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 do i find the percent of packages that will be delivered in less than 16 hours

Couruer service has found that its delivery time is normally distrbuted with a mean of 14 hours and a standard deviation of 3 hours. what percent of packages will be delivered in less than 16 hours?
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7 views

Additive and multiplicative model

I'm new in study of survival analysis. There are some basic-questions that I wanna ask and it will be great if somebody could help me to aswer these following questions: What the differences between ...
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12 views

p value and hypothesis testing

In a random sample of 99 games the home team won 57 games. Test the null hypothesis That home team wins one half of all games against the alternative that the home team wins a majority games. How can ...
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2answers
26 views

Simultaneous density function of two continuous variables, X and Y.

I'm having issues with calculating the simultaneous density function of two continuous variables, X and Y. I took a screenshot of the information: How should I start? I know that if the two ...
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0answers
8 views

How to combine 2 different variables (time&cost) into a fitness function?

We are working on a simulation of uber taxi drivers, and to measure the effectiveness of additional functionality (like picking up extra passengers along the way) we would need to create a fitness ...
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18 views

Hoeffding’s inequality extension

In Hoeffding’s inequality we assume that the random variables $X_i$ ,$i=1,..,n$ are i.i.d. and bounded . Is there any extension to Hoeffding’s inequality for the case that $X_i$ are identically ...
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1answer
26 views

Steve Nash’s expected value from his one-and-one free throw situation is 1.72 points. What is his free-throw percentage?

The one-on-one free throw situation works like this - for the first throw, if you make it, you get to do it again. If you miss, you don't get another chance. If you make it the second time, you get ...
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12 views

Hypothesis test, p value problem self study

In a random sample ,p=57/99 Hypothesis test H0 : P = 1/2 H1 : P > 1/2 I want to test the H0 and want to find the p value and threshold of significance level. I am trying to solve this problem. I ...
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3answers
24 views

How to denote an average of data satisfying a given condition

I have to write an expression for the following. I have values, one fpr each of the last 5 months: ...
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10 views

bias reduction when the bias depends on the true parameter

Let's say we estimate a parameter, $\theta$, by $\hat{\theta}$. For this estimator we have the following property that $$\hat{\theta}\to_{p}\theta+f(\theta)$$ where $\to_{p}$ denotes convergence in ...
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1answer
30 views

How to use Bayes' Theorem to forecast crude oil prices? [on hold]

So I'm new with Bayes' Theorem, but I have to use it to predict future crude oil prices. I have been looking at examples online, but I got very confused by it. I can understand the simple examples ...
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32 views

Urn problem with replacement where the amount of balls drawn is random [on hold]

Suppose we have an urn with 52 balls, all are numbered from 1 to 52. We draw twice from the urn with replacement. The amount of balls drawn every turn is random with the probabilities p_0,p_1...p_8 ...
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What is a easy way to draw from distribution $f(x|u) \propto x^{\alpha-1}I(x<u)$?

We know $u$ and want to draw $x$ from the conditional density $f(x|u) \propto x^{\alpha-1}I(x<u), \alpha>0$. One way is that first draw $r$ from uniform(0,1), and then set $x=u ...
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13 views

What statistical methods can I use to compare the differences between the two time series?

I conducted gas emission measurements from two buildings for a year (one is the building without treatment; the other is the one with treatment). They are two time series data. I plotted the data and ...
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3answers
72 views

Suppose a city with Three type of coins ?!

in a city we have tree type 1 dollar, 2 dollar, 3 dollar of coins. we want to pay for a 20 dollar product. how many ways we can pay for a 20 dollar product, if the seller has no money and number of 1 ...
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1answer
34 views

Marginal distribution of uniform distribution conditioned on poission?? [on hold]

Let $N$ be a Poisson distribution with parameter $\lambda$. Conditioned on $N$, let $X$ be uniformly distributed. What is the marginal distribution of $X$? ( this is one of my final exam problems ...
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1answer
31 views

Geometric Random Variables

I have a question that involves a certain criteria of a random variable as shown: The random variable $X$ has the distribution $Geo(0.2)$ and I would love it if someone could explain what the ...
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0answers
56 views

What is the probability that 5 randomly chosen cards in a deck add up to 40 or more? [on hold]

I have made a probability game, where you have to pull out any 5 cards without looking (from a deck of 52 cards), and if all five cards add up to 40 or more, they player pulling the 5 cards from the ...
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3answers
27 views

Proof variance of Geometric Distribution

I have a Geometric Distribution, where the stochastic variable $X$ represents the number of failures before the first success. The distribution function is $P(X=x) = q^x p$ for $x=0,1,2,\ldots$ and ...
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1answer
57 views

What's the probability that the first four children born are boys and the last two children born are girls?

