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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44 views
Convergence of probability densities
I appreciate if you can give me some guidance on how to approach this question:
Suppose $f_n(x) \text{ and } g(x)$ are densities such that for all x, $f_n(x) \rightarrow g(x)$ as $n \rightarrow ...
3
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1answer
238 views
Median of order statistics
I recently learned that to find the pdf of the median of say $X_1,X_2, X_3$, you first find the Cdf via $$ P(M \le x) =P(\text{at least 2 are}\, \le x) = P( \text{exactly 2 are}\, \le x) + P(\text{all ...
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1answer
47 views
Meaning of Fisher's information
If I am correct, Fisher's information at parameter $\theta$ is defined to be the variance of the score function at $\theta$. The score function is defined as the derivative of the log-likelhood ...
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2answers
206 views
Expected value of applying the sigmoid function to a normal distribution
Short version:
I would like to calculate the expected value if you apply the sigmoid function $\frac{1}{1+e^{-x}}$ to a normal distribution with expected value $\mu$ and standard deviation $\sigma$.
...
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1answer
34 views
Bayesian learning
Imagine we assume there are two different types of coins:
Coin A: a fair coin, p(heads) = 0.5.
Coin B: biased to heads at p(heads)=0.7.
We then want to learn from samples which coin we are ...
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1answer
62 views
how to find the quartiles
I am confused on the method to find the 1st and 3rd quartiles.
Suppose given a data set 2,5,6,9,10,11,13,17 with an even number of data. What is the formula to ...
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4answers
68 views
What does it mean to integrate with respect to the distribution function?
If $f(x)$ is a density function and $F(x)$ is a distribution function of a random variable $X$ then I understand that the expectation of x is often written as:
$$E(X) = \int x f(x) dx$$
where the ...
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2answers
32 views
95% confidence interval for the mean becomes irrelevant when t-test is insignificant?
For the results I have been given, I have been asked to interpret the 95% confidence interval for the mean. However, the t-test shows a value more than .05, meaning that the differences I am exploring ...
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1answer
886 views
How can I calculate the standard deviation knowing an event probability and a number of trials?
I'm writing a test for a probabilistic data structure that I've implemented. Since its probabilistic, its performance is different every time, and in particular, the performance varies much more ...
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1answer
2k views
is it possible to calculate the standard deviation with a given mean and sample size?
I have been going in rounds with this problem... I may be thinking "complicated", any advice?
I have the mean and total sample size (=number of data points) and I need to know what is the standard ...
4
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4answers
115 views
Expected value for $f(x)= \frac{\Gamma (\alpha+\frac{1}{2})}{\Gamma (\alpha)} \frac{\beta^\alpha}{\sqrt{\pi}} \frac{x^{\alpha-1}}{\sqrt{1-\beta x}}$
$$f(x)= \frac{\Gamma (\alpha+\frac{1}{2})}{\Gamma (\alpha)} \frac{\beta^\alpha}{\sqrt{\pi}} \frac{x^{\alpha-1}}{\sqrt{1-\beta x}}$$
where $0<x<\beta$.
So these are three terms all multiplied ...
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0answers
26 views
Covariance between sample variance and sample sum of squares
I am trying to find the cov(A,B), where A is the sample variance, and B is the sample sum of squares. I am new here, and don't know yet how to enter the formuale in the question box, but I think they ...
3
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2answers
508 views
Examples of Student's T-distribution in real world empirical data?
I have recently stumbled onto some empirical (forecasting error) data that should be normally distributed. However, the normal distribution fits relatively poorly due to the abundance of data points ...
3
votes
1answer
162 views
A question about finding the convergence of a distribution
Let $X$ have the Gamma$(s,1)$ and given $X=x$, let $Y$ have the Possion distribution with parameter $x$. Show that $$\frac{Y-E(Y)}{\sqrt{\operatorname{var}(Y)}}\longrightarrow W$$ where ...
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0answers
16 views
Do data points outside standard deviation persist in data subsets?
I have a data set of dates and values that are being plotted. I create a simple trend line y=a+bx using linear regression, and then I create a 'trend channel' (I ...
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1answer
91 views
Grouping ranges
I have this set of numbers, that is a range of prices (ie 6-10) and a quantity of products (5)
...
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1answer
64 views
Some Algebra 2 help [closed]
The table shows the distance and costs of flights for a given airline with the same starting destination and different ending destinations.
Distance (miles) vs Cost($)
1281 473
2796 683
790 ...
