Tagged Questions

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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10 views

determinant of the covariance matrix of a normal distribution

Suppose a $p \times 1$ vector $x \sim N_p(\boldsymbol 0, \boldsymbol \Sigma_1)$. Now, There is another covariance matrix $\boldsymbol \Sigma_2$. We know that $|\boldsymbol \Sigma_2| < |\boldsymbol ...
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2answers
15 views

If 4 balls are drawn without replacement, what is the probability that at least 3 black balls are drawn? [on hold]

There are 9 black balls and 10 red balls in an urn. If 4 balls are drawn without replacement, what is the probability that at least 3 black balls are drawn?
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14 views

Lack of memory of a geometric distribution, proving a general case.

I have to prove this for a general value so $P(X > j+k | X>j) = P(X > k)$ Using the conditional probability I get that $P(X > j+k | X>j) = \dfrac{P(X > j+k) \wedge P(X > ...
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0answers
11 views

Statistic z scores [on hold]

. Transport Canada was investigating accident records to find out how far from their residence people were 2 when they got into a traffic accident. They took the population of accident records from ...
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3answers
15 views

Standard deviation…

I have this random variable $X = \{-1, 0, 1\}$ with uniform repartition $p(X = -1) = p(X = 0) = p(X = 1) = \frac{1}{3}$. Expected value is $$E[X] = \sum_{i\in\{-1,0,1\}} x_ip_i = 0$$ Then variance ...
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2answers
18 views

simple probability with marbles - requery

There is a post about how to calculate probability with marbles. I doubt the answer and i am asking for a more detailed explanation if possible. Picking marbles without replacement and without ...
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0answers
32 views

Probably, expected eatings on a roulette wheel

The probability that a roulette wheel stops on a red number is $\frac{18}{37}$ For each bet on “red” you are returned twice your bet (including your bet) if the wheel stops on a red number, and lose ...
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1answer
22 views

Solving for $\lambda$ in an exponential distribution given an average

Studying for a mid-term, and not sure how to go about the following problem. Given $t = 700$ as an average, I have to solve for lambda. I'm thinking since t is determined, I don't need any integrals ...
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0answers
8 views

For the general linear model, what is the distribution of $C\hat{B}$? Note $C$ is $q \times (k+1)$.

For the general linear model, what is the distribution of $C\hat{B}$? note $C$ is $q \times (k+1)$. So as far as I know, the general linear model is $\hat{Y}=X\hat{B}+e$. I don't understand why the ...
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0answers
35 views

Probability with expected value for diagnostic tests

Two percent of the population has a certain condition for which there are two diagnostic tests. Test A, which costs $1 per person, gives positive results for 80% of persons with the condition and ...
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1answer
10 views

Probability: How to find what proportion is between the 2 values

Assume that head sizes (circumference) of new recruits in the Canadian armed forces can be approximated by a normal distribution with a mean of 22.8 inches and a standard deviation of 1.1 inches. ...
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14 views

Please help. Confidence Intervals for variance and standard deviations

Find the 99% confidence interval for the variance of the number of hours spent using the internet per week if a sample of 37 survey respondents has a standard deviation of 4.3 hours per week. Could ...
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1answer
35 views

Probability - Diagnostic Tests, expected cost per person

Assume that for a randomly selected person: $P (D) = 0.02$, $P (R\mid D) = 1,$ $P (R\mid D') = 0.05$ So that the inexpensive test only gives false positive, and not false negative, results. ...
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2answers
37 views

Expected Value and Variance - Finding expected winnings

A game is played where a fair coin is tossed until the first tail occurs. The probability $x$ tosses will be needed is: $$f(x)=(0.5)^x;x=1,2,3,\ldots$$ You win $2^x$ dollars if $x$ tosses are ...
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0answers
11 views

Minitab help/ getting standard deviation from Poisson with mean

Suppose that a random variable $X$ has a Poisson distribution with mean $µ$ = 25. Generate 1,000 observations from the distribution of X and obtain the sample standard deviation I have no idea how ...
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0answers
18 views

Poisson Distribution word problems

During rush hour the number of cars passing through a particular intersection23 has a Poisson distribution with an average of 540 per hour. (a) Find the probability there are 11 cars in a 30 second ...
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2answers
23 views

Paternity probability calculator based on blood group and eye color

I am currently writing a paternity probability calculator. I am struggling with finding the correct statistical approach to determining probability based on blood type and on eye colour. For example, ...
2
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1answer
16 views

Normal distributions sums

I read this property about normal distribution If $X\sim\mathcal N(\mu_X,\sigma_X^2)$ and $Y\sim\mathcal N(\mu_Y,\sigma_Y^2)$ are independent, then $$ X+Y\sim\mathcal ...
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1answer
12 views

Mean of a sampling distribution.

