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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Asymptotic Normality of MLE when data is modelled with covariates

Say I have data vector $X_1,\ldots,X_n$ which I want to model with some parametric distribution function $f(X_i;\theta,Z_i)$ and covariates $Z_i$. In this case, how can I prove the asymptotic ...
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1answer
14 views

MGF of a random sum of random variables

I am doing this quesiton right now and I don't even know how to start it up. I know that it's in relation to a sum of a random number of random variables, but I don't know how to continue on from ...
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2answers
22 views

Order Statistics Expected Value

Can someone help me with this question? I have arrived with the distribution function equal to $g(y) = \frac{n}{\theta }\left(1-\frac{y}{\theta }\right)^{n-1}$. But couldn't solve the integral for the ...
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1answer
50 views

Linear Algebra question relating to eigenvectors

Let A be an m x m positive definite symmetric matrix with eigenvalue-eigenvector pairs $(\lambda_1,e_1),....,(\lambda_m,e_m).$ The eigenvectors are orthonormal. Let $C = e_1e_1'+....+e_me_m'$. ...
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23 views

Listing the values of particular probability distributions

Three balls are selected at random from a bag containing 2 red , 3 green , and 4 blue balls. Define the random variables R = the number of red balls drawn, and G = the number of green balls drawn. ...
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1answer
21 views

Continuous Random Variables and Probability Distributions [closed]

Let X denote the amount of time a book on two-hour reserve is actually checked out, and suppose the cdc is: $F(x) = (x^2)/4$ (when $0\leq x<2$), $1$ (when $2\leq x$) find $E(X)$ The answer is ...
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11 views

Question regarding Balanced Incomplete Block Design

Question: Consider a BIB design with a treatments, b blocks and c < a number of plots in each block where a,b,c ≥ 2. Let $n_{ij} = 1$ if an observation is made on the ith treatment in the jth ...
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1answer
10 views

Additivity of the Chi-squared distribution

Can additivity of the $\chi^2$-distribution be used when substracting variables? For example: Does X-Y=Z where X~$\chi^2$(3), Y~$\chi^2$(1), Z~$\chi^2$(2)?
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35 views

Finding the most likely serve speed of a tennis player

U.S. Open Tennis Championship a statistician keeps track of every serve that player hits. The statistician reported that the mean serve speed of a particular player was 95 miles per hour. Suppose ...
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18 views

Find density function from likelihood ratio test

We have statistical test: $$H_0 : X \sim f_0(x) = \frac{1}{(1+x)^2}, x \in (0,+\infty) \\ X_1 : X \sim f_1(x) = ?$$ Suppose that the power of this test (likelihood ratio test) is equal $$1-\beta = ...
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32 views

Find the pdf of $X+Y$ (discrete case)

Given the marginal pdfs: $$p_X(k)= e^{-\lambda}\frac{\lambda^k}{k!}\text{ and }p_Y(k)= e^{-\mu}\frac{\mu^k}{k!},\quad k=0,1,2,\ldots$$ find the pdf of $X+Y$. I know for $W=X+Y$, $p_W(w)= \sum_x ...
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1answer
15 views

Bounds of integral in Power function

Here is the question: Let $X_1,X_2$ be iid uniform $(\theta,\theta+1)$. For testing $H_0:\theta=0$ versus $H_1: \theta>0$, we have two competing tests: $\hspace{15mm}\phi_1(X_1):$Reject $H_0$ if ...
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2answers
28 views

Question about random variables operations

As we can see in the picture above, what we call random variables looks like much more like a function, in the way that there is an input and then this random variable perfoms a process and gives as ...
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21 views

Derivative of stochastic process

I have a set of data of a random process (one sample path). The process is sampled every 10 min and each sample is a 10 min average from a sensor. I can compute the statistics of the random process, ...
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1answer
16 views

Question on the correlation between two dependant variables

I'm working on this question and it's stumping me. Let Sn = X1 + ... + Xn (with n>=1) be a random walk with X1,...,Xn be iid RV's. E(Xk)=mu Var(Xk)=sigma^2. Find the covariance of Sn and Sm Can ...
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1answer
34 views

