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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Get standard deviation of a distribution made by subtracting two distributions with known standard deviations

I have some data D1 (N=4) D2 (N=4) Unfortunately, I don't have the data points. I only have the statistics: (mean d1,mean d2) = (2.5,8) (SD1,SD2 (standard deviation))= ...
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1answer
81 views

Histogram normalization

I need to generate random numbers from Gaussian distribution and to draw an equalized histogram. I've generated them in Matlab using Box-Muller transformation. Since I wasn't sure how to equalize the ...
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7 views

Statistics sampling on over-lapping categories.

I got a finite population. (e.g. a pool of files) The objective of the audit is to determine whether the files are in order. I need to audit the files for 5 different category on the population. Each ...
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1answer
98 views

Find unbiased estimator for $\theta$ when PDF of X is $e^{\theta-X}$?

I know the MLE for $\theta$ is $min{[X_i]}$ but I can't check if that's unbiased because I don't know how to solve (U=$min{[X_i]}$ here) $\int_\theta^\infty u*f_U(u)du$ = $\int_\theta^\infty ...
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1answer
37 views

Find the approximation of an inverse of sum of two special matrices?

Given $K$ is a psd symmetric covariance matrix. Is there any way that we can approximate $(I+K)^{-1}$ by $K$. Where $I$ is the identity matrix. Best,
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33 views

Probability in game shop items

In a game I can visit a shop regularly to buy an upgrade for my characters. The shop offers three possible items. These items are chosen from three different pools. Pool A, Pool B and Pool C. If I ...
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1answer
36 views

How to prove that A and B are independent if and only if P(A|B) = P(A).

How to prove: A and B are independent if and only if P(A|B) = P(A). What I'm confused about is proving the assertion in both directions... How does one approach that?
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21 views

How would one calculate the probability of 3 heads in 5 fair coin tosses, conditional on the first three tosses are heads?

How would one calculate the probability of 3 heads in 5 fair coin tosses, conditional on the first three tosses are heads? The probability of 5 heads in the 5 tosses, conditional on the first three ...
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14 views

Usefulness of the factorization theorem?

first time poster - I apologize in advance if my question is unclear! I am not quite sure if I understand the usefulness of the factorization theorem in statistics. While I get the essence of the ...
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11 views

Estimate Pathways given Transition Counts between Nodes

We know how many entities enter the system through S1. Entities can move between sections in both directions via S1, S2, S3, S4. Can we estimate likely unique paths per entity given counts in each ...
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33 views

Alternative answer to choose $k$ people out of $n$, then choose 1?

The question is fairly simple, we first choose $k$ people out of $n$, that is $C(k,n)$ as the combination function, then we choose 1 person out of $k$, we have $k$ choices. The total number of choice ...
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33 views

A little confusion about calculating variance?

given $a = 10% $, $b = 50%$, $c = 25%$ and $d = 15% $ calculate the variance? ( this is a made up question to make thinks clear for me) what i have done is 1st eliminate percentages? then i ...
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27 views

Moment List for Standard Normal Distribution

I am stuck trying to find the moment list for a standard normal. I have been told I can find it the similar way for exponential distributions using taylor series. I know the MGF = e^((1/2)(t^2)) for ...
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30 views

Subject order for probability and statistics?

I haven't taken math since grade 11, which was almost 10 years ago, meaning my math skills are relatively low now. I'm now planning to go to university and will have 9-10 months to gain a firm ...
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7 views

Variance and spectral density of time series

As an epidemiologist who is interested in time series of events, I have a question which pertains to time domain and/or frequency domain measures. I have read that variance is mathematically equal to ...
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53 views

Fisher information matrix of transformation of a vector of parameters

I have a term for the Fisher information matrix (FIM) of a vector of parameter $\theta$ (column vector with $3$ elements). Let us denote it as $J_\theta$. $\phi$ is a scalar, defined as: $\phi=[1, ...
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31 views

MODE of mixed distribution

Suppose $f_X(x) = \left\{\begin{matrix} 0.5 & x = 0 \\ x & 0 < x \le 1 \end{matrix}\right.$ What is the MODE of this distribution? I think it should be $0$ coz it has the highest mass, ...
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34 views

