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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If X and Z are independent and Y and Z are independent random variables, is cov(XY, Z) = 0?

Let $X$, $Y,$ and $Z$ be random variables. (There are no restrictions on these variables, but you may assume that these are continuous random variables if you want.) Suppose that $X$ and $Z$ are ...
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13 views

P(k) = a( 1 2 ) k . [on hold]

Find a so that the sum of all probabilities is 1, and hence P(k) is a legitimate formula for a probability distribution. ANd the probs distrubtion goes from 1 to2.. How do I start this problem
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6 views

Matching of points in two discrete linear sequences with potentially missing points

This is a question that I've been thinking about in my research lately. I've gone down the route of a few linear-optimization techniques, but nothing particularly spectacular has come up. Anyway, ...
3
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1answer
31 views

Determining an upper bound

I have a function $$f(\lambda)=n\ln(1-p+pe^{\frac{\lambda}{n}})-\lambda p$$ I need to prove that $$f(\lambda)\leq \frac{\lambda^2}{8n}$$ using Taylor expansion. I have used the taylor expansion for ...
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1answer
10 views

Moment generating function of sample mean of bernoulli random variables

Let $p \in (0,1)$ and $n \in \mathbb{N}$. We consider a sample of $n$ i.i.d. Bernoulli variables $X_1,\dots,X_n$ with parameter p. Computer $E[e^{\lambda\bar{X_n}}]$ such that $\bar{X_n}= \frac{1}{n} ...
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7 views

What is the Bernoulli class conditional distribution?

What is the Bernoulli class conditional distribution? I am trying to implement a procedure for computing a naive Bayes classifier for binary features with a Bernoulli class conditional distribution. ...
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6answers
45 views

Probability of getting $5$ heads on $10$ (fair) coin flips?

Even before attempting the problem, I immediately defaulted to an answer: $\frac{1}{2}$. I thought that this was a possible answer since the probability of flipping a head on one flip is definitely ...
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9 views

Sampling Distributions. Statistics [on hold]

I'm stuck in this problem: Problem Picture I did the literal a and b, but the rest of them I don´t understand, the reason why.
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probability that a customer who purchases up to $5$ songs from $4$ music genres prefers jazz and buys at least $3$ songs [on hold]

Customers can choose from $4$ music genres: jazz, rock, new age, country; and can purchase up to $5$ songs. The events are: $A =$ customer prefers jazz and buys at least $3$ songs $B =$ the customer ...
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10 views

Why does ergodicity not neccesarily imply ergodic for the mean?

I'm trying to answer a question where I have an ergodic and covariance stationary process $\{x_t\}$, and without imposing further moment conditions need to prove $\frac{1}{n} \sum\limits_{t=1}^n x_t^2 ...
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9 views

using standard deviation in decision making [on hold]

i have a decision need to be taken in a research, i have made some calculations, mean, median and standard deviation and really need your guys help to make sure that my selection is correct. for ...
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1answer
11 views

Hypothesis test of a sample that contains both male and female.

If I want to do a hypothesis test of a sample that contain $49$ women and $51$ men. The hypothesis test is only regarding the women which has a given sample standard deviation of $12,032$ and sample ...
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10 views

Statistical calculation for neural firing rates with negative rate on numerical simulation

I am now working on a biological neural network simulation (NEST-Simulator) project with a problem of calculating firing rates. Background: The data set as result of simulation is a set of events in ...
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25 views

Dealing with Recurrence Relations of Random Variables

Let $(Y_n)_{n\in \mathbb N} $ be some sequence of independent random variables, and $(X_n)_{n\in \mathbb N} $ another sequence, defined recursively as follows: $$X_{n+1} = \alpha X_n + \beta Y_n ...
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1answer
41 views

References for information theoretic statistical tools

Strange statistical concepts like spaces of probability distributions, "metrics" like Fisher information or relative entropy, and convergence with respect to these quantities are necessitated in my ...
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1answer
19 views

Limiting Distribution $\Delta-$method

Let $Y_n\sim \chi^2(n)$. What is the limiting distribution of $U_n= \dfrac{\sqrt{Y_n}-\sqrt{n}}{\sqrt{2}}?$. What I know is that if $X_i\sim \chi^2(1)$, I can write $Y_n = \sum\limits_{i=1}^n X_i$. ...
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1answer
19 views

Finding the MLE for an open interval.