I'm having some problems with determining how to calculate a question about the gender proportion in newborns in some random family. A family consists of 6 children. The probability of a boy being ...
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1answer
34 views

Pareto distribution

So I'm given a Pareto distribution with parameters $\alpha >0$ and $k>0$ which is the form of \begin{equation*} f(x) = \frac{\alpha k^\alpha}{x^{\alpha+1}},~x > k. \end{equation*} I found ...
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24 views

Combining Sample Means and Variances

Let us assume we have several samples of unknown size but known mean value $\mu_{i}(x)$ and known variance $\sigma_{i}^{2}$. Now we want to calculate the mean value and variance for the total ...
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20 views

Solving an integral to solve a statistical problem

In solving a statistical problem, I want to know $$\mathbb{P}( (Y_2 \leq q_\alpha | Y_1 \geq q_\alpha) $$ where $Y_1 = X_1 + Z$ and $Y_2 = X_2 + Z$ and $Z \sim N(0,\sigma_z^2), X_i \sim N(0, ...
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13 views

Does Weighted avg of Weighted avg makes sense..?

Scenario is like we have a city in which there are many schools. All of them conducted a mathematics exam and it depends on the school for each class how much marks the exam would be. Suppose school ...
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3answers
30 views

Symmetry in Probability Around a Particular Phenomenon in Time?

This has been hurting my brain substantially, recently. I'm not sure if I'm failing to make connections or if I see connections but am weary of their relevance. In my text the author claims that ...
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7 views

What is Bootstrapping in statistics? How can I use it to determine error in the mean, variance, kurtosis and skewness of a data set?

From what I understood from searching randomly is that it has something to do with resampling. What does this resampling mean? Is it selecting random data from a distribution or is it getting data ...
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12 views

Explaining the reasoning behind the initial derivation steps for the Gamma distribution

When deriving the gamma distribution, how is it justified that $P(W>w)$ is the summation of separate exponential distributions? I understand that for the gamma distribution, we have a series of ...
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1answer
25 views

How to prove that the maxium likelihood estimator of $\theta$ is aysmptotically unbiased and cosistent

In a class we looked at this example: Let $X_1,...,X_n\sim U(0,\theta)$. Then the maximum likelihood function is $\mathcal{L}(\theta) = \begin{cases} \dfrac{1}{\theta^{n}} & \text{if } ...
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1answer
18 views

Independence - Probability and Statistics

Any help on this problem is greatly appreciated! I'm completely stuck School board officials are debating whether to require all high school seniors to take a proficiency exam before graduating. A ...
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1answer
656 views

How to quantify the differencen between 2/4 and 20/40?

Assume I have two methods to do prediction. The first method makes 4 predictions and 2 out of 4 are correct. The second method makes 40 predictions and 20 out of 40 are correct. The prediction ...
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1answer
32 views

If I flip $n$ coins, what is the probability of at least $3/4$ of them coming up heads?

I'm looking for an equation that will work for any positive integer n. I know that multiple coin flips approaches a normal distribution, but I'm looking for answers that deal specifically with random ...
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29 views

$\overline{X} \rightarrow c$ in probability $\Rightarrow g(\overline{X}) \rightarrow g(c)$ in probability

Prove that if $\overline{X} \rightarrow c$ in probability and if g is a continuous function, then $g(\overline{X}) \rightarrow g(c)$ in probability. Once I think in the situation $|g|<M$, it is ...
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1answer
25 views

Lagrange primal and duality properties [on hold]

max Lprimal(x,lamb,mu) >= f(x*) + lamb g(x*) + mu h(x*) s.t. lamb >= 0 min Ldual(lamb,mu) = min max Lprimal(x,lamb,mu) What is the purpose and properties of the lagrange dual function, why do I need ...
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0answers
18 views