Method or Algorithm to produce near-accurate probability selection

Are there any known algorithms or methods to accurately identify, using random occurrences, the most probable next occurrence(s) as a pre-determined length of number set, by providing an already ...
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1answer
13 views

Continuous averaging of solar cell efficiency

I have a stream of numbers - efficiency ratio of solar cells - in range 0-100%. The efficiency is defined as (produced/expected)*100. I'm measuring the ratio every minute, and I'd like to have a ...
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30 views

counting techniques with poker

Two cards are fixedly selected from a standard deck of 52 playing cards. The first card is replaced before the second card is selected. Determine the probability that the denominations of the two ...
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43 views

Can I use Monte Carlo method to approximate PDF of sum of dependent random variables?

Suppose that I have two dependent random variable, X and Y, can I use Monte Carlo method to estimate the PDF of $X+Y$? As a special case suppose that $X=Z_1+Z_2+Z_3$ and $X=Z_2+Z_3+Z_4$, and lets say ...
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46 views

Goodness of fit through parameter estimation

A common way to test the goodness of fit of a particular model is within the framework of hypothesis testing with a likelihood ratio test. If your hypothesized model depends on a parameter, you can ...
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1answer
10 views

Sum of 2 uncorrelated numbers

Can we always RMS, two uncorrelated error to get their total contribution/error to the system? what is the idea/math behind it.
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1answer
35 views

How to measure similarity of two data

I am measuring the similarity of two data with same size is 20. That is ...
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1answer
10 views

Determining 13 random numbers given MAE of more than 13 guesses as to those 13 numbers

If you are trying to find 13 random numbers and are allowed to guess all 13 at a time and are then told the mean absolute value of your error, is there a way to determine the 13 correct numbers if you ...
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1answer
23 views

Probability of taking a random sample of 24 measurements and getting a mean of at least 103.6 of true population

A random sample of size n = 24 measurements is drawn from a normal population. The sample has a mean of 103.6 and a standard deviation of 12.5. If the true population is 100, find the probability of ...
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62 views

One queue vs multiple queues: how to calculate mean and variance?

I am having trouble understanding the solution to a queueing problem. You are at an amusement park. There are two ticketing areas. The first one has 1 ticketing machine and a 10-person line. The ...
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1answer
91 views

Finding Moment Generating Function of Normal Distribution

I need to show that the moment generating function of $Y$ is $$M(t)=(1 − 2σ^{2}t)^{−1/2}$$ where $X$ ∼ $N$($0$, $σ^{2}$) and that $Y$ = $X^{2}$. I know the moment generating function of $Z$ is ...
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1answer
25 views

condition probability quick question

Suppose I had a batch of old computer screens. 30% of these are bad 70% are good. The lifetime of both of these screens are indepdent exponential random variables. The mean for the bad screens is 150 ...
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30 views

t test vs f test

For conducting statistical tests concerning the parameter $\beta_1$ (the slope of the estimated linear regression function), why is the $t$ test more versatile than the $F$ test? This is a question ...
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83 views

Probability of shooting a 10 in archery?

I was interested in calculating the probability of shooting a 10 in archery with the scores that one has already shot. (Yes, that may include misses) My approach to do this was to simply assume a ...
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1answer
41 views

How to prove a statistic is sufficient using conditional probability argument?

I'm trying to tackle a question, but don't know how to start it: Components of a certain type are shipped in batches of size k. Suppose that whether or not any particular component is satisfactory ...
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27 views

If $P(Y^{2} \geq k) = 0.95$ is it true that $P(\sqrt{k} \leq Y \leq 1) =0.95$

Suppose $Y$ is a random variable defined on $[0,1]$, and that we know $Y^{2} \sim \mathrm{unif}[0,1]$. Suppose I determine the value of $k$ for which $P(Y^{2} \geq k) = 0.95$. Would it be correct ...
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22 views

Biased estimator for the standard deviation

Question: If a distribution is equally likely to take the values 1 or 4, show that $s_{n-1} $ forms a biased estimator of sigma. I started this question by finding that: $E(X)$ = 2.5, $E(X^2)$ = ...
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1answer
11 views

Find a constant so that the random variable has a t-distribution.

