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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Confidence interval of a binomial distribution?

I'm trying to use the Clopper Pearson Interval for a binomial distribution, but am not sure how to find it, despite having the formula: What does this mean ...
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50 views

Beginner question in probability

i'm a beginner in statistics and probability and i need help with a given problem please. We are given a guy who has a machine and a button, the outcames when he presses the button are: a)Music ...
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146 views

calculate Limiting distribution $\displaystyle\frac{\sum_{i=1}^n X_i}{\sum_{i=1}^n Y_i}$

let $X_1,X_2,\ldots,X_n$ are random sample of bernoulli distribution with parameter of $\displaystyle\frac{\theta_1}{\theta_1+\theta_2}$ and $Y_1,Y_2,\ldots,Y_n$ are random sample of geometric ...
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3k views

Chance of randomly guessing 21 questions right out of 50 with 4 multiple choice.

Lets say a person decided to randomly fill in a scantron of 50 questions with 4 choices each. After submitting it to be graded, the result was 42% correct. How would we figure out the probability of ...
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1answer
298 views

Proof that a median minimizes 1-norm. [duplicate]

I was wondering whether there is an easy way to show the following: We have a data set $x_1,...,x_n$ and $m$ is a median if for at least half of the n data points we have that $x_i \le m$ and for ...
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88 views

Absolute values of a linear combination of three random variables

From my stat homework: with $X \sim N(3.2, 6.5)$, $Y \sim N(-2.1,3.5)$, $Z \sim N(12.0,7.5)$ (all are independent random variables) find probability that: $$ |X + 6Y + Z| \geq 2 $$ I have $(X + 6Y + ...
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77 views

Central Limit theorem Clarification.

Central Limit Theorem (Sum Version) Let $X$ be a population random variable with finite mean $\mu_X$ and finite variance $\sigma^2_X$ and let $(X)_{i=1}^n$ be a random sample of $X$. Let $S= ...
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872 views

Calculating the second moment of binomial random variable

I don't understand how they got the equality E[Y+1]= np[(n-1)p+1]. http://imgur.com/SNeeyYO
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Random var. Y with pdf $f_Y(y) = 4y^3$. Show that $-2\ln (Y^4)$ ~ $X_{(2)}^2$.

Let Y be a random variable which has pdf $$f_Y(y) = \begin{cases}4y^3, & 0 < y < 1, \\ 0, &\text{elsewhere}.\end{cases}$$ Show that $-2 \ln (Y^4)$ ~ $X_{(2)}^2$. Could anyone get me ...
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33 views

Derivation of $ \frac{n-1}{n}\sigma $

Near the last step for deriving the unbiasness check of $\hat{\sigma}$, we come to a point where $$ = \frac{1}{n} \bigg( \sum_{i=1}^{n} \sigma + \mu^2 -n (\frac{\sigma}{n} + \mu^2 ) \bigg) $$ ...
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285 views

Finding the expected value of poisson distribution

The number of breakdowns per week for a type of computer is a random variable Y having a Poisson distribution with mean μ. A random sample Y1; Y2, .... Yn of observations on the weekly number of ...
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101 views

Sufficient statistic.

If $X_1$ follows Binomial distribution with parameter $m$ and $p$ where $m$ is the number of trials and $p$ is the probability of success , that is , $X_1\sim B(m,p)$ and $X_2\sim B(n,p)$ then how can ...
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191 views

Two-tailed or one-tailed test?

The study using 139 students studies the maximum amount of alcohol consumed. Based on data are there differences between males and females in the maximum amount of alcohol consumed? (Use alpha= .05 ...
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62 views

Need help finding expected value and variance [closed]

Let $Y$ be a random variable with $E(Y) = \mu$ and $Var(Y) = \sigma ^2$ . Let another random variable $X = Y +c$, where $c$ is a known constant. Show that $E(X) = \mu +c$, and $Var(X) = \sigma ^2$ . ...
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108 views

what is the difference between maximum likelihood estimation and usual probability inference? [closed]

Can somebody tell me a clear difference between MLE from the usual probability inferences?
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38 views

Statistics: If $X_1$ and $X_2$ are both normally distributed then explain why $X_1 - X_2$ can be standardized with mean 0 and standard deviation of 1

I am currently studying hypothesis testing for two populations and I would like a math major or someone experienced to explain to me why this particular statistic has a mean of 0 and a standard ...
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49 views

In alternative hypothesis in $t$-test, why is $t_{1-α}$ used as the criteria $t$ for rejection and not $t_α$?

