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
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, \...
0
votes
1answer
49 views

Conditional Variance

I'm having a bit of trouble conceptualising the step which i have highlighted.
0
votes
1answer
64 views

Double integrals for reconstructing probablistic model

I am trying to reconstruct this probabilistic model, \begin{equation} \begin{split} \frac{1}{\mu}\int^{\infty}_{0}P(N \geq n\, |\, L=l, T=t)\,e^{-\frac{l}{\mu}} dl &= \frac{1}{(n-1)!\mu}\int^{\...
0
votes
2answers
196 views

Probability of being selected in a raffle

There's a raffle with 1,000 names in a bucket. 600 of those names are in there once, and 200 are in there twice. So, just to reiterate, there are 800 unique names in the raffle, and 1000 names total. ...
0
votes
1answer
18 views

Hypothesis testing question trying to find the p value

for each of the data sets find the p value This is what I have so far ts = (735-854)-0/38 = -3.13 |ts| = 3.13 I am having a hard time getting the P-value. Please help.
1
vote
0answers
339 views

How do I measure the goodness of cosine similarity scores across different vector spaces?

I am a computer scientist working on a problem that requires some statistical measures, though (not being very well versed in statistics) I am not quite sure what statistics to use. Overview: I ...
1
vote
1answer
62 views

Finding p-value when df and test stat is given. PLEASE HELP.

For a chi-square goodness of fit test with 10 degrees of freedom, the test stat is 20.000, then how to find the p-value? In other words, what is ?: X^2?,10 = 20.000? The X^2 table does not give the ...
0
votes
1answer
77 views

How do I find $\theta$ with bootstrap?

I have two vectors of known values $x$ and $y$. And the relationship between them is $y=\sin(\theta \cdot x)+\epsilon$, $\epsilon \sim N(0,1) $ . The question is how do I estimate $\theta$ with ...
1
vote
1answer
66 views

Probablility of a Dice Game

Player A rolls $m$ dice, while Player B rolls $m + 1$ dice. If Player A rolls $a$ $n$'s and Player B rolls $b$ $n$'s, then Player A wins if $a > b$ . Otherwise, Player A rolls up to $k$ of the $m$ ...
1
vote
1answer
172 views

Probability of Renewal Processes

Suppose that there are two brands of replacement components, Brand X and Brand Y, and that for political reasons a company buys a replacements of both types. When a Brand X component fails it is ...
1
vote
0answers
417 views

Generate correlated random numbers precisely

Let's assume I want to generate k samples of n random numbers, that are correlated according to a given correlation matrix C (e.g. $n = 3$): ...
1
vote
1answer
37 views

proving a complete sufficent statistics

Suppose that $ X_1,\ldots,X_n$ are iid poisson($b$); $c = b^2$ and $S_n = \sum X_i$ To Show that $S_n$ is a complete sufficient statistic for $c$. I can prove using exponential family that $S_n$ is ...
1
vote
0answers
76 views

MLE of discrete uniform distribution

Assume that $X$ is a discrete random variable with uniform distribution on the set $\{1,2,3,\ldots, N\}$, where $N$ is an unknown positive integer. Find the MLE $\hat{N}_k$ of $N$, assuming that $...
1
vote
2answers
63 views

Uniform Distribution and Distribution function technique

Let $X_1$ and $X_2$ be independent random variables having the uniform density with $\alpha = 0$ and $\beta = 1$. Find expressions for the function $Y =X_1 + X_2$. (a)$y \le 0$ (b)$0<y<1$ (c)$...
29
votes
2answers
535 views

Statistics Primer for the Unwary Mathematician

I have a new position in a biology department (after being housed in a maths department) working on cognitive and population modeling. People in my lab are asking for help with applying statistical ...
0
votes
1answer
283 views

Distribution function technique and exponential density

I'm having quite a bit of difficulty with the distribution function technique. If $X_1$ and $X_2$ are independent random variables having exponential densities with parameter $\theta_1$ and $\theta_2$...
0
votes
1answer
54 views

a question which is somhow related to law of large number

suppose that $\mathbf p = [p_1, p_2, ..., p_n]'$ is a random vector. (' == transpose) and each element of $\mathbf p$ like $p_i$ is a Gaussian random variable with zero mean ($\mathbb E(p_i)=0$) and ...
1
vote
0answers
22 views

Multiple measurements per person per treatment

Suppose I wish to assess reaction time of individuals before and after treatment. Now to analyse the results I could use a paired t-test or if I had additional treatments, I could use a repeated ...
0
votes
1answer
200 views

if the CDF is non-invertible or does not have a closed form solution(e.g. Normal CDF), how can we generate random data from such a distribution?