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0answers
27 views
Statistics( find the no. of students)
Question:1 The mean weight of 500 male students at a certain college is 75kg and the standard deviation is 7kg.Assuming that weights are normally distributed, the number of students weigh between 60 ...
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2answers
32 views
Quadratic and geometric average
I'd like to find the find the quadratic average and the geometric average. To do this I have these informations :
The standart deviation, the arithmetic average and the number of values.
I know the ...
1
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3answers
86 views
What is the $\operatorname{cov}(X, \max(X,Y))$ and $\operatorname{cov}(X, \min(X,Y))$ where $X,Y \sim N(0,1)$?
having trouble with this one. The exact questions is the $\operatorname{cov}(X, \max(X,Y))$ and $\operatorname{cov}(X, \min(X,Y))$ where $X,Y \sim N(0,1)$.
i think the way to calculate it is to get
...
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2answers
61 views
Cumulative Distribution of X/Y
Let X, Y be independent exponential variables with rates $\alpha$, and $\beta$. Find the c.d.f. of X/Y.
So far, I let Z = X/Y.
I can then show $f_Z(z) = \int_{-\infty}^{+\infty} |x|f_{X,Y}(x,xz) ...
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1answer
234 views
Statistics - Z-score confusion.
For z score, you are taking the sample value subtracting population mean and dividing it by std deviation. Is that correct so far?
Now, the "sample value" is defined by an equation. In my scenario, I ...
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0answers
15 views
On $F(X^−)$ where $F$ is cdf of $X$.
$F$ is cdf of r.v. $X$. Define $\tilde{X} := F(X^−)$, where $F(x_0^-) := \lim{x \to x_0^-} F(x)$. This is what I sometimes see from statistic books especially about nonparametric statistics.
I ...
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2answers
229 views
Statistics: normal distribution, finding truncated mean.
Say $X$ is a random variable arising from a normal distribution with mean $10$ and variance $4$ $(N(10, 4))$ truncated at $X=6$. How do I find the truncated mean of the distrubution?
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1answer
31 views
Probability mass function
Why is the probability function of a discrete random variable called a probability "mass" function? What does the word "mass" mean here?
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1answer
227 views
Bus stop probability question
People arrive at random times and independently at a bus stop and wait for the bus to arrive. The bus arrives at this stop once every hour. Thus, the waiting times of the people follow a uniform ...
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2answers
44 views
Calculating the MSE for assessment
Let $X_1, \ldots, X_n \sim \mathcal{N}(\mu, \sigma^2)$ be the sample, when $\mu$, $\sigma$ are unknown.
We suggest assessment for $\sigma^2$:
$$S^2 = \frac{\displaystyle\sum_{i=1}^n (X_i - ...
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1answer
212 views
Multiple Regression over an experimental dataset
I want to do a multiple regression over an experimental result shown as 3D-Plot and heatmap in following Images. Sorry as a new user i am not allowed to post them directly but it is just a link to ...
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3answers
35 views
What does it mean to take an integral of a probability?
My understanding is that you have some function y=f(x) to represent a probability density function, correct? For instance for a uniform random variable it looks like a giant rectangular block. I don't ...
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1answer
43 views
How to combine covariance matrices?
I have a data set of points in three dimensions.
I'm calculating the barycenter (mean) and $3\times3$ covariance matrix from this data set.
I store the average, the $3\times3$ matrix (where really ...
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1answer
17 views
relating $\hat {p}$ to $\bar{x} $
I'm trying to develop an intuitive understanding of the relationship between $\hat{p}$ and $\bar{x}$. Please point out anything that is incorrect.
For a Bernoulli distribution, if $X_1, X_2, ... X_n$ ...
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1answer
21 views
Stuck on 'differentiating the integral from above' for computing of a PDF
I am stuck on a math derivation that has to do with statistics, so I am putting the statistical context here for context. In short, I am stuck on understanding how the answer to the PDF was attained.
...
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0answers
25 views
Appropriate statistical test to test if probabilities are accurate
I have some data that looks like this:
Prob Outcome
0.09 0
0.10 0
0.10 0
0.11 1
0.84 1
0.99 1
0.86 1
0.78 1
0.86 1
0.00 0
etc.
...
4
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1answer
363 views
Calculation of the moments using Hypergeometric distribution
Let vector $a\in 2n $ is such that first $l$ of its coordinates are $1$ and the rest are $0$ ($a=(1,\ldots, 1,0, \ldots, 0)$). Let $\pi$ be $k$-th permutation of set $\{1, \ldots, 2n\}$.
