let $Z = \sqrt{X^2+Y^2}$. (a) Derive the marginal pdf of $Z$ and then using the marginal pdf to compute ${\rm E}[Z^2]$ (b) Can you propose ...
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130 views

LR Test for Exponential Family of Distributions

LR Test for Exponential Family of Distributions: The exponential family of distributions, both discrete and continuous, based on a parameter θ is defined by: f (x |theta) = c(x)d(theta)exp[a(theta)...
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56 views

What exactly are order statistics?

I'm very confused by what these are. My definition reads: Let $Z_1,\dots,Z_n$ take the values of random variables $X_1,\dots, X_n,$ arranged in increasing order so that $Z_1\leq \cdots\leq Z_n$. We ...
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82 views

expected distance

Let P be a point chosen at random from the unit disk. The unit disk is the disk of radius 1 centered at the origin. It is described by the condition $x^2 + y^2 \le 1$. "At random" means "according ...
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53 views

random number generator statistics

A random number generator has produced the following sample from the uniform distribution on $(0,1): Y_1=.12, Y_2=.87, Y_3=.61.$ Use this and the recipe in (b) of the Special Transformation Theorem to ...
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195 views

Probability applied to economics

The following two questions are based on the wondrous Statistical topic probability. After attempting both questions I have yet to answer either correctly. If anyone has encountered similar problems ...
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110 views

Ancillary statistics and the use of Basu theorem

Let Y1, Y2, ....., Yn be the order statistics of a random sample of size n from the distribution exp(theta) If R= ( n Y1 / M) where M = summation Yi . To show that R and M are independent ?? ...
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1answer
911 views

Mean time spent in transient states/Markov chain

I dont get this in my book: For transient states $i$ and $j$ , let $s_{ij}$ denote the expected number of time periods that the markov chain is in state $j$ , given that it starts in state $i$. Let $\...
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178 views

A point chosen at random from a disc

I've been working on this question and have managed to complete parts (i) through (iii) but am struggling with the last two parts. For (iv) I end up getting this when trying to find the distribution: ...
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99 views

Normally distributed sum of values

Mike's bowling scores are normally distributed with mean 110 and standard deviation 13, while Jack's scores are normally distributed with mean 135 and standard deviation 10. If Mike and Jack each ...
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37 views

Joint continuous random variable conditional probablity

Okay so here is the problem: The PDF is $$f(x,y)=6(1-x)\;\; 0\leq y \leq x \leq 1$$ The question asks find $P(0<Y<0.25\;|\;X=0.5)$. I approached the problem like this $$P(0<Y<0.25\;|\;X=0....
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1k views

Uniformly Distributed Random Variables Minimum

Let A,B,C,D,E be independent random variables, each of which is uniformly distributed in the interval [0,18]. Let X=min{A,B,C,D,E}. What is the expected value of X? Enter your answer as a decimal. ...
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Find a 99% confidence interval for the population mean for a sample of size 36 that yields a mean of 78 and a standard deviation of 6…

Find a 99% confidence interval for the population mean for a sample of size 36 that yields a mean of 78 and a standard deviation of 6. Assume sigma, the population standard deviation, is unknown. A) (...
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265 views

Proving Unbiased estimators

Hello all, Here is a question I am struggling to understand, ...
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53 views

Determining how accurate an ellipse fit is

So I have an image of bacteria particles which are often shaped very irregularly with many grooves. Im trying to fit ellipses onto these particles so I can get a better, more smooth analysis of the ...
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1answer
24 views

How to know when samples tend toward lowest number?

I am working on a computer simulation. In a perfect world, I only want to know what the minimum result value is from the simulation. (I have no a-priori knowledge about the range of numbers or their ...
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387 views

The golden ratio in statistics of literature

Let a book, for example, or a poem... It consists in words and letters and symbols like : ;,!... Let $W_b$=the number of words of the book. Let $L_b$=the number of letters of the book. The number $N_b$...
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32 views

Hypothesis testing using chi-square distribbution

Four players meet weekly and play eight hands of cards. Over a year, one of the payers finds that he has won x of the eight hands with frequency fx given in the following table: x 0 1 2 3 4 5 6 7 8 ...
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469 views

Green balls and Red balls, probability problem

I'm studying for my exam and I came across the following draw without replacement problem : ...
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2answers
61 views

Why does $E[X]$ not equal the integral of $f(x)^2$

If $X$ is a random variable with the pdf $f(x)$ and $Y=g(X)$ how come $E[Y]$ is the integral of $g(x)f(x)$ but $E[X]$ is the intergral of $xf(x)$ ??
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143 views

What's the MLE of lambda for $f(x)= \frac{1}{λ}\exp{\frac{−x}{λ}}$?

