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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How to get the lowest passing grade in normal distribution

$$ Pr[X\le a]=0.05 $$ $$ P[{Y\le {{a-64}\over 7.1}]} = 0.05 $$ I tried up to here but I don't know what to do now... Hint please.
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2answers
775 views

POISSON Distribution help

I already know the answer to this question. This was on a quiz which I got wrong. I can't figure out how to do this. The correct answer is "0.1396" but I can't work it out. If someone can explain how ...
6
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1answer
95 views

About cutting Almonds

Every year, during Christmas baking, I chop almonds, which causes me to puzzle over the same question, and I don't quite know how to approach it. I start out with N almonds. Let's assume they are all ...
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5answers
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If I roll two fair dice, the probability that I would get at least one 6 would be…

11 out of 36? I got this by writing down the number of possible outcomes (36) and then counting how many of the pairs had a 6 in them (1,6) (2,6) (3,6) (4,6) (5,6) (6,6) (6,5) (6,4) (6,3) (6,2) (6,1). ...
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2answers
1k views

If you roll four fair dice, what is the probability that you will end up with two 1's and two 3's?

Can someone help me with this? I'm completely stumped on how to go about it and no one I know can help. I just need to be shown how to do it.
2
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1answer
307 views

What is the maximum entropy distribution for a continuous random variable on $[0,\infty)$ with given mean and variance?

I know that for a given logmean and logstdev its the lognormal, but what about where we directly specify the mean and variance? The above seems to depend on the log-transformation to the maxent for ...
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0answers
147 views

Computing Replacement rate in a renewal process

I found this problem on some slides that I'm studying as I prepare for a final: "A machine in use is replaced by a new machine either when it fails, or when it reaches the age of $T$ years. If the ...
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0answers
39 views

Method Recovery and Uncertainty

Can anyone help to solve? Without Standard deviation, i cannot calculate the uncertainty of method recovery. thank you.
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2answers
64 views

The weighted mean results in smaller value than the simple mean.

If weights attached to larger items are smaller and those attached to smaller items are larger, then the weighted mean results in smaller value than the simple mean. My try: Let the weights be $w_1&...
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0answers
83 views

MSE For a Single Calculation (intel processor errors)

This is the question, from a practice final for a stat course: The Intel Pentium Processor chip has been discovered to make small errors occasionally; that is, errors of +1 or –1 (in $10^{-4}$ ...
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1answer
175 views

Probability (Independent Events need explanation)

Here's my question. If I'm given two events, both independent of each other. How do I know if $A \cap B$ is empty or if I have to multiply $Pr(A)\;Pr(B)$ to find $A \cap B$? I had a question as such, ...
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1answer
3k views

Finding smallest and largest possible values for the probability of the union of 3 events

Three events $A, B, C$ each occur with the same probability $p$, which takes any value between 0 and 1. Find the smallest and largest possible values for the probability of the union $P(A\cup B\...
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1answer
1k views

Real Analysis and Statistics

What level of real analysis do you think is desirable for the study of statistics? I know that for many statisticians with applied focus, rigorous mathematics tend to give them a headache and I am ...
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1answer
669 views

What is my unit of observation for this time series data set?

My professor gave us a data set to run a regression and I have a data set which lists years from 1959-2007, gross private investments (in billions of dollars), and gross private savings (in billions ...
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2answers
42 views

Expectation of function of two random variables

I have the random variables X and Y, with joint density function $f(x,y)$ over the plane $-\infty < x < \infty$ and $-\infty < y < \infty$. I am trying to find the expectation of $(X-Y)^2$....
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0answers
149 views

How would you explain the pdf of the normal distribution to high school students (11th/12th graders)

I will be teaching the normal distribution in January and I need to know how to effectively explain the concepts that does not in any way confuse students or make them feel that the material is ...
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2answers
202 views

Finding probability that the function of a random variable is less than another random variable

X and Y are random variables, whose joint density function is $f(x,y)$ for $\infty<x<\infty$ and $\infty<y<\infty$. I am trying to find $P[X^2<Y]$. Here's how I plan on solving the ...
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4answers
1k views

Expected value of number of cards drawn from a deck to get 5 spades.

