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.

learn more… | top users | synonyms

0
votes
1answer
11 views

Find the probability density function of $Y=X+Y$

If the joint density of $X_1$ and $X_2$ is $$f(x_1,x_2) = 6e^{-3x_1-2x_2}, x_1 >0; x_2>0$$ Find the probability density function of $Y=X_1+X_2$ We did this as a class example but never ...
0
votes
2answers
15 views

There's something fundamental that I'm missing regarding the Standard Uniform Distribution (Continuous)

So if we have the standard uniform distribution $$X \sim U(0,1)$$ $$f(x) = 1 \text, 0 < x < 1$$ So now I don't understand how the probability of having any point between 0 and 1 is just 1. ...
0
votes
0answers
11 views

How can I calculate percent error with a denominator of 0?

I am using the MAPE formula, ABS(actual-forecast)/actual *100, but this raises problems when the actual values are zero. Is it possible to add 1 to the denominator and calculate this way? Would I get ...
0
votes
0answers
11 views

How to determine whether a split wins or loses in blackjack?

I would like to determine whether or not the act of splitting results in a Win, Push, or a Loss. From what I have seen so far, other simulators simply divide the two hands and record the W/L of each ...
1
vote
0answers
13 views

Is $X(T) = A \sin(\omega_0 t + \Phi)$ mean ergodic?

This is an example of a tutorial but I think has not been solved properly. Please help me! $X(T) = A \sin(\omega_0 t + \Phi)$ $A$ and $\phi$ are independent $A$ is uniformly distributed over ...
0
votes
1answer
13 views

The logic behind this algorithm or how to teach this estimation

I'm trying to understand the logic behind the estimation of the calculation of the median from group data: (see this link) I understood the algorithm process, my question is about the logic behind ...
0
votes
1answer
22 views

Variance Explanation

Suppose the expected number of blowouts for 15 trucks is $\mu=4$, the variance is $\sigma^2=3$, and the standard deviation $\sigma \approx 1.72$. What does the variance with respect to this problem ...
0
votes
1answer
11 views

Why or how can we say errors(residuals) are independent and they follow the normal probabilties in regression analysis?

While I am studying linear regression analysis and I have encountered a sentence salying "errors are independent and follow normal probabilities". I can only guess what it says but I can't trust my ...
0
votes
0answers
11 views

Multivariate gaussian and average covariance matrix

Suppose we have a (possibly infinite) collection k-variate gaussian distributions $\{(\mathcal{N}(\mu_{\lambda}, \Sigma_{\lambda}))\}$ ($\lambda$ is just a label), and for each distribution $\mu \in ...
1
vote
1answer
29 views

Question about package of cookies that is a random variable

The weight in grams of package of cookies is a random variable with expected value of $300$ grams $\color{blue}{A)}$ assume that X is normally distributed with standard deviation of $15 $ grams ...
0
votes
1answer
18 views

Simple Regression

The question asks for the slope and intercept, but I don't have a correlation coefficient or the raw data (just sample size, mean, and standard deviation).
0
votes
1answer
31 views

Slope and Intercept using only sample size, mean, and standard deviation.

I have been tasked with finding the simple linear regression model slope and intercept of two sets of data, but the only have access to the sample size, mean, and standard deviations of each set of ...
8
votes
1answer
81 views

Math Intuition and Natural Motivation Behind t-Student Distribution

I am trying to understand with basic mathematical background how the $t$-Student distribution is a "natural" $pdf$ to define. So I hope that this not too-general a question, but given that the ...
0
votes
1answer
17 views

Fisher Information: While calculating the expectation of score function, why do we integrate with dx?

Let $$S=\frac{d}{dy} \left[\log f(x|y)\right]$$ where $y$ is the parameter and $S$ is score function Now in text books, ...
4
votes
2answers
75 views

“Mastermind”-esque safe opening problem.

