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 to get more profit in stochastic process?

Suppose there is a system, for each step, I cost something but I didn't know how much I cost, and the system return to me something, which follow Guassian distribution and the expectation is what I ...
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Deriving a formula for a confidence interval

Derive a formula for a $(1-\alpha)100\%$ C.I. for $\mu_x -\mu_y $ for data that has the following properties: A random sample $X_1,X_2...X_n \ are \ i.i.d ~N(\mu_x, \sigma^2 ) $ Another random ] ...
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3answers
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How to prove expected value of uniform random variable?

I tried this: $$\int_a^b t~dt = \frac{t^2}{2}\Big]_a^b = \frac{b^2-a^2}{2} = \frac{(b+a)(b-a)}{2}$$ Isn't it supposed to be $\frac{b+a}{2}$ or something like that? Obviously if I multiply the ...
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computing p-value with small n

As part of the quality-control program for a catalyst manufacturing line, the raw materials (alumina and a binder) are tested for purity. The process requires that the purity of the alumina be greater ...
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37 views

Is the Monte Hall Problem a Ludic Fallacy?

"Is the Monte Hall Problem a Ludic Fallacy?" by James Greiner Suppose Fat Tony is on a game show, and he's given the choice of three doors: Behind one door is a car; behind the others, goats. (and, ...
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Confusion with Z-Score

Having some issue with the concept of Z score. When exactly do I use $Z = \frac{\bar X - u}{\sigma}$, and when do I use Z = $Z = \frac{\bar X - u}{\frac{\sigma}{\sqrt{n}}}$. I get very confused ...
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4 cards are shuffled and placed face down. Hidden faces display 4 elements: earth, wind, fire, water. You turn over cards until win or lose.

Question: 4 cards are shuffled and placed face down in front of you. Their hidden faces display 4 elements: water, earth, wind, fire. You turn over cards until win or lose. You win if you turn over ...
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incorrect rejection of a true null hypothesis?

We have a contest 1 weeks ago. One question is a bit strange for us as follows: $X\sim B(4,p). $ for test $H_0:p=0.2$ versus $H_1:p>0.2$. if $X=4$, $H_0$ assumption is rejected. calculate ...
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22 views

Method of moments estimation for $\theta$

I read one example in my notes, but I couldn't find out how the answer in my notes is derived. If $x_1,...,x_n$ are realizations of a random variable distributed with the following PDF: $f(z; ...
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How sample size affects confidence interval.

Suppose the weight of n primary one students has sample mean of 20KG. If n = 40, a certain percentage of confidence interval for the population mean is (15.5,24.5). Find the confidence interval if we ...
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F-test and T-test produce the same results

I am modelling a stochastic process by two different methodologies and I expect the results of each to be normally distributed with identical means and stdevs. To test that the distributions after ...
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35 views

Finding the probability using a normal distrubtion.

I have a stats question that says, "An airline flies airplanes that hold 100 passengers. Typically, some 10% of the passengers with reservations do not show up for the flight. The ...
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Relationship between distributions of correlations $\rho(X^1,Y^1)$ and $\rho(X^2,Y^2)$ if $X^2=WX^1$, $Y^2=WY^1$ and $W$ is a known stochastic matrix?

I have been stacked for a while with the following problem: Consider two samples of iid observations $X^1=\{X_1^1,\dots,X_n^1\}$ and $Y_1=\{Y_1^1,\dots,Y_n^1\}$ where $X_i^1 \sim \mathcal{N}(0,1)$ and ...
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1answer
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Uniformly distributed independent random Variables [on hold]

Let X and Y be independent random variables each uniformly distributed on (0,1). Find $P(Y\geq X | Y\geq \frac{1}{2})$. The answer is $\frac{3}{4}$ But I don't know how they got it :( Please help as I ...
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17 views

Calculate P-Value

In a certain area, regulations require that the chlorine level in wastewater discharges be less than 100 $\mu$/L. In a sample of 85 wastewater specimens, the mean chlorine concentration was 98 ...
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Argmax distribution of Brwonian motion plus linear drift