So the problem says: Let $X = (X_{1},...,X_{n})$ be a random sample, where $X_{i} \sim Unif (0, \theta _{0})$, where $\theta _{0} \in (0,\infty)$ is unknown. Find the maximum likelihood estimator $T$ ...
3
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1answer
46 views

How to find $z$-score

I have some probabilities, but I have to find the $z$-score. I am not sure how do to this when I am told I have to use slope-intercept. Where do I plug the numbers in exactly? Here is one of my ...
2
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1answer
22 views

Likelihood function for a distribution with both discrete and continuous components

Suppose $X_1, X_2, \ldots, X_n$ are $IID$ normal RVs with mean $\mu$ and variance $1$. However, we observe only $Y_i$'s where $Y_i = \max (0, X_i)$. I would like to know how to write likelihood ...
2
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2answers
33 views

If $X\sim\operatorname{Poisson}(u)$ and $\theta = \mathbb{P}\{X=0\} = e^{-u}$, is $\hat{\theta}_1 = e^{-X}$ an unbiased estimator?

If $X\sim\operatorname{Poisson}(u)$ and $\theta = \mathbb{P}\{X=0\} = e^{-u}$, is $\hat{\theta}_1 = e^{-X}$ an unbiased estimator? Here's what I tried, is this right? $$ \begin{align} ...
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6 views

An applied example of Moment Generating Function in R? Or other software

I'm trying to understand MGF, I get the theory but I'd like to find an example I can relate using a software, like R studio or Matlab. Any example of one would be really appreciated!
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1answer
17 views

Median of Medians

Given a set A with median Am = 10 and set B with median Bm = 20 is it true that the median of the combined set C is $10 \le$ Cm$\le 20$ ? My first thought was that this wasn't true so I tried to find ...
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25 views

Estimate the number of “solvable” polynomials with degree $d$ and coefficient limit $l$

Enumerate the irreducible polynomials with degree $d$ and integer coefficients $[-l,l]$ and check how many of them have a solvable galois-group. We can assume that the leading coefficient of the ...
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10 views

How to find variance of weighted sum in terms of inputs and weights ? Variance of inputs and weights are given. [on hold]

Let $X = [x_1, x_2, ... , x_M ] $ and $ W = [w_1, w_2, ... , w_N] $ $y_i = {\sum_{j=1}^N}{\sum_{i=1}^{M} {x_iw_j}}$ Variance of $X$ and $W$ are given. Can we find the variance of $Y$ in terms of ...
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1answer
30 views

If $Y = (\mathcal{N}(\mu_1,\sigma_1^2) + \mathcal{N}(\mu_2,\sigma_2^2))^2$, what is $\Pr(Y>\mathrm{E}[Y])$?

Given $X_1 \sim \mathcal{N}(\mu_1,\sigma_1^2)$ and $X_2 \sim \mathcal{N}(\mu_2,\sigma_2^2)$, with $X_1$ independent of $X_2$, as well as $Y = (X_1 + X_2)^2$, what is $\Pr(Y>\mathrm{E}[Y])$? ...
0
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1answer
30 views

The density of a random variable $X$ is $f(x)$ proportional to $x^{-1/2}$ , what is the mean of $X$?

The density of a random variable $X$ is $f(x)$ proportional to $x^{-1/2}$ for $x \in [0,1]$$ and $f(x) = 0$ for $x \notin [0,1]$. Then, the mean of $X$ is $\frac 12$ $\frac 1{\sqrt2}$ $\frac 13$ ...
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2answers
28 views

Obtaining probability density function $f_Y(y)$ when we know joint probability distribution $f(x,y) = 1/(x+1)$

Suppose joint probability density function is $f(x,y) = 1/(x+1)$ for $0<x<1$ and $0<y<x+1$. I try to calculate marginal density function $f_Y(y)$ by $$f_Y(y) = \int_{y-1}^1 ...
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1answer
43 views

Why the Sum of all possible outcomes does not equal to one, in this case?

The question is an extension from an example (click this--> Introduction to Probability and Its Applications by Richard Scheaffer, Linda Young. The link points to the exact question/solution. Edit:- ...
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2answers
13 views

Let $X = -10Y + 10$. Let $r_1$ be the correlation between $X$ and $Z$ and $r_2$ be the correlation between $Y$ and $Z$.

Let $X = -10Y + 10$. Let $r_1$ be the correlation between $X$ and $Z$ and $r_2$ be the correlation between $Y$ and $Z$. Then, which of the following is the best answer? $r_1 = r_2$. $r_1 = 10r_2$ ...
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16 views

probablitly using bayes theroem

Three numbered urns contain colored balls as described in the table below. One of the urns is picked at random and a ball is drawn from the urn; the ball is red. What is the probability the ball can ...
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Which distribution would be the most appropriate?