In general common rejection region of t-test, there are three possible alternative hypotheses and rejection regions for the one-sample t-test: For alternative hypothesis $H_1: μ1 ≠ μ2$, the ...
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272 views

Can the singular values be plotted?

I have a set of 3 x 3 co-variance matrices that I need to plot. Using singular value decomposition (svd) in Matlab I managed to obtains a vector of singular values for each matrix as stated here. I ...
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56 views

Weak Law of Large Numbers - Why Use $Pr()$ Statement With Limits?

In the literature describing Law of Large Numbers, the law is described in terms of $X_1,..,X_n$ random variables which are i.i.d. with each $X_k$ having mean $\mu$ and variance $\sigma^2$. My ...
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53 views

Technical name when professor decrease the numerator.

When a professor re-size the class's mean and standard deviation, this is called curving the grade. I was wonder what the technical name is when the professor decrease the total number of possible ...
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59 views

how to compute this expectation value

A random variable $X \sim N(0,1)$, compute $\Bbb E(X^n)$ . I manage to do this by characteristic function. Now I try to compute this by moment generating function or do it directly. So I have 2 ...
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60 views

Suppose X~Poisson(lamda) and Y~Poisson(sigma) and X and Y are independent. What is the distribution of X+Y?

if x and y are independent , the distribution of them would be double Poisson?
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78 views

Jacobian Transformation help

Let X1 and X2 ~ i.i.d N(0,σ^2) Find the joint pdf of Y1:= X1^2 + X2^2, Y2:= X1/√(X1^2 + X2^2) I tried using the Jacobian transformation: X1 = √(Y1) * Y2 .......... X2 = Y1-√(Y1)*Y2 and then I took ...
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67 views

Proving Expected Value

Prove that $$E[X] = \int_0^\infty (1-F_X(x))\,dx - \int_{-\infty}^0 F_X(x)\,dx$$ I am guessing I need to have $f'(x)$ somewhere in here? How can I start?
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Does the central limit theorem apply for random variables with densities which are not asymptotic?

I’m probably missing something, but how can for example the mean of samples of nutritional needs (assuming i.i.d. random variables) be normally distributed when there is a certain minimum in the ...
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30 views

Setting up statistics problem

Assume $\beta_{U,T}$ is the underlying slope of straight line associating $U$ with $T$. We know that $X=U+f$ and $Z=T+e$ are measurable instead of $U$ and $T$, where $e$ and $f$ are uncorrelated ...
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257 views

Why is $(x'x)^{-1}x' = x(x'x)^{-1}$

If $(AB)'=B'A'$ then $(x'x)^{-1}x'$ should be equal to $x((x'x)^{-1})'$ . However most econometrics textbooks say that this is equal to $x(x'x)^{-1}$ . What happened to the transpose of $(x'x)^{-1}$? ...
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99 views

Infinite Probability Space of Real Numbers

I have a question on my homework that I'm very befuddled about. Find a value of a constant $c$ such that the formula $p(n)=\frac{c}{3^n}$ defines a probability distribution on the set ...
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282 views

Can you explain what EXACTLY happens when you multiply $\sigma^2$ to $N(0,1)$?

I understand that when you apply a transformation $\sigma Y+\mu$ to $Y$~$N(0,1)$ ,we get a new random variable that is distributed $N(\mu,\sigma^2)$. However, I dont know through which mechanism this ...
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Adding two normal distribution

Suppose that $X_1, X_2, X_3$ are i.i.d. normal random variables with mean $0$ and variance $1$. And Suppose that $Z \sim N(1, 2^2)$ and is independent of all $X_i$. Define $Z_i = Z + X_i$ for $i = 1, ...
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140 views

What is -2loglikelihood?

I don't understand the term "loglikelihood"? I'd like to have a practical understanding of this word, and of why this is important. Besides this whenever we calculate some statistic like chisquare or ...
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80 views

Should be simple inductive proof

Establish the following recursion relations for means and variances. Let $\overline{X}_n$ and $S_n^2$ be the mean and variance, respectively, of $X_1,\dots,X_n$. Then suppose another observation, ...
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Distribution of the minimum of a random sample

Suppose $x_i$ is a random variable with CDF $F(.)$ with a bounded support. I get a random sample $S_n=\{x_1,x_2,\dots,x_n\}$. Define $x_{\text{min}}=\min(S_n)$. How can I find the pdf of this random ...
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49 views

Is there a way to find $P(x \mid y \,\,\text{and}\,\, z)$ given $P(x \mid y)$ and $P(x \mid z)$?

