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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expectation calculation problem small problem

a Continuous, positive random variable X, whose PDF is proportional to $(1+x)^{-4}$, where $0<x<\infty$, determine $E(X)$ i tried to solve it directly by integrating from 0 to infinity to get ...
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5 views

Bayesian update multivariate normal based on one-dimensional signal: simple rule

Is there a simple rule to update the linear combination of normal distributions based on a one-dimensional signal? The unconditional joint density of $(\eta,\theta)$ is multivariate normal ...
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21 views

expectation calculation problem

I got the answers for this and i know its 1.05 but the way it explains is very difficult to understand so im seeking for some help here. A system made up of 7 components with independent, identically ...
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2answers
14 views

Applying the basic formula for binomial distribution

I'm pretty confused on how this works. In my class my teacher states that: Let $X$ be a random variable with $S_X = \{0,1\}$. $X$ follows a Bernoulli distribution if $P(X = x) = p^x(1-p)^{1-x}$ for ...
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1answer
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Distribution of random variables (normal and standard normal)

Suppose that $X_i \sim N(\mu, \sigma^2)$ for $i = 1, \ldots, n$ and that $Z_i \sim N(0,1)$ where all of the random variables are independent. Denote $s^2_Z$ as the sample variance of $Z_1 , \ldots, ...
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30 views

How to find $E[Y|X=1]$?

A fair die is repeatedly rolled. Let $X$ and $Y$ denote, respectively, the number of rolls required to obtain a $1$ and a $2$. How do I find $E[Y|X=1]$? edit: for using this I got 1*6 = 6 total ...
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Prove by induction: $E[\sum_{i=1}^nc_iU_i(X)]=\sum_{i=1}^nc_iE[U_i(X)]$ Please just check what I've done

Prove by induction: $$E[\sum_{i=1}^nc_iU_i(X)]=\sum_{i=1}^nc_iE[U_i(X)]$$ Let me show you what I've done. I think I'm right: $$n=1,$$ $$E[c_1U_1(X)] = c_1E[U_1(X)]$$ Okay so maybe this one looks ...
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4 views

A mix between the Horvitz-Thompson and ordinary estimator

I have asked this question on mathoverflow, but got no answer. Here I have corrected some mistakes and wish to hear any ideas that may bring at least numerical result: The data I have two samples: ...
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1answer
16 views

Selection of Distribution model

An expressed parcel delivery company offers a First Class service for which it is promised that 80% of all parcels are delivered within 24 hours of dispatch. It is suspected that the true successful ...
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28 views

Standard deviation and mean question [on hold]

The pulse rates of healthy adults are normally distributed with mean 72 and standard deviation of 5. i) Approximately ninety-five percent of the adult population can be said to have their ...
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1answer
30 views

formal proof that p-values are uniformly distributed

I'm trying to prove that $p$-values under the null hypothesis are uniformly distributed in $[0, 1]$ for an absolutely continuous test statistic $X$. Proof: By continuity of $F_X$, it is sufficient to ...
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19 views

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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21 views

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
19 views

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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1answer
17 views

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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60 views

Is the Monte Hall Problem a Ludic Fallacy? [on hold]

"Is the Monte Hall Problem a Ludic Fallacy?" by James Greiner ---- EDIT ---- My Questions: 1) Does cheating mathematically reduce your odds? 2) Regarding the Monte Hall Problem, how is it any ...
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2answers
21 views

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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1answer
29 views

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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1answer
53 views

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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1answer
25 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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27 views

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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16 views

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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1answer
42 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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11 views

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
31 views

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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1answer
20 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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10 views

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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2answers
30 views

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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1answer
15 views

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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2answers
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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
32 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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38 views

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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3answers
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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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14 views

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
21 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
20 views

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
21 views

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
27 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
36 views

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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1answer
22 views

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 ...