I want to know the the density function or the tail of the density funcion of the following random variables: $$\underset{{t\in [0,+\infty]}}{\arg \max} \quad {W_t-t}.$$ Thank you very much
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Finding distribution of distance from origin

A shot is fired at a circular target. The vertical and horizontal coordinates of the point of impact (taking the centre of the target as origin) are independent random variables, each distributed ...
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39 views

Question about the definition of independent events

If you have two events A and B that are independent, then it is said that $p(A)p(B)=p(A\cap B)$, and illustrated in a venn diagram as two areas that do not overlap. The opposite goes for dependent ...
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You are making cookies and add N chips to dough randomly, and split it into 100 equal cookies, again at random. How many chips should go into dough?

Question: You are making chocolate chip cookies. You add N chips randomly to the dough and you randomly split the dough into 100 equal cookies. How many chips should go into the dough to give a ...
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2answers
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Joint Random Variable: Given f(x,y), find P(X>Y)

There are 2 continuous random variables, X and Y. Say the joint pdf of (X,Y) is f(x,y). How do you find the P(X>Y) generally? Like I am not sure where to start with.
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question on expectation of random variable

If I have some discrete probability mass function for random variable $X$, like for example, 0 has probability 0.2, 1 has probability 0.3, 2 has probability 0.5, is the expected value $E(\sin(x)) = ...
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How do you compute a 90% and 95% confidence interval for a guesstimation problem?

Question: How would you estimate the weight of Mount Everest? Give a 90% and 95% confidence interval. I would define what Mount Everest is. Including its boundaries (length, width) and estimate the ...
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Is there a way to derive the skewness formulae for different distributions?

I would like to know if there is a way to derive the formula to calculate skewness for different distributions, as they are not included on the formula sheet in the coming exams. For example, ...
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1answer
31 views

Book Suggestion for Statistics

I recently finished my Masters in Mathematics, more incline to analysis and algebra. I do not know if I should blame my school for laying a poor foundation in statistics or I should blame myself for ...
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Basic Probability - is the textbook wrong here?

The question is "You have eight songs on your mp3 player. You set your player to play all the songs on the list in random order without repeating any songs. Suppose that four of the songs on your ...
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Probability Question: What percentage of the bag of balls is marked?

In a bag of reds and black balls, $30\%$ were red, and $90\%$ of the black balls and $80\%$ of the red balls are marked balls. What percentage of the bag of balls is marked? I thought I would have to ...
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Statistic question on biased estimators [on hold]

Can anyone help with this question please? A distribution is equally likely to take the values of 1 or 4. Show that Sn-1 (in the book Sn-1 is generally used for unbiased estimator of a population) ...
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1answer
22 views

How to calculate MISE - Mean Integrated Squared Error?

I might have misunderstood something, but it seems like taking a definite integral from expectation or expectation from definite integral. But the first stage will return a number in both cases. I ...
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1answer
19 views

Interval estimate to infer the population mean with a 95% confidence level

An industrial designer wants to determine the average time it takes for an adult to assemble a toy. 24 people were randomly chosen to assemble the toy and the time taken (in minutes) were as follows: ...
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Why is the joint density function of X and Y 1/L^2? [on hold]

Why is the joint density function of $X$ and $Y$ $\frac{1}{L^{2}}$ If an accident occurs at a point $X$ that is uniformly distributed on a road of length $L$. An ambulance is at location $Y$ at the ...
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1answer
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How to prove $\sqrt{Y}$ can be variance stabilizing transformation of poisson distribution?

I am studying constant variance checking when conducting ANOVA. I know that $\sqrt{Y}$ is one of the common transformations for a Poisson distribution, but I can't prove it. I also read Anscombe ...
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Measuring data/Statistics/Scoring

I am not sure how to make this a problem and solve it. It's statistical data that I am trying to figure out for a paper I'm writing. Basically, I am scoring candidates with 9 questions. Here's an ...
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Intuitive explanation of requirement for achieving the Cramer Rao Lower Bound

this question relates to the requirement for achieving CRLB. I know that for a random sample $Y_1, \ldots, Y_n$, an estimator $U$ of $g(\theta)$ is MVUE (i.e. it is unbiased and also ...
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1answer
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A state legislator wishes to survey residents of her district to see what proportion p of the electorate..