I read this interview question for a trading job and it seems quite difficult. What is the technique to solving it? You have a safe with six digits and a light. You can input a code, if you have ...
2
votes
2answers
28 views

Statistics question on basil bush random variable

The height, $H$, in meters of a basil bush is a random variable with the probability density function $f_{_H}(t)=e^t,\;0\leq t\leq H_0$ such that $H_0$ is the maximal height. $\color{blue}{(1)}$ I ...
4
votes
0answers
50 views

Finding the expected value and variance of ${X^3}$

For a random variable $X$, $(X^3-1)$ is uniformly distributed in the interval $[0,7]$ I need to find the expected value and variance of $\color{blue}{X^3}$ and I know that: cumulative ...
2
votes
0answers
18 views

Estimating the Average and Standard Deviation of a Population based on a Sample with Missing Data with Known Ranks

I need a way to shows me how the parameters of PDF, log-normal in this case, can be estimated based on a set with missing data points at the tail end of a sample. For example, Consider we had 20 ...
0
votes
1answer
31 views

sufficient conditions for a stochastic process to be wide sense stationary

From the page Stationary process, I have the following definition: WSS random processes only require that 1st moment and autocovariance do not vary with respect to time and from the page ...
0
votes
2answers
27 views

The random variable $ Z = 1-F(X)$

I will formulate the theorem (with no proof) if $X \in \mathbb{R}$ is a random variable with continuous distribution function $F$ then the random variable $Z = 1-F(X)$ has a uniform distribution on ...
1
vote
1answer
40 views

Finding the expected value and variance of $X$

For a random variable $X$, $(X^3-1)$ is uniformly distributed in the interval $[0,7]$ I need to find the expected value and variance of $X$ and I know that: cumulative distribution function: ...
0
votes
1answer
21 views

Finding the probability density function and the distribute accumulate function

For a random variable $X$, $(X^3-1)$ is uniformly distributed in the interval $[0,7]$ I need to find the probability density function and the cumulative distribution function of $X$ My attempt: ...
-3
votes
0answers
18 views

Minimum sample size [on hold]

I have a survey of $n$ distinct opinions and their outcomes rounded to percentages. How can I compute the minimum number of sample size? For example IF resuls are 0.33, 0.33, 0.33 then the minimum ...
-1
votes
0answers
26 views

different combinations of numbers [on hold]

Can anyone help me pick 1,000 combinations of six numbers from two separate pools of numbers - five different numbers from 1 to 75 and one number from 1 to 15?
2
votes
3answers
46 views

compute temporal average of $\sin(\omega_0t+\Phi)\sin(\omega_0t+\omega_0\tau+\Phi)$

assuming that $\Phi$ is uniformly distributed over $(0,2\pi)$ compute: $$E[\sin(\omega_0t+\Phi)\sin(\omega_0t+\omega_0\tau+\Phi)]$$ I have solved the problem as continues: $$\begin{align} ...
0
votes
0answers
21 views

Finding upper critical value with Chebyshev's inequality

Consider $X$ is a Poisson random variable with distribution $X$~$Pois(\theta)$. I define the mean in my hypothesis as $\lambda$ and nominal significance level $\alpha$. Null hypothesis $H_0 : ...
1
vote
1answer
40 views

Why this process is nonergodic?

I am studying a tutorial on stochastic processes and there's an example in it which I don't understand anything of it. First of all there is this criterion for a mean-ergodic random process: For ...
1
vote
2answers
26 views

maximum likelihood estimator for theta [on hold]

I was wondering if someone could please just get me started on this question i'm just a bit stuck: $$ f(y_1,y_2,\ldots,y_n\mid \theta)\propto \exp\left[\frac{−1}8 \sum_i (y_i−\theta)^2\right] $$ Any ...
0
votes
0answers
18 views

Possible to eliminate mutual information between random variables by reducing the number of them?

Say you have a set of random variables that have some mutual information structure. Could be that they all have nonzero MI between them. Or perhaps there are some clusters of variables with ...
0
votes
2answers
11 views

Statistical Dependency Transitivity

I came across this question here on Stack Exchange, and it didn't address something that I then became curious about. If $X_1, X_2$ are dependent and $X_2, X_3$ are dependent, then are $X_1, X_3$ ...
3
votes
1answer
52 views

Covariance of 1-D random process is $n\times n$!!!!

I'm reading a tutorial on stochastic processes. There is an example in the tutorial as follows: General Moving Average random process given as $X[n]=\frac{(U[n]+U[n-1])}{2}$ where $E[U[n]]=\mu$ ...
0
votes
1answer
35 views

Moment generating functions…which distributions to use?

Q: You hired a terrible programmer and the moment generating function for the distribution of software bugs is M(t) = (1 - $\theta$t)$^{-\alpha}$. Groups of bugs can be detected within $\mu$ = 47 ...
2
votes
3answers
47 views

I have some questions related to Fisher's book 1925.

I was studying Fisher 1925 and while reading i had some trouble with this part. Fitting the Normal Distribution From a sample of $n$ individuals of a normal population the mean and the standard ...
2
votes
1answer
31 views

Is tossing a die in 10 consequent days an ergodic process?