What's the MLE of $\lambda$ for $$f(x)= \frac{1}{λ}\exp\left({\frac{−x}{λ}}\right)$$ Values of x are 5,7,9,3,6,8 Is it just the mean of $x$? Thanks.
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stats - limiting distribution of $X_i$

Suppose $P(X_{n}=i) = \frac{n+i}{3n+3},$ for $i = 0, 1, 2$. Find the limiting distribution of $X_{n}$. Is the cdf $F_{x_{n}}$ like: " if x = 0, $F_{x_{n}} = \frac{n}{3n+3}$; if x = 1, $F_{x_{n}} = ...
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Conditional probability - finding a single event

Context: $P(L) = 0.1$, $P(M | L) = 0.45, P(M | L') = 0.51$ $M =$ Male, $M' =$ Female, $L =$ Left handed, $L' =$ Right handed. 1. Find $P(M)$. So I knew I had $P(M | L)$ and $P(M | L')$ so I ...
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111 views

Coefficient of variation - Help me decode this text

Doing some exercises I hit upon this wall of text: Coefficient of Variation should only be computed on ratio scales (i.e., data where there is a "true" zero, like temperatures in Kelvin, or ...
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113 views

Standard Deviation and Mean

take a look at this question: The sample from Population A has a mean of 35 and a standard deviation of 1. The sample from Population B has a mean of 45 and a standard deviation of 15. Which of ...
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41 views

Randomized Block Design for repeated trials - Please help

Suppose I measure reaction time of 4 students, initially before any treatment, then after treatment 1, and finally after treatment 2. (Note all treatments will eventually be applied to all students) ...
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60 views

Question on finding the Jacobian

I have a question which is as follows: The random variable U has the pdf $n \dfrac{u^{n-1}}{\theta^{n}}$ , $ 0 \le u \le \theta$ This is not independent of the parameter θ. Let Y=U/θ. Use the ...
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36 views

some knowledge about comare means of two group

I'm studying a bout statistic. I can compute very well. But i don't really know well. I want to know clearly about theory. And i have this problem: An experimenter wishes to compare two ...
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stats - limiting distribution

Let $0 < p < 1$, and let $X_{n}$ have p.d.f. $f_{n}(x) = ( 1 – p ) ( n + 1 ) ( 1 – x )^{n} + p n x^{n – 1}$, for 0 < x < 1, zero elsewhere. Find the limiting distribution of $X_{n}$. ...
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202 views

Prove that $COV(h(x),g(x)) \leq 0$ means the different direction for $h,g$

(Covariance Inequality) Prove that if $g$ is nondecreasing and $h$ nonincreasing, then $$ E(g(X)h(X)) \leq E(g(X)) E(h(X)) $$ I know that it is equivallent to prove $COV(g(X),h(X)) \leq 0$ if $h$ ...
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1answer
109 views

Probability: 30 balls in a bucket, homework

i need some help with some homework, first time i am doing probability and statistics, id like to know if my 2 answers below are correct, and how i can solve the remaining 2. There are 30 balls in a ...
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65 views

stats - determine limiting distribution

Let $Y_{1} < Y_{2} < ... < Y_{n}$ be the order statistics of a random sample from a distribution with pdf $f(x) = e^{-x} , 0 < x < \infty$, zero elsewhere. Determine the limiting ...
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56 views

Order statistics of random variables

Let $\{I_1, I_2, \dotsc, I_N\}$ be $N$ i.i.d random variables. I know that the smallest orders statistics and the largest one are defined respectively as follow: $$I_{(1)}=\min(\{I_1, I_2, \dotsc, ...
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49 views

What is this pattern?

I have a pattern of numbers from some simulation that I'm trying to piece backwards into some sort of formula. All I can see right now is that they maybe have to do with integrals? The numbers (which ...
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115 views

Combining confidence intervals for sums of generic random variables

So my fiancee is a civil servant and asked me for help with the following problem. She has been given a collection of upper and lower bounds on expenditure for a collection of projects like: ...
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1answer
100 views

How to choice between paired sample and Independent sample

I don't know how to decide to choose between pair and Independent to compare mean of two groups. Here is the excersise from a book: ...
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1answer
113 views

Fishers Exact test alternative to chi-squared?

Background*strong text* Am analysing results from a questionnaire (n=130). The respondents fall into four categories (A / B / C / D), and each question/statement has 5 possible answers (Likert Scale ...
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1answer
17 views

Combining Random Variables to get the Variance

So $$\operatorname{Var}(aX + bY) = a^2 \operatorname{Var}(X) + b^2 \operatorname{Var}(Y).$$ But I have also seen: $$\operatorname{Var}(aX - bY) = a \operatorname{Var}(X) + b \operatorname{Var}(Y).$$ ...
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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?
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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. ...
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103 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 ...
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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?!
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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)...
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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.
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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 ...
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83 views
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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}) ...
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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: $f_{X,Y}(x,y) = f_{Y,X}(y,x)$ ?