Suppose $\hat{p}=1/\overline{X}$ is an estimator of the parameter $p$ of a population variable $X\sim\text{Geo}(p)$. Suppose $p=0.36$ and $n=25$. What is the mean of the sampling distribution? This ...
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1answer
33 views

Probability - Airplane overselling tickets

Few days ago, I came across a question for probability in one of the interview. Question : The same small commuter plane has 30 seats. The probability that any particular passenger will not ...
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1answer
16 views

Bivariate Continuous Distributions

What is the marginal density of $X$ and $Y$ given the probability density function, ${f(x,y)= \lbrace3x ,\;\;0\le y\le x\le1}$
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1answer
39 views

Exam grades and bell curve

What is the mathematical explanation for the tendency of exam grades to conform to a bell curve? Initially, I was thinking it should be explained via the central limit theorem, but it's not clear to ...
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1answer
18 views

Poison distribution variance,probability. and mean.

Let $X$ be the poisson random variable such that $P(X = 2) = 9P(X=4) + 90P(X=6)$ a) find the mean and variance of $X$. b) find P(X $\geq 1$) c) find P(X $\leq 10$) Ok so for the first question I ...
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0answers
18 views

Lottery Ticket Probability [on hold]

At a certain retailer, purchases of lottery tickets in the final 10 minutes of sale before a draw follow a Poisson distribution with mean = 15 if the top prize is less than 10,000,000 and follow a ...
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0answers
27 views

CI for the expected value of the sum of two dependent normal RVs

Let's consider 2 dependent, normally distributed R.V.s, $X_1$ and $X_2$. The means, $\mu_1$ and $\mu_2$ are known, as well the covariance matrix $\Sigma$. Let's consider the following random ...
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0answers
14 views

Question about regression model

Suppose you fit (estimate the parameters of) a regression model, obtaining $\hat{Y}$, $\hat{B}$, and $\hat{E}$. And you fit a second regression model , using $\hat{Y}$ x matrix from previous model ...
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1answer
21 views

Showing a group of observation is standard normally distributed

Let $X_1,X_2\dots$ be a sequence of independent RVs such that $X_{n}$ is binomial with parameters $2n - 1$ and $1/2$. Define $$Y_{n}=\frac{2(X_{1}+X_{2}+\cdots+X_{n})}{n} -n$$ Show $P[Y_{n}<t]\to ...
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0answers
10 views

Random Sample taken [on hold]

A random sample of 300 people are taken. What is the probability that at least 100 of them are over 180cm in height given average height = 175 and standard deviation = 10?
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1answer
25 views

how to show for a simple regression with an intercept and one independent variable$ R^2 = r ^2$ , where $r$ is the ordinary correlation coefficient.

how to show for a simple regression with an intercept and one independent variable $R^2 = r ^2$, where $r$ is the ordinary correlation coefficient. Here is where I'm at. $R^2= ...
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0answers
18 views

Clarification on proposition in paper

I'm referring to proposition 1, page 309 of [1] The proposition itself reads: Let $\pmb x_1,\ldots,\pmb x_n,\ldots$ be iid with $\pmb x_i=\pmb B\pmb u_i+\pmb t_0$ where $\pmb u_i$ has ...
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2answers
32 views

Can 2 different random variables have the same CDF?

I'm looking for proof that two different random variables can have the same Cumulative Distribution Function; in other words, I'd like to disprove that a CDF uniquely defines a random variable. ...
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2answers
24 views

Co-relation Coefficient

$X$ and $Y$ are jointly continuous random variables. Their probability density function is: $$f(x,y) = \begin{cases}2x & \mbox{if } x\in [0,1], y\in[0,1] \\ 0 & \mbox{ otherwise ...
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1answer
19 views

Finding the probability of a randomly selected event?

I know I'm over-thinking the following question, I just need to know how to start! In a certain population of women 4% develop symptoms of a classic disease, 20% are smokers, and 3% are smokers and ...
1
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1answer
27 views

Joint probability density function probability

$X$ and $Y$ are jointly continuous random variables. $$f(x,y)=\begin{cases}kx & x\in[0,1], y\in [0,1]\\0 & :\text{otherwise}\end{cases}$$ a) What value of $k$ makes this a density ...
2
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1answer
23 views

Expectation of vector valued functions

Let $t_1,\ldots,t_m$ be $m$ random variables that are independently and identically drawn from a Bernoulli distribution with a constant parameter $p$. Now, we define some functions of ...
2
votes
1answer
34 views

Find the value of k which makes f a density function.

Observe the following probability density function for a continuous random variable X $$f (x) = \begin{cases} k\sqrt x (1-x) &\text{ for }x\in(0,1)\\ 0 &\text{ otherwise} \end{cases} $$ Find ...
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0answers
13 views

Bayes rule with discrete prior

Assume following discrete prior on $m$ where $m \in \left\{2, 3, 4, 5\right\}$ and $p(m)= .14, .13, .2, .32$ accordingly. If $f(x|m)= \exp[(n/2\sigma^2)(x-m)^2]$. What is the posterior values for ...
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0answers
11 views

Beyes coin problem

assume coin with probability p. Probability is unknown but there are possible values of {.01, .02, .03} with probabilities {.26, .05, .03} 1) if there are observed 12 heads and 13 tails, with 25 ...
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1answer
31 views

Drawing Probability Density Function

Can someone help me to draw this pdf? I really don't have an idea how to convert a function to pdf. Thank you p(x | c) = 1/3 for 1 <= x <= 4 and P(c) = 0.5
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0answers
38 views

Probabilty Models and distribution techniques

Coliform bacteria are distributed randomly and uniformly throughout river water