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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Help with deriving minimum length confidence bounds for a F distribution variance $\sigma^2$ …

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 ...
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9 views

Unbiased estimator exponential distribution

Is the reciprocal of the sample mean an unbiased estimator of the exponential distribution parameter? How can we get it unbiased? Hint: use out, that the sum (convolution) of independent exponential ...
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10 views

Sufficient statistic for normal distribution

Let X from a Normal distribution $(\theta,1)$. a) Find a sufficient statistic for $\theta$. b)Is $S_n^2$ sufficient for $\theta$ My answer for part a) The joint p.d.f= $1 \over (2\pi)^{n/2}$$e^{{-1 ...
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1answer
8 views

Estimating parameter of random sample

5 random numbers were recorded: 100,32,76,52,17. If we know that these are elements of random sample took from set {1,2, ..., N}, how do we estimate the parameter N?
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1answer
16 views

Lack of memory property of probability distributions

According to wikipedia lack of memory property applies to geometric and exponential distributions. I was trying to apply it to binomial distribution. Am I modelling my question correctly? So imagine ...
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3answers
370 views

Probability of Parking Spot Being Empty

A parking spot is unoccupied 1/3 of the time... But, you find it empty for nine consecutive days in a row. Find the probability that it will be empty on the tenth day. Read more: ...
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7 views

One and Two Tailed Independent T Test Questions

The city council for a small town has been receiving complaints from local law enforcement that citizens have been extremely uncooperative when pulled over for minor traffic violations. To remedy ...
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1answer
9 views

Derive the Cramer-Rao lower bound (CRLB) for any unbiased estimator of $\mu^2$.

Let $Y_1, Y_2, . . . , Y_n$be a random sample from a normal distribution with mean μ and variance 1. Derive the Cramer-Rao lower bound (CRLB) for any unbiased estimator of $\mu^2$. Could anyone ...
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1answer
17 views

Sum of two independent non-identical uniform random variables

Let's say we have two independent random variables, $X$ is uniform on $[0,1/2]$ and $Y$ is uniform on $[1/2,1]$. If we look at the distribution of $X+Y$, is it triangular distribution between ...
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8 views

Questions related to Rao–Blackwell theorem

In this exercise, we illustrate the direct use of the Rao–Blackwell theorem. Let $Y_1, Y_2, . . . , Y_n$ be independent Bernoulli random variables with $p(y_i | p) = py_i (1 − p)1−y_i , y_i = 0, 1.$ ...
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find E($\bar{Y^4})$ by using moment generating function for a normal distribution with mean μ and variance 1.

Let $Y_1, Y_2, . . . , Y_n$be a random sample from a normal distribution with mean μ and variance 1. I would like to find E($\bar{Y^4})$ by using moment generating function. The setup I have right ...
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1answer
19 views

Calculate the Marginal Probability

f($X_1$, $X_2$| $p_1$, $p_2$) = $p_1^{x_1}$(1-$p_1)^{({n_1}-{x_1})}$$p_2^{x_2}$(1-$p_2)^{({n_2}-{x_2})}$ $p_1$~Unif(0,1) independently $p_2$~Unif(0,1) $n_1$=34 $n_2$=56 Calculate the marginal ...
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1answer
15 views

Moment Generating Function of the Chi-Squared Distribution

The questions wants us to show that the MGF for the chi-squared distribution is equal to I know that to show that I need to evaluate this integral. I'm not sure where to begin to evaluate it. ...
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8 views

Statistics: When to use pooled vs non-pooled vs paired

I'm looking for a simple method for knowing when to use pooled vs non-pooled vs paired tests in a given statistics problem. For example, if the standard deviations are exactly the same, I'm told I can ...
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0answers
26 views

Need help finding probability distribution [on hold]

In Cairo $30\%$ of residents listen to the local fm radio. $10$ residents are chosen at random: a) state the distribution of the random variable b) find the smallest value of $s$ so that $\Pr(X \ge ...
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11 views

Find critical Value

A Basketball scout randomly selected 144 players and timed how long each player took to perform a certain drill.The times in this sample were distributed with a mean of 8 minutes. The population ...
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23 views

Probability of events [on hold]

1) 5 cards are selected from a 52-card deck for a poker hand. a) How many simple events are in the sample space? b) A royal flush is a hand that contains the A,K,Q,J, and 10, all in the same suit. ...
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1answer
18 views