MLE of $n(\theta,a\theta)$ family

Question: A special case of a normal family is one in which the mean and variance are related, the $n(\theta,a\theta)$ family. If we are interested in testing this relationship, regardless of the ...
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1answer
18 views

Problem Calculating Moments using a Moment Generating Function

Problem: Find (a) the moment generating function of the random variable $X$ having density function $ x = \begin{cases} \frac{x}{2} & 0 <= x <= 2\\ 0 &\text{ otherwise } \\ ...
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1answer
20 views

Statistics question Conditional Probability

Question: Of three cards, one is painted red on both sides; one is painted black on both sides; and one is painted red on one side and black on the other. A card is randomly chosen and placed on a ...
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23 views

Continuous Distribution how to find percentile

The problem is to find the 90th percentile of the distribution with the following density: $$f(x)=2x, \; 0\leq x<1$$ I found the mean to be $\frac{2}{3}$ and the variance $\frac{1}{8}$. But how do ...
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1answer
22 views

Identifying the k points in 2D geographic space which are 'most distant' from each other

I have a set of DNA samples from Y plants in a given geographic area. I'm going to be doing DNA sequencing on individuals in this population (and a number of other, separate populations), however due ...
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17 views

Power of statistical test with uniform random variables

Let $U_i$ be iid and $U_i \approx U(0,1)$. $$\begin{align} H_0: X & \approx \min(U_1,U_2,U_3) \\ H_1: X & \approx \min(U_1,U_2) \end{align}$$ Find power of this test if $\alpha = ...
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855 views

Why get the sum of squares instead of the sum of absolute values?

I'm self-studying machine learning and am getting into the basics of linear regression models. From what I understand so far, a good regression model minimizes the sum of the squared differences ...
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1answer
24 views

Inclusion probability [closed]

I have a problem to solve this exercise. Given the population, $(u_i=1,2,3; P_i=\frac{1}{6},\frac{2}{6},\frac{3}{6}) $. Specifications: 1) $n=2$ 2) sampling without replacement 3) $u_{ij}=u_{ji}$ ...
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28 views

Why is the covariance a measure of how much two random variables change together?

In descriptive statistics, the empirical covariance of a point cloud $\left\{(x_i,y_i) : 1\le i\le n \right\}$ is defined as ...
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20 views

How mean, median and mode changes with change in one element of set.

I have N numbers. Let mean of these numbers be A. Mode of these numbers be B.Median be C. Now if I change one element in these, how will my mean, median and mode change ? Can I calculate it directly? ...
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1answer
47 views

Expected value for Head/Tails

There are $N$ coins placed in a line. A coin may be facing head/tail direction with $0.5$ probability. Now I need to find number of pairs of coins $(i,j)$ such that $i<j$ and on index $i$ , I ...
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2answers
15 views

When does Mean Square Error increase?

As far i know, we want the model to include as few regressors as possible because the variance of the prediction $\hat y$ increases as the number of regressor increases. But from the hald cement ...
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1answer
100 views

Determining a consistent estimator/asymptotic relative efficiency

Question: Let $X_1,\ldots,X_n$ be i.i.d. as $N(0,\sigma^2)$. a) Show that $\delta_1 = k \sum_{i=1}^n |X_i|/n$ is a consistent estimator of $\sigma$ if and only if $ k = \sqrt{\pi/2}$. b) Determine ...
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1answer
26 views

Confusion about Notation for Bayesian Statistics

I'm currently trying to learn Bayesian Statistics but I keep losing time trying to figure out what exactly is meant by notation. Could someone answer the following for me? Let's say $X \sim ...
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99 views

Assume that three fair coins are to be tossed. What is the prob of the following event? Exactly two heads appear or all of the tosses are the same [closed]

I said previously that I needed help with one question but I actually need help with this one as well.
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probability n! please help it's the only question i don't know how to do on the homework [closed]

write n! in terms of (n-1)! I am not sure what it is asking. I have tried everythng.
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1answer
48 views

A cumulative distribution function is discontinuous at some points. Then Random Variable X is discrete?

Here is the distribution So, is the random variable X discrete or continuous? Note: first condition is if x<0 Edit: I have never come across this kind of distributions before. Extensive ...
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1answer
24 views

Computing the probability of two variables of the same sample.