The question: What is the expected number of cards required to be drawn in order to draw 5 spades. What I have: Let $X=X_1+\cdots+X_{43}$ (43 because we're examining the case when 4 spades have been ...
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1answer
233 views

95 % confidence interval

In a random sample of 41 cyclists two years ago, 18 tested positive for drugs. A random sample of 45 is conducted this year and 15 test positive. (a) Give a 95 percent confidence interval for the ...
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1answer
41 views

Uniformly Distribution Problem

A random variable X drawn from a uniform [0,3] distribution and a random variable y is independently drawn randomly drawn from a uniform [0,4] distribution. The joint probability density f(x,y) is ...
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1answer
50 views

Exponential Distribution with Probability

$$f(y) = \begin{cases} \int_0^y\frac1\beta e^{\frac {-t}\beta}dt = -e^{\frac {-y}\beta}+1 & \text{for } 0 ≤ y < ∞,\\ 0& \text{for } elsewhere\end{cases}$$ $P(Y>3) = 1 - P(Y ≤ 3) = 1 - (-...
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1answer
98 views

Finding probability that two discrete random variables are equal

I have two random and independent variables, X and Y, whose probability distribution functions are given by $f(x) = (1/10)(9/10)^x$ and $f (y) = (1/5)(4/5)^y$, for X = 0, 1, 2, ... and Y = 0, 1, 2,......
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1answer
533 views

Finding new probability density function with change of variable Y=sqrt(X)

Say we have a given distribution, such as X~No(a, b). I am trying to find the pdf and mean for $Y=\sqrt{X}$. I know the steps for finding the PDF, but since Y can only take on positive values, then ...
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1answer
40 views

Least Square regreesion

I have 300 equation with only 7 variable. i need to calculate these 7 variable in a way that in all 300 equations it fits using least squares approach. how can i compute these variables? Best Regards,...
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0answers
46 views

Question about $t$-distribution

The random variable $T$ has a $t$-distribution with $20$ degrees of freedom. Find the value of $t$ such that (a) $P(|T|>t)=0.98$ (b) $P(|T|>t)=0.05$ For part (b) am able to find $t=2.086$ ...
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1answer
56 views

Why is it answer Stratified random sampling

Forty seven math majors, 22 music majors and 31 history majors are randomly selected from 585 math majors, 279 music majors and 393 history majors at the state university. What sampling technique is ...
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1answer
665 views

Combining random variables

On combining random variables and their means, this article states: Suppose you have two variables: $X$ with a mean of $μ_{x}$ and $Y$ with a mean of $μ_{y}$. The mean of the sum of these ...
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3answers
2k views

Easy GRE question: Statistics

I'm not sure how to set this statistics problem when they give me a group of arbitrary values. Can someone help? A group of 20 values has a mean of 85 and a median of 80. A different group of 30 ...
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2answers
87 views

Where do the definitions in statistics come from? [duplicate]

The most statistics I ever took was a few lessons on it back in high school. What always bothered me is how arbitrary the definitions seemed. For instance, I remember having trouble with the ...
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1answer
804 views

Standard deviation - a general confusion.

I am confused by this question: A casino knows that people play the slot machines in hopes of hitting the jackpot, but the most of them lose their dollar. Suppose a certain machine pays out an ...
3
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1answer
316 views

If $X$ is independent of $Z$ and $Y$ is dependent with $Z$ is it possible for $X + Y$ to be independent of $Z$?

Suppose $X, Y$ and $Z$ are three non-degenerate random variables. Suppose that $X$ and $Z$ are independent and that $Y$ and $Z$ are not independent. Is it possible for $X + Y$ to be independent of $Z$?...
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1answer
196 views

calculating mean squared error for the Mean.

Exam Question There are two independent random variables $X_{1}$ $\&$ $X_{2}$ that are having normal distribution with mean $\mu$. Further Var$(X_{1})=1$ and Var$(X_{2})=2$.an unbiased estimator ...
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24 views

Variance of ratios of correlated values

I want to estimate a scaling factor $s$ between two vectors of scalars $a$ and $b$. In the ideal case it would be constant for all components, so $a = s \cdot b$, however there is some noise in the ...
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2answers
72 views

How do I solve for $Ex(Y)$?