Likelihood of Two Binomial Distributed RV's

We are given that Let X1~Bin(n1 = 34, p1) and X2~Bin(n2 = 56, p2) In general, what is the likelihood, L(p1, p2) = f (X1, X2 | p1, p2) for the data X1 and X2 I believe that I am supposed to use a ...
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9 views

State null and alternative hypothesis [on hold]

Experience in investigating insurance claims show that average cost to process a claim is approximately normally distributed with a mean of \$80. New cost-cutting measures were started and a sample ...
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0answers
8 views

equal-variance whitening transform

Out of all the whitening transformations, PCA gives us the one that maximizes the discrepancy in variances, i.e. the components in the PCA basis have the biggest and the smallest variances. How does ...
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24 views

Linear regression, reversing it back then.

This is my first Question. I am performing linear regression upon set of floating point between 0 and 1. there are few hundred points.once the slope and intercept is found for first iteration,the ...
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2answers
15 views

Why normal approximation to binomial distribution uses np> 5 as a condition

I was reading about normal approximation to binomial distribution and I dunno how it works for cases when you say for example p is equal to 0.3 where p is probability of success. On most websites it ...
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0answers
11 views

Beta marginal distributions and Dirichlet distribution

I know Dirichlet distribution has Beta marginal distributions (sum to 1). I am not sure if the other direction is also correct. That is, for example, if we have 3 beta distributions: $X_1 $ ...
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1answer
34 views

How to compute the unique MLE from an Exponential Family of Distributions?

Let $$ f(x;\theta)=\frac{1}{\pi} \frac{e^{\theta x}\cos(\theta \pi/2)}{\cosh(x)}, x\in{\mathbb{R}} $$ be a family of densities and which is clearly exponential family. Then what is the Maximum ...
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1answer
19 views

Small population $t$-test

If the population is small, can I use the normal distribution or I need to use $t$-student like if it was a sample? eg: In a class of $8$ students, $4$ male and $4$ female, I need to compare the ...
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2answers
25 views

Distribution of ratio of uniform and exponential random variables

This is a homework question, I feel like I'm doing it right, but I can't seem to get the answer to match up. I have a uniform RV from 2 to 4, and an exponential with mean 4, so $X \sim ...
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2answers
16 views

How to weight three different variables to create a ranking

I have data on the graduation rate, tuition, and average student income (a measure of accessibility) for about 7,000 colleges and universities in the U.S., and I'm trying to compose an interactive in ...
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17 views

Normal distribution exercise [on hold]

In a factory, compacts are filled with a cosmetic powder. We consider the weight of the powder follows a normal distribution $N\sim(\mu, 1.21)$. The value of $\mu$ depends on the setting of the ...
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1answer
59 views

Is there any statistical method to compare two curves?

Is there any statistical method to visually compare two curves? What is the best and correct way to compare two similar curves and calculate the error/difference in percentage? I have created a ...
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0answers
15 views

Help defining the statistical number that pi posses a philosophical question in the first 15 characters. [on hold]

I am looking for assistance calculating the statistical possibilities that pi posses a philosophical question of life in the first 15 characters. Additionally, it has the Fibonacci Sequence and the ...
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17 views

Find spikes in data

I have some datasets and I need to find spikes in them. Imagine the data looks like trading data. If the spike is big enough, I need to log it, otherwise, proceed in the analysis. I tried with a ...
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1answer
17 views

Functional Choice for p in a Bernoulli Distribution

Why is the functional choice $p = \exp(x)/(1+\exp(x))$ to model $p$ a good one in a Bernoulli distribution? Is it because it is limited at $0$ as $x$ approaches $0$ and $1$ as $x$ approaches ...
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1answer
28 views

probability and applied statistics 3 [on hold]

given two urns, suppose urn 1 contains four black and seven white balls. urn 2 contains three black , one white , and four yellow balls . we select an urn at random and then draw a ball . what is the ...
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Estimating variance in least squares in regression

How to show $\hat\sigma^2=$$1\over n-2$$[S_{yy}-$$S^2_{xy}\over S_{xx}$]. I don't know the matrix form in regression.I don't even understand how to begin this
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Given two players competing, what is the probability of one getting ahead x wins vs the other getting ahead y

We have 2 players, A and B, competing. The probability that A wins a match is p, making the probability that B wins a match (1-p) = q. The game is won by player A as soon as he gets one more win than ...
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2answers
22 views

Estimate standard deviation of sample

You have a random sample of 25 objects with mean weight of 24 grams, estimate the standard deviation of the sample. In addition, you know it's supposed to be 25 grams with a deviation of 1 gram, but ...
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1answer
20 views

properties of least square estimators in regression

$Y_i=\beta_0+\beta_1 X_i+\epsilon_i$ where $\epsilon_i$ is normally distributed with mean $0$ and variance $\sigma^2$ . The least square estimators of this model are $\hat\beta_0$ and $\hat\beta_1$. ...
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Is this method to find mean already discovered?