A state legislator wishes to survey residents of her district to see what proportion $p$ of the electorate is aware of her position on using state funds to pay for abortions. a) What sample size is ...
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1answer
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Reading P value from ANOVA table generated by R [duplicate]

I generated an ANOVA table in R and every boxes are in number except the P value shows "3.387e-05". What does that really mean?
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1answer
25 views

math statistics [on hold]

A computer lab contains 12 computers . The probability that any one of them will requi re repair on a given month is 0.15 ; assuming a binomial distribution a. [ 4 ] F ind the probability that ...
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1answer
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Probability - something with a small chance of occurring, but is repeated multiple times.

For example, if you have a $1.5\%$ chance of obtaining admission to any school and you apply to $15$ schools what is the chance that you'll get into a least $1$ school? Is this as simple as $1.5\%$ ...
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Definition of true density

I am reading a paper and it talks about true densities, I mean they talk about obtaining densities from data and later compare them with the true density I want to know how can I obtain a true density ...
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2answers
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How to prove $E[e^{e^y}]=\infty$? y is a normal random variable

The question is, given $Y\sim N(\mu,\sigma^2)$, how to prove$E[e^{e^Y}]=\infty$? I tried to look Y as some kind of Ito's process and apply Ito's formula to it but it doesn't make sense. Next I tried ...
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1answer
28 views

Computer Component with Gamma Distribution?

I comes to a question of one old-exam as follows: if the life of one computer component (in year) has Gamma Distribution (if I translate correctly) with ...
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46 views

Logic behind the combo of cards in a hand that contain only clubs

In looking at a stats problem where you want all combos of a 5-card hand that contain at least one club, the approach I have is to find the combos of 5-card hands that do not contain clubs, and then ...
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1answer
28 views

Comparison of parameter: two different populations

I was wondering what the best way is to check for the equality of two parameters for a regression with no constant including possibly a confidence interval and p-value. $$H_0:\beta_1=\beta_2\ \vert\ ...
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2answers
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How to find the bias, variance and MSE of $\hat p$?

If $X_1,\dots,X_n$ are iid $\mathrm{Binomial}(3,p)$, then the maximum likelihood estimator of $p$ is $$\hat p = \frac{1}{n}\sum_i X_i$$ Find the bias, variance and MSE of $\hat p$? We are asked to ...
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1answer
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Normal distribution with dice

I'm wondering how to control the normal distribution that comes from summing dice rolls only using different numbers of dice, different combination of types of dice (d4, d6, d8, d10, d12, d20) and ...
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1answer
41 views

Kruskal Wallis - Effect size

I analyse 4 algorithms and 3 sets of metrics for each algorithm in which I apply the non-parametric Kruskal-Wallis test for each metric to detect any differences in performance between these ...
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Help me with Statistics..? [on hold]

Cars pass a point on the highway at a Poisson rate one per minute. If five percent of the cars on the road are dodgers, then a)What is the probability that at least one Doge passes by during an ...
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Difference in Chebyshev inequality.

I am following a lecture, that provides the following formula of the Chebyshev inequality: $$P\left \{ \left | \frac{S_n}{n} - p \right | > \varepsilon \right \} \leqslant \frac{1}{4n\varepsilon ...
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2answers
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Testing $H_0 : \mu_x \neq \mu_y $, in a company that markets two brands of latex paint.

A company markets two brands of latex paint regular and a more expensive brand that claims to dry an hour faster. A consumer magazine decides to test this claim by painting ten panels with each ...
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Distribution of P[Y=n] = P[n-1<X<n] for X exponentially distributed

From an assignment, we have "Let X be an exponentially distributed random variable with probability density function. $f(x) = λe^{−λx}$, for $x > 0$" I've worked out that for $P[Y=n] = P[n-1 < ...
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Find the probability of the following events:

Suppose that a survey of married couples shows that $20%$ of the husbands watched football and $8%$ of wives watched. Also, if the husbands watched, the probability that the wives watched increased to ...