Given a random variable $Y=\frac{c(X_1+X_2)}{\sqrt{(X_3)^2+(X_4)^2}}$, find c such that Y has a t-distribution. Seems like I need to divide the numerator by $2$ and the denominator by $\sqrt{2}$ so ...
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15 views

What is the test statistic mentioned in the paper by Dai and Singleton?

I have read the paper with title 'Specification Analysis of Affine Term Structure Models' (2000) by Qiang Dai and Kenneth J. Singleton. On page 22 there's a table which include test statistics, these ...
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37 views

Expectations using Indicator Functions

Consider the following problem. a, b are constants in [0,1] and x is a random variable drawn from a continuous distribution F with support on [0,1]. Let $1_m$ represent the indicator function ...
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54 views

Probability that a portion of people will receive a flyer problem

High school dropouts make up 12.1% of all Americans (give or take 1.7%) aged 18 to 24. A vocational school that wants to attract dropouts mails an advertising flyer to 25000 people between 18 and 24. ...
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1answer
72 views

Expected Value Problem of a weird coin game

Consider the following game of chance. A fair coin is tossed until the first tails appears. You place an initial bet of k. If the 1st tails appears on the nth toss, you receive a total of $2^n$ (2 to ...
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153 views

Integral of the product of two Normal distribution CDF (erf)

How do I solve the following? $$ \lim_{x \rightarrow \infty} \int_0^{x} \left[ 1 + \text{erf} \left( \frac{\epsilon - a}{b} \right) \right] \left[ 1 + \text{erf} \left( \frac{\epsilon - c}{d} \right) ...
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40 views

Expected Z score for tail end of a standard normal distribution

Good day, For a Standard Normal Distribution (Mean: 0 and Variance: 1), I need to find the expected z-score for the section that makes up the final 2.5% (0.025) of the distribution. So, finding the ...
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1answer
100 views

variance validation

The scores on a placement test given to college freshmen for the past five years are approximately normally distributed with a mean $μ = 74$ and a variance $σ^2 = 8$. Would you still consider $σ^2 = ...
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52 views

Expectation using indicator method and definition do NOT have same answer

Problem: A fair die is rolled 10 times. Find the expectation of the number of faces which fail to appear in the ten rolls. I understand I can do this with indicators: X = I1 + I2 + I3 + I4 + I5 + I6, ...
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32 views

Ball bearings. Find expectation

The volume (in ${cm}^3$), $V$ , of ball bearings manufactured by a certain process has pdf $f_V(v) = \frac1v$, $1 < v < e$ $0$, otherwise Find the expected value of the radius of the ball ...
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37 views

How do I find the average waiting time and standard error of the kth person in queue?

Given the sample average wait time for the 1st person in queue is Wq, and the sample variance of the wait time for the 1st person in queue is Vq, how do you calculate the average wait time for the Kth ...
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1answer
19 views

The relationship between CDFs of two random variables

Suppose that for random variable $X$, we have $Pr(X>b)>p$ where b and p are real numbers and $0 <= p<= 1$. Now suppose that for random variable Y, we have $Y>aX+cX^2$. Can we conclude ...
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1answer
25 views

Let k be a natural number. What is the probability of it being equal to a random number in group?

We have 6 groups: A takes values from [0, 1, 2, 3, 4, 5, 6] B takes values from [0, 1, 2, 3, 4, 5] C takes values from [0, 1, 2, 3, 4] D takes values from [0, 1, 2, 3] E takes values from [0, 1, ...
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23 views

Design a statistical test for random number generator?

Design a statistical test for random number generators, based on the following result. Then apply the test to any generator you want and explain the results. A coin is flipped consecutively until ...
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10 views

When will statistic significance be achieved when the required security level is 99,9 percent

Given some one foretells that a coin will show either head or tail after tossing it and that we do a statistic on it. If the person is correct in predicting the outcome about 70 percent of times ...
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1answer
26 views

let a and b be constants and let yj=axj+b for j=1,2…n. What are the relationships betwee…

let $a$ and $b$ be constants and let $yj=ax_j+b$ for $j=1,2....n.$ What are the relationships between the means of $y$ and $x$, and the standard deviations of $y$ and $x$? I'm not sure what they are ...