Maximum density linear combination chi squares

I have a positive linear combination of chi square variables \begin{equation*} X=\sum_{i=1}^k \lambda_i \chi^2(r_i), \end{equation*} the degrees of freedom satisfy $r_i>1$. I need an upperbound ...
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24 views

Confidence interval of exponential random variables

I have a sequence of random variables $X_1, X_2, ..., X_n$ such that $X_i = e^{-(x_i-Θ)}$ I have to construct a confidence interval of the form $[Θ−c,Θ]$,where $Θ = \min _i{X_i}$. For $n = 10$ how ...
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1answer
103 views

Confidence Interval for Pareto Distribution

A random variable is said to have probability density function $$f_X(x)=\frac{\alpha k^\alpha}{x^{\alpha +1}},\quad \alpha , k>0 \; \text{ and }\; x>k.$$ 1. Compute the MLE estimators ...
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2answers
14 views

Finding the probability of time between two events for a poisson process

If two events occur at a rate of 1.8 per hour on average, and this occurrence follows a poisson process, what is the probability that there is at least 1 hour between two events? My approach for this ...
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12 views

Stat MGF question [on hold]

Suppose we are given a random variable Y with probability density function f(x) = exp(((xƟ -b(Ɵ))/Φ) + c(y,Φ)) show that the moment generating function is: exp((b(Ɵ+Φt) -b(Ɵ))/Φ) Any ideas on how ...
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1answer
37 views

Is there a method to check if two curves (non-linear) are identical

I have two data sets of pollutant concentration on simultaneous days. I have to check whether these two curves follow similar pattern or not ( there might be some time lag between both) on daily ...
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0answers
38 views

Skellam CDF Increasing vs Decreasing in a parameter

I'm working with the following Poisson difference distribution: $$\text{Prob}\{X_1-X_2 \geq 0\} $$ where $X_1 \sim$ Poisson $(\mu_1)$ is independent from $X_2 \sim$ Poisson $(\mu_2)$. I need to ...
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1answer
166 views

Why should statistics be considered mathematics? [on hold]

Statistics should be completely separate from mathematics because it is not exact as pure mathematics , so my question is why they are still together? Statistics is a science in my opinion, and it is ...
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1answer
24 views

Confusion about the sample mean and random variables

As I understand the sample mean you just add a bunch of random variables that constitute a sample from their common distribution and divide by the number of those same random variables. When I apply ...
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1answer
26 views

What does “2- place real function” mean?

What does "2-place real function" mean? This comes up in the context of copulas, as here.
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25 views

Estimating the size of my population

I have a following problem: Imagine you have a hat with many different balls in it and you want to estimate, how many balls are totally in the hat. The only think you are allowed to do is to take one ...
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0answers
205 views

Pareto distribution,fisher information, confidence interval [on hold]

Having a bit of problem at these questions, greatly appreciated if anyone can solve them. For the notation, k^ is k with a hat on top of it, don't know how to do that on a keyboard. The rest is ...
1
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1answer
71 views

Density functions and estimators

A random variable is said to have probability density function $$f_X(x)=\frac{\alpha k^\alpha}{x^{\alpha +1}},\quad \alpha , k>0 \; \text{ and }\; x>k.$$ 1. Compute the MLE estimators ...
2
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1answer
34 views

Confidence Interval Question - Confused on Approaching the Problem

My question is almost exactly the same as another one here on math stack exchange, but it isn't as explanatory as I'd like it to be with some parts. I was unsure of whether or not it'd be "okay" to ...
2
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1answer
27 views

Probability of co-occurence

Of total $N$ people, $m$ people are good at mathematics and $c$ people are good at computer science. What is the expected number of people good at both mathematics and computer science? Or what is the ...
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1answer
28 views

proof of the convergence of confidence intervals

The confidence interval can be derived intuitively by replacing the standardized mean with the standard normal and variance with sample variance, but is there a formal limit? I'm trying to prove if ...
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1answer
79 views

Fisher information matrix of MLE's

I know what it means to compute the fisher information matrix of a vector of parameters. However, how does one compute the fisher information matrix of a vector of MLE's? Specifically, I am working ...