Given the CDF of a distribution to generate random data from that distribution by using the inverse transformation of the CDF. Then if the CDF is non-invertible or does not have a closed form solution(...
1
vote
1answer
96 views

statistics inequality

Let $\theta$ be a discrete pararmeter and $\gamma_{n}$ be an estimator. Prove that for any $c>0$ we have that $$\text{E}[(\gamma_n-\theta)^2] \ge\Pr[|\gamma_n-\theta|>c]\cdot c^2$$
2
votes
1answer
78 views

standard deviation of sums of numbers of different sets

Suppose that I have $n$ different sets of numbers , each containing $m$ different numbers and I can only form sums of $n$ numbers by choosing only one element of each set. Is there an easy way to find ...
2
votes
2answers
141 views

Computing $\langle\sin(\gamma_i)\rangle= \int_{(S^2)^N} \sin(\gamma_i)p(\Theta)dS$

I'm trying to evaluate the following integral, which I know must be zero, $$\langle\sin(\gamma_i)\rangle= \int_{(S^2)^N} \sin(\gamma_i)p(\Gamma)dS$$ Where, $$\langle \vec{a}(\vec{r_1},...,\vec{r_N})\...
0
votes
1answer
986 views

Fast way to recalculate average and standard deviation as new values arrive

Say I have a stream of values arriving all the time, and I want to get the average and standard deviation of only the last $n$ values. If I already have the average $V$ for values $v_1, ..., v_n$, ...
0
votes
1answer
157 views

exactly, and at least : Probability

thanks for looking at my question. Any help would be appreciated! at a university 60% of the students are male and 40% are female If ten students are selected at random, what is the probability that ...
2
votes
3answers
240 views

Combinations/probability calculations using ball/bag analogy

I'm wondering how to approach this question? I'm analysing data for a research project, but I feel like it falls into the category of choosing combinations of balls in a bag. Any help would be much ...
1
vote
3answers
635 views

Why MLR (monotone likelihood ratio) implies stochastic increasing?

the following argument holds: for $\theta_1<\theta_2$, $\dfrac{f(x\mid\theta_2)}{f(x\mid\theta_1)}$ is increasing in $x$. Then, $F(x\mid\theta_2)\leq F(x\mid\theta_1)$ for all $x$. Intuitively, ...
1
vote
1answer
170 views

Is a 99% upper confidence bound the upper limit of a 99% confidence interval?

I have to find a "99% confidence bound" for a standard deviation. This is not hard. The only question I have is whether this is finding the $\chi^2_{.99}$ value or just the upper bound for the 99% ...
1
vote
1answer
219 views

Probability Dice Game Question

I have the following problem to solve that deals with probability (something I haven't done since Grade 8 (6 years ago)) This is a one player game and it is described for $q$ sided dice. You start by ...
5
votes
1answer
71 views

Inequalities that show if a distribution decays slowly

Often, one is often interested in theorems/inequalities of the following kind: Let $X$ be a random variable then the probability that $X$ is close to typically $\mu$ (or larger than some constant) is ...
7
votes
5answers
1k views

What does multiplication mean in probability theory?

For independent events, the probability of both occurring is the product of the probabilities of the individual events: $Pr(A\; \text{and}\;B) = Pr(A \cap B)= Pr(A)\times Pr(B)$. Example: if you ...
0
votes
2answers
286 views

Hypergeometric Distribution Question

In a hypergeometric distribution, you can compute the probability of drawing without replacement, which is useful in a number of studies and statistics. However, I'm having a problem that I've had for ...
1
vote
4answers
129 views

Probability of choosing a subset of elements where each element has a different probability

I am trying to write a C++ program to do this but nobody on Stackoverflow can seem to help me so I thought I'd try to do it myself with some help from you guys. My post on Stackoverflow can be found ...
1
vote
1answer
79 views

Probability of 2 identical events

My professor said that probability of 2 identical events in a very short amount of time (dt converges to 0) is 0. However, I did not agree with him about this. Is there a proof for that assertion? ...
0
votes
1answer
86 views

Expanding variance

Could someone please expand on line 2 and 3 of: Thank you.
1
vote
1answer
75 views

Two Expected value definitions of the geometric random variable

Ok so I'm looking at my book and it defines the geometric distribution to be $\sum_{i=1}^{\infty}p(1-p)^{n-1}$. My book says the expected value of a geometric random variable is $\dfrac{p}{q}$. It ...
1
vote
1answer
62 views

Probability density function help

If a random variable is given by $Y=aX+b$ where $X$ is a random variable does that mean $aX+b$ is $Y$'s pdf?? and if i wanted to find $E[Y^2]$ would this just be the same as finding $E[(aX+b)^2]$ and ...
-1
votes
1answer
86 views

Calculating probability of a dice with different numbers

I was given a problem in class: ...
2
votes
2answers
219 views

Expected value given that distribution is positive vs. conditional expectation

Referring to Expected value of normal distribution given that distribution is positive Where is the difference between $E(X$1$_A)$, where $A=[X>0]$, and $E(X∣A)$? Both seem to express the ...
2
votes
2answers
90 views

Expectation value with condition

how can i show that: $E[XY \vert X ] = XE[Y \vert X]$ for two random variables $X$ and $Y$ sorry this must be wrong what i meant was $E[ E[XY \vert X ] ]= E [XE[Y \vert X]]$
0
votes
1answer
107 views

Joint PDF of all n Order Statistics

If $X_1,\ldots,X_n$ is a random sample from a continuous distribution with pdf $f_{\theta}(x)$, why is the joint PDF of the order statistics $X_{(1)},\ldots,X_{(n)}$ the following: $$\large f_{X_{(1)},...
2
votes
2answers
180 views

Traversing an array and counting the number of distanct number from the given elements in an array.

You are given an array $A[0 \ldots n-1]$ of $n$ numbers. Let $d$ be the number of \emph{distinct} numbers that occur in this array. For each $i$ with $0 \leq i \leq n-1$, let $N_i$ be the number of ...
1
vote
0answers
51 views

statistical analysis of discrete (non-uniform) p-values: cryptographical random data test

i'm doing a statistical analysis of a well-known cryptographic algorithm and have hit an anomaly. i need to prove that what i have found is statistically significant. i am taking block sizes of 256 ...