Problem: The mean of a variety of apple is 400g with a standard deviation of 50g. If we choose 2 random apples of this variety, what would be the probability that the first one weights 150g more than ...
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1answer
15 views

Regression Line Through Origin

Suppose we have the regression model: $y_i$ = $\beta_0$ + $\beta_1x_i$ + $\epsilon_i$ where $y_i$ = ($Y_i$ - $\bar{Y}$) and $x_i$ = ($X_i$ - $\bar{X}$). I need to determine if the regression line ...
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4answers
140 views

Formal proof that mean minimize squared error function

On an important book of Machine Learning, I've found this proof. We want to minimize the cost function $J_0(X_0)$ defined by the formula $$J_0(x_0) = \sum_{k=1}^n \|x_0 - x_k \|^2.$$ The solution to ...
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1answer
33 views

When computing the CDF from a PDF, why is the integral bound a different variable? $F(x) =\int_{-\infty}^x f(t)\,dt$

Right now, I know the variable in the integral must be x, otherwise, the final result does not match the published CDFs of popular distributions. But I don't know why conceptually this is. If the CDF ...
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1answer
27 views

How to use Chi-Square for a rolling die

I'm watching this video. The guy said that this kind of test allows us to check whether the experiment occurs by chance or something wrong with the way the experiment is being done. I have done very ...
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12 views

What will happen if we adopted 1 x IQR to determine an outlier?

What's the percentage of all values that would be considered outliers if we adopted the 1 x IQR rule?
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4answers
707 views

Gambling problem

Question Robert will win $\$1$ with probability $\frac{1}{4}$, win $\$2$ with probability $\frac{1}{4}$, and lose $\$1$ with probability $\frac{1}{2}$ in a bet. Each bet is independent. Determine ...
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7 views

Calculating average with special summary statistics

We want to compute the mean of a data set D. The data set is not accessible. Instead we repeatedly gain access to average and size of a subset of D, which contains a data point d $\in$ D and some ...
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1answer
26 views

find out how 'peaky' histogram is

I have a histogram made of number of detections made by an application: 9 instances of detecting 1, 1 instances of detecting 2 and so on. Say I have these histograms: first histogram: 9 1 1 second ...
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17 views

How to measure the similarity or divergence of two distributions with different supports?

Suppose $X$ and $Y$ are two random variables with the distributions $F_X$ and $F_Y$ on the same support $\Theta$.Then KL divergence $D_{KL}(X||Y)$ is a way to measure the statistical distance between ...
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1answer
100 views

Poisson Distribution Question about the value of $\lambda$

I am stuck on a Poisson problem and for the life of me I can't tell what I'm doing wrong, I'm assuming I'm misinterpreting what $\lambda$ is supposed to be. The question reads: Past insurance company ...
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1answer
21 views

Why John Tukey set 1.5 IQR to detect outliers instead of 1 or 2?

To define outliers, why we cannot use: Lower Limit: Q1-1xIQR Upper Limit: Q3+1xIQR OR Lower Limit: Q1-2xIQR Upper Limit: Q3+2xIQR
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IQR vs. Z-score method, which one should I use to detect outlier?

I don't know when should I use IQR and Z-score method to detect outlier for a single numerical variable. Which one is appropriate for what situation?
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76 views

Do mathematicians suffer from higher rates of madness?

After seeing this post I watched the BBC documentary Dangerous Knowledge. Overall I felt it worthwhile, but couldn't help bristle at the explicit connections made between mathematical abstraction and ...
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14 views

Whats the reason of log-odds?

I'm following a course of bayesian inference. Out of the middle they are talking about log odds. For example, say that we have a uniform prior distribution, $\theta$ ~Beta(1,1). $$g(\theta) = ...
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1answer
32 views

Poisson Process

Assume emails arrive in your inbox modeled by a Poisson process with a mean of 4 per hour. Let $X$ denote the the time unitl your next e-mail, which is therefore exponentially distributed. Find the ...
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1answer
17 views

Value in between given percentiles

If I have a set of data such as 3500,4500,5700,7100 that represent the 25th, 50th, 75th, the 90th percentile how would I go about finding a percentile in between two of the given points? For instance, ...