What standard distribution would be suitable for the random phenomenon at hand, and what are the knowns and unknowns? e) The size of an automobile insurance claim I'm thinking that the distribution ...
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1answer
26 views

An explanation of how this solution is derived

I am having difficulty understanding the solution to this problem. Since the solution is in the form of Bayes theorem I expected something along the lines that looked similar to Bayes theorem. ...
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2answers
22 views

Likelihood Function for the Uniform Density. $ (\theta-1,\theta+1)$

Let the random variables $X_1,X_2,...,X_n$ iid $U[\theta-1\,,\theta+1]$. So the likelihood function therefore has the form: $L(\theta|X)=\prod_{i=1}^nf(X_i|\theta)=\frac{1}{2^n}I(X_1, . . . , X_n ...
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21 views

Probability density function above a given value. $\{ f(x) > c\}$

Say $X$ is a stochastic variable with a distribution $\nu$ and $f$ is the corresponding Lebesgue-measurable density. If I want to calculate a set $$A = \{ x \in \mathbb{R} \ | \ f(x) > c \}$$ for ...
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20 views

Minimum of four exponential variables

Four accidents occur independently, with each accident following an exponential distribution with a mean of 22.5. What is the expected value of the minimum of the four accidents? Attempt: ...
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50 views

Suppose $P(|X| < 1) = 1$ and $P(|Y| = 2) = 1$.

Suppose $P(|X| < 1) = 1$ and $P(|Y| = 2) = 1$. Then which of the following is true? The standard deviation of $X$ is smaller than that of $Y$. The mean of $X$ is smaller than that of $Y$. The ...
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9 views

How to calculate mean of busy-cylce

Let $\{X(t)\}$ be birth-death process on a finite state {0,1,2} with non negative birth rates $(\lambda_0,\lambda_1)$and death rates $(\mu_1,\mu_2)$. Suppose $\mathbb{P}(X(0)=0)=1$ and $s_0=\inf ...
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2answers
28 views

Prove that if $X_n \xrightarrow{P} c$, then $E(X_n) \to c$ for $X_n$ uniformly bounded

I have been trying to prove that for a random variable that is uniformly bounded, i.e. $|X_n - c| <M$ for all $n$, convergence in probability to $c$ implies that $$E\left(X_n \right) \to c$$ ...
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8 views

How to find expectation of birth-death process

Let $\{X(t)\}$ be birth-death process on two-state space {0,1}. Let birth rate $\lambda=2$ and death rate $\mu=12$. How to calculate $\lim_{t\to\infty} \mathbb{E}[X(t)]$?
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Proof of the DKW inequality

My goal is to prove the following inequality, known as the Dvoretsky-Kiefer-Wolfowitz inequality (1956) : Let $(X_i)_{i \geqslant}$ be iid random variables. Let $\displaystyle F_n(x)= ...
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1answer
24 views

Calculate the estimators of $E[X]$ and $Var[X]$ using the method of moments

$(X_1,\dots, X_n)$ is a random sample extracted from a uniform distribution on the interval $$(\alpha-\beta, \alpha+\beta) \ \ \ \ \alpha \in \mathbb{R}, \beta \in \mathbb{R}^{+}$$ Demonstrate ...
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2answers
25 views

In the context of ordered statistics, each of Y(1),Y(2),…,Y(n) a single observation or distributions that are I.I.D?

In statistics one aspect of the I.I.D. concept that bothers is when I think about it in the context of ordered statistics. As most of you already know, $Y_1,Y_2,Y_3,...,Y_n$ are I.I.D. when the ...
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27 views

Please help me create a mathematical model that can be used to compare two different strategies [on hold]

I have been pondering this question: "Is it better to be balanced, or focus on one area?" Let's say that I can decide how to divide my investment between two products. I want to know if it is better ...
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statistics help [on hold]

A charity solicits donations by phone. From long experience the charity’s director reports that 60 percent of the calls will result in refusal to donate, 30 percent will request more information via ...
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25 views

If $P(A) = 0.2$, $P(A \cap B) = 0.1$, $P[(A \cup B)'] = 0.3$, what is $P[(A \cap B) \mid (A \cup B)']$?

Suppose events $A$ and $B$ are such that $P(A \cap B)= 0.1$ and $P[(A \cup B)'] = 0.3$. If $P(A)=0.2$, what is $P[(A \cap B) \mid (A \cup B)']$? I tried solving it by using the conditional ...
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2answers
45 views

Covariance/Correlation Proof

I'm having a little problem with a statistics problem I am working on. I'm not really sure where to start to prove the two statements. Any help would be greatly appreciated. Let $x$ and $y$ be ...
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2answers
31 views

Notation Bayesian Statistics $\propto$

I often read the following notation: $\propto$. How is this sign called and what is the definition of it?
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18 views

How to analyse data set

I have a data set from work I'm not sure how to analyse. We have a bunch of people we've contacted (about $1,000$) at a random time during a project they're conducting. That time is expressed as a ...