Suppose a plant has $X$ offspring in a year with $P(X = x) = \frac14$ for $X = 1, 2, 3, 4$ and, independently, each offspring has from one to four offspring in the next year with the same discrete ...
2
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1answer
70 views

How to explain last part of Two-sample Kolmogorov–Smirnov test

Sorry for the dumb question. I am trying to understand the last part of Two-sample Kolmogorov–Smirnov test.. The part is: $$ \sqrt{\frac{n + n'}{nn'}} $$ I am thinking that $n$ and $n'$ are the two ...
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1answer
60 views

Likelihood Function for Censored Model

I have a model that results in the following data generating process: $$x=\begin{cases}\begin{array}{c}y-\theta\\0\end{array} & \begin{array}{c}if\ y>\bar{y}(\lambda_1)\\if\ y\leq\bar{y}(\...
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1answer
2k views

Rolling standard deviations

I am trying to calculate standard deviations on an array of numbers. My psuedo code looks like this: ...
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1answer
771 views

Sufficient statistic for uniform distribution

Given random sample $\left\{ { X }_{ 1 },{ X }_{ 2 },...,{ X }_{ n } \right\} $ from $ U(0,\theta)$. Let ${Y}_{i}$ be the order statistics. Then the sufficient statistic for $\theta$ is ${ Y }_{ n }...
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1answer
89 views

statistics to find mean ,median mode

for the frequency distribution given below $v=-\dfrac16,-\dfrac14 ,0,\dfrac12,\dfrac13$ $f= 12, 16, 21, 8, 27$ find mean ,median mode what percent of the population is non negative what ...
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1answer
83 views

normal distribution in a biostatistics

Among a large group of coronary patients it is found that their serum cholesterol levels approximate a normal distribution. It was found that 10% of the group had cholesterol levels below 182.3 mg per ...
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0answers
113 views

Classification problem: admissible rule is a Bayes rule for some prior $\pi$

I have a classification problem where I want to place an observation $X$ into a population described by a pdf equal to either $f_1$ or $f_2$. Given $P_{f_i}(\frac{f_1(X)}{f_2(X)}=j)=0$ for all $j\in [...
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1answer
127 views

Geometric explanation of Inverse Transform Sampling

I am familiar with the mathematical setup. Given the CDF of a distribution, we can sample from it using inverse transform sampling as follows, 1) sample u uniformly from a uniform disibution. 2) ...
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1answer
349 views

find the average of a student [closed]

Five students in a closed room want to find average of their exam scores without revealing their personal score to each other. How can they do it? Hint: they can write something on paper and pass it, ...
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5answers
141 views

How to find the minimum of the function?

How to find the minimum of the following function $$ {\rm f}\left(w\right) = {1 \over 2}\sum_{i = 1}^{n}\left({1 \over 1 + {\rm e}^{-x_{i}\,w}} -y_{i}\right)^{2} $$ where $x_{i}, y_{i} \in \left(0, ...
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1answer
86 views

The sum of two geometric distributions with same p

Let X and Y be independent random variables, and k is the number of failures before a success $P(X=k)=P(Y=k)=p(1−p)^k$ How is it possible to arrive at the probability function $Z = X + Y$ $P(Z=z)=(...
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1answer
304 views

Textbook Recommendation: Statistics

I am still a freshmen taking statistics and am disappointed with a second semester statistical theory course which just raves on from typical distributions to bi-variate random variables to random ...
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1answer
194 views

Finding aggregate score from incomplete data

$$ \begin{align} n & = \text{number of reviews}\\ x & = \text{review score}\\ \bar{x} & = \text{aggregate score} \end{align} $$ I have a specified number of reviews for a product, I have ...
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1answer
62 views

Beginning statistics, simple confidence interval problem

I am in a first statistics class and we have been covering confidence intervals. This seems like a confidence interval problem but I'm a bit thrown by it. The problem states: "A machine fills $100$-...
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2answers
666 views

Finding expected number of white balls drawn before any red ball [closed]

A box contains 12 balls: 6 white, 4 black, and 2 red. Draws are made without replacement. Find the expected number of white balls drawn before any red ball is drawn. Similarly, how do I find the ...
4
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2answers
239 views

$L^1$ norm of product of independent random variables

I am trying to show that $\|XY\|_1 = \|X\|_1\|Y\|_1$ for $X,Y$ independent random variables, where $\|X\|_1 = \int{|X| d\mathbb{P}}$. I have a feeling that this result is intuitive, but could anyone ...