I am a 10th class student and in our syllabus, we have three methods for finding mean of grouped data: Direct method. Assumed mean method. Step deviation method. Out of these, the Step deviation ...
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Confidence Intervals Theory

The problem in the textbook is: Let $0 \le γ \le α$. Then a $100(1 – α)\%$ CI for $μ$ when $n$ is large is $$ \left(\bar{x} – zγ\frac{s}{\sqrt{n}}, \bar{x} + zα-γ \frac{s}{\sqrt{n}}\right) $$ ...
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2answers
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Forth Moment of Sum of Normal with Equal Correlation

I have $X_1,\dots,X_n$ identically normal distributed $N(0,\sigma^2)$ and $\operatorname{corr}(X_i,X_j)=\rho $ for all $i\neq j$. I'd like to compute \begin{equation} E\left(\sum_{i=1}^nX_i\right)^4. ...
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1answer
29 views

If $X$ ~$Uni(-1,1)$ show that $X$ and $X^2$ are not independent

I provide my approach in solving this but I amd not entirely sure whether I am correct. Since X~uni(-1,1) $f_X(x)=1/2$ and $F_X(x)=(x+1)/2$. $F_{X^2}(x)$=$Pr[X^2≤x]$=$2F_X(√x)$=$(√x+1)/2$. Hence ...
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0answers
14 views

confidence coefficient z value

I'm having a bit of difficulty understanding a concept in my notes and was wondering if someone could help me. This is probably a really simple concept that I've just completely overcomplicated but ...
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10 views

Writing in Lag form and finding the characteristic polynomial, MA(2) with constant

I'm wondering how to write an MA(2) model with a constant in lag form such that I can calculate the characteristic polynomial and get the roots (to see if it's stable). The model is given by $Y_t = ...
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Statistic test, t-test,binomial sign test , Pearson and Spearman

I have the following situation: (i) It was observed that the average PSI on rainy days is lower than the average PSI on sunny days (PSI is a measure of air pollution). (ii) After introducing airbags ...
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1answer
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Taking the standard deviation over multiple assignments?

Lets say that a group of students all take 5 tests, with the average and standard deviations as follows: test 1: mean 43/50, SD 8 test 2: mean 23/30, SD 4 test 3: mean 56/70, SD 12 test 4: mean ...
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1answer
28 views

Conditional expectation and Rao Blackwell

Consider a family of densitites $f(x,\theta)=\frac{\exp(-\sqrt{x})}{\theta}$. Let $X_1$ be a single observation from this family. I have shown that $\sqrt{X_1}/2$ is an unbiased estimator. Now ...
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Is it possible to calculate the autocovariance of a DT WSS signal knowing only it's mean and it's linear estimator?

Let's also assume that we know that $C_{xx} [0] = K$ Where $K$ is some constant. I'm trying to figure out if that's possible and if yes how I would get around doing it.
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30 views

Find E[MSLOF]. Please help.

Find the expected mean squares error of lack of fit. Trial: $$SSLOF=\sum_{1}^mn_i(\bar y_i-\hat y_i)^2\\=\sum_{1}^mn_i(\bar y_i-\bar y)^2-\sum_{1}^mn_i \hat\beta_i^2(x_i-\bar x)^2$$ and ...
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16 views

Heavy tailed sum of iid light tailed random variables

I know that to get one, the number of summands has to be random with a heavy-tailed count variable. I am wondering how you prove the resulting sum is heavy-tailed and in particular wondering if there ...
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1answer
29 views

Clique factorization

I'm reading about Clique factorization in wikipedia: http://en.wikipedia.org/wiki/Gibbs_random_field#Clique_factorization but I'm unable to understand this: What is $X_C$ here? Ok I understood ...