Define
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1answer
51 views
Asymptotics for infinite sum with erf
I'm interested in approximating the infinite sum
$$
\sum_{i=1}^\infty Z\left(\frac{\alpha i\pm1}{\beta}\right)
$$
where $\alpha,\beta$ are constant and
$$
Z(a\pm ...
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0answers
42 views
how to find the pooled proportion estimate? [closed]
A youth prevention organization is examining the effect of parental smoking on the decision of their teenagers to smoke. A survey of 1000 teenagers was conducted. The random sample consisted of 500 ...
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1answer
32 views
finding Moment generating function and CDF with pmf
The random variable X has the pmf f(-1)=1/4, f(0)=1/8, f(1)=1/4, f(2)=3/8
a) How would you draw the c.d.f with points (-2,F(-2)), (-1,F(-1)), (0,F(0)), (1,F(1)), (2, F(2)), (3,F(3))
b)Write the MGF of ...
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0answers
30 views
Hypothesis Testing Bayesian Way
I'm having trouble with the following problem:
Suppose a machine is composed of 2 components (1 and 2, independent from each other). Each component has a exponential failure probability distribution ...
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0answers
29 views
Statistical Analysis- Pre and Post-survey Data Analysis [closed]
I am doing a research project on whether or not students are aware of the benefits of graduating from high school. I gave students a pre-survey, then gave a presentation, then gave the same students ...
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0answers
14 views
Upper bound for the number of measurements above a certain value.
So I have a practice prelim problem. If the sample mean $\bar{x}$, number of observations $N$, and root mean square $\sigma = \sqrt{\frac{1}{N}\sum_{n=1}^N x_n^2}$ are all given, how can we derive ...
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0answers
20 views
Underlying assumption of t-test and standard error
I am having a hard time explaining why my significance test based on randomization is different from a t-test.
I have 20 different data points and I know each comes from a Gaussian distribution. I ...
4
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1answer
119 views
Compute $P(X>40\; |\; X>10)$ where $X$ has an exponential distribution
Please could someone advise if I have interpreted this problem correctly
Let $X$ have an exponential distribution with a mean of $i = 20$
(1) Compute $P(X>40 \;| \;X>10)$
I believe the ...
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30 views
Computing Conditional Probability (Poisson Random Variables)
The number of red and blue cars that go through a given intersection in an hour is a Poisson-distributed random variable with $\lambda$ = 10. What is the probability, conditionally, that at most ...
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30 views
Homework Help. Probability Density Functions.
$X$ is $N(10,1)$. Find $f(x|(x-10)^2 < 4)$
This is a homework question. I can only figure out that X is normally distributed with mean 10 and variance 1.
Can you please explain what is meant to ...
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2answers
364 views
expectation of $ \left(\sum_{i=1}^n {x_i} \right)^2 $
If $x_i$ is exponentially distributed $(i=1,...,n)$ with parameter $\lambda$ and $x_i$'s are mutually independent, what is the expectation of
$\left(\sum_{i=1}^n {x_i} \right)^2$
in terms of $n$ and ...
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3answers
59 views
Homework Question. Joint Probability Distribution.
Here is the question.
The joint PDF of X and Y is given by $f_{XY}(x,y) = {\frac 14} e^{-|x|-|y|}$. Find $P(X \le 1 ,and, Y \le 0)$
Solving the problem I first found the marginal probabilities of X ...
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0answers
43 views
Evaluate the right side [closed]
$X_i$ [where $i=1,2,\dots n]$ is normally distributed with mean $P$ and variance $1$, that is $X_i~N(P,1)$
then how
$$-\frac{1}{2}\sum^n_{i=1}X_i^2–\frac{1}{2}nP^2=-\frac{n}{2}P^2$$
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20 views
Probability that difference sample mean from population mean is 1.96 sd [closed]
Probability that difference sample mean from population mean is 1.96 sd
a)68%
b)95%
c)47/5%
d)99%
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1answer
102 views
Finding the cumulative distribution function for the double exponential
Im considering the double exponential with parameter $\lambda$,
$$g(x)=\frac{\lambda}{2}e^{\lambda x}, x<0; \frac{\lambda}{2}e^{-\lambda x}, \geq 0$$
Just simple one looking for the c.d.f, I ...
1
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1answer
18 views
manual random selection, how?
Thas a question for my test in one hour...how do I do that? I have tried to find it online but no way...any help please?
Suppose that a simple random sample of 20 of the Aucklanders surveyed is to be
...