IT maybe an elementary question but I'm totally new to the concept. In Wikipedia, ergodicity is defined as follows: In statistics, the term describes a random process for which the time ...
0
votes
1answer
20 views

what's the difference between variable and process from a statistical point of view?

I'm reading a tutorial stochastic process: ergodicity and temporal averages and I'm totally confused. It is said that: Suppose an IID random process whose marginal PDF is Gaussian with mean ...
1
vote
1answer
18 views

Ensemble average of square of fluctuations proof

The ensemble average of a random variable $x$ is denoted as $X$ or $\left \langle x \right \rangle$, and is defined as: $$ X = \left \langle x \right \rangle = \lim_{N \to \infty} \frac{1}{N} ...
0
votes
0answers
7 views

Comparison of Cramer Rao bound - deduction and conceptual question

The CRB gives the variance of the estimation error of the estimates and a lower value is preferred. I have computed the cramer rao bound (CRB) of the estimates of the coefficients $\mathbf{h^T}$ for ...
0
votes
0answers
41 views

what is the difference between statiscal averagre and average?

I'm reading a book on synthetic aperture radar and it is said that: The term $\sigma^{\circ}$ is the averaged radar cross section per unit area, also called the scattering coefficient or ...
1
vote
1answer
19 views

Linear regression relationships

Velocity $= X$, distance to stop $= Y$ $\beta_0= -17.5791$, $\hat{\operatorname{se}}(\beta_0)=6.7584$ $\beta_1 = 3.9324$, $\hat{\operatorname{se}}\beta_1 = 0.41.55$ degrees of freedom $=48$ (a) is ...
0
votes
1answer
15 views

What am I plugging in wrong to my normal distribution calculator?

I am trying to find the probability of the following question: Cans of regular Coke are labeled as containing 12 oz. Statistics students weighed the contents of 7 randomly chosen cans, and found the ...
5
votes
1answer
43 views

A good, self-study statistical computing book

I'm looking for a book an introductory statistical computing that has proofs for the methods as well as examples. I'd like proofs that are about the same level as (or lower than) proofs in Statistical ...
1
vote
1answer
29 views

Is there any difference between statistical learning and machine learning?

Straight to the point, I'm a math student and I have a course this year called Statistical Learning. From the description, the course contains: Large datasets analysis, regression, principal ...
0
votes
0answers
24 views

Weighting the data by the history

I have a input stream 3D data that comes every time frame. Each point is defined by 3D vector of x,y,z. There is a evaluation function [say f(x)] that computes if the point at time t is valid or ...
-1
votes
1answer
30 views

Normal distribution calculations

We have a gaussian distribution $$ X \sim N(\mu,\sigma^2)$$ where $\mu = 4$ and $\sigma^2 =1.5$ . Probability is given by : $P(x<c)=0.35$ $c$ needs to be calculated. And we got ...
2
votes
2answers
34 views

What is the variance of the volumes of particles?

According to Zimmels (1983), the sizes of particles used in sedimentation experiments often have a uniform distribution. In sedimentation involving mixtures of particles of various sizes, the larger ...
0
votes
2answers
33 views

Class Coin Toss Experiment

My classmates and I are doing a coin toss experiment (i.e. toss coin 100 times). I have already determined that I have a fair coin, since I tossed $43$ heads, and this falls into a $95$% confidence ...
-2
votes
0answers
21 views

What is the proof behind the mean confidence interval for a Binomial Distribution?

How do we obtain the range to be as [$\mu-$$zσ$, $\mu+$$zσ$]? Is it when $n$ is sufficiently big?
0
votes
1answer
31 views

Pairing birthdays [on hold]

How large a group of people would you need to provide a better than 50-50 chance that everyone will have at least one birthday (just month/day) partner?
1
vote
0answers
25 views

How Kriging, Bochner theorem and Positive definite (PD) function are related?

This question referes to the link: https://en.wikipedia.org/wiki/Kriging I can understand the relation between Bochner's theorem and PD function. But could not properly understand and connect all ...
2
votes
1answer
38 views

Cramer-Rao lower bound for normal($\theta, 4\theta^2$)

I am trying to find the Cramer-Rao lower bound for unbiased estimators of $\theta$, given a sample $X_1,\ldots, X_n \sim \textrm{normal}(\theta,4\theta^2)$. I am calculating the CRLB as $$ ...