For example, if P(cavity | infrequent brushing) = 0.524 and P(cavity | toothache) = 0.662, is there a way to find P(cavity | infrequent brushing and toothache)?
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Binomial distribution convergence

Let $Y \sim\binom{n}{\pi}$ Suppose $n \rightarrow \infty$ and $\pi \rightarrow 0$ such that $n\pi \rightarrow \mu$, where $\mu$ is a constant. derive the limiting distribution of Y. $f_Y(y)= ...
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Moment Estimate of theta

Consider a random variable $X$ whose pdf is $f(x;θ)=θx^{θ−1}$ for $0<x<1$ and zero otherwise. i) Show this is a density function ii) determine the moment estimate of theta on the basis of a ...
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119 views

Covariance and Expectation of Discrete Random Variables

I have to get back to basics here and need some help on checking some solutions. Given 2 discrete random variables $X$ and $Y$ and $\mathbb E (Y|X) = \mathbb E (Y) = \mu$ and $\mathbb E (X) = ...
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55 views

How to prove that the following inequality holds true?

To prove: $\frac{1}{2}(x_n-x_1)^2 < \sum\limits_{i=1}^n (x_i - \bar{x})^2 $. I simplified the LHS to: $\frac{1}{2} (x_n^2-2x_1+x_1^2)$ and the RHS to $\sum\limits_{i=1}^n x_i^2 ...
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19 views

About the variance and a connected integral

Is given a positive measure $\mu$ such that $\mu(\mathbb{R}^+)<+\infty$. Is it generally true that: $$\int_0^\infty x^2 d \mu < + \infty \space\space ^{?}\iff^{?} \int_0^\infty \left(x ...
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441 views

Need help with expectation of summation for power of Gaussian variable.

I am trying to derive a formula, and getting stuck on a part of the derivation. Basically what I have is the following: $$ P = \sum_{n=0}^{N-1} \sum_{k=0}^{N-1} \mathbb{E} \Big[x^2[n] x^2[k] \Big] $$ ...
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166 views

Conditional Expected value of number of rolls in a die

A die is rolled repeatily. Let $X$ be the random variable that denotes the number of rolls to get a 4 and $Y$ be the random variable that denotes the number of rolls to get a 1. What is $E[X|Y=7]? My ...
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24 views

what is the intuitive understanding when you integral for $x$ in pdf $f(y|x)$?

what is the intuitive understanding when you integral for $x$ in pdf $f(y|x)$ as we all know that $f(y|x)$=$f(x,y)/f(x)$, and also if you integral for $x$ in joint pdf $f(x,y)$ then you get marginal ...
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63 views

What do we know about the pdf of $\bar{X}$

We have $n$ independent random variables $X_i$ all with mean $\theta$ and variance $\sigma^2$. The sample mean is given by $$\bar{X} = \frac{1}{n} \sum\limits_i^n X_i$$ and the means square error is ...
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51 views

$E[m(X)h(X)]=0$ for every $m(X)$ implies $h(X)=0$?

I have a question as follows: $m(X)$ and $h(X)$ are functions of a random variable X. If we know that $E[m(X)h(X)]=0$ for every $m(X)$, does this imply that $h(X)=0$, a constant?
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172 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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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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343 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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1answer
49 views

Why does $p(X\;|\;Y) = \displaystyle\frac{p(Z,X\;|\; Y)}{P(Z\;|\;X,Y)}$?

I'm reading about expectation maximization and on one point in my paper it is said that: According to probability theory: $$p(X\;|\;Y) = \displaystyle\frac{p(Z,X\;|\; Y)}{P(Z\;|\;X,Y)}$$ Where $p$ ...
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315 views

Mean and Variance of exponential function

Given this function with j and k as unknown parameters. What is the expression of Variance and Mean of this exponential function? $$f_{j,k}(y)=\frac{\sqrt{j}}{\sqrt{2 ...
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2answers
236 views

Calculating dice probability and risk-reward value

Having trouble how to model this cost-risk question with Probability. Assume that you are given 5 standard six-sided dice, what's the probability you get at least three 2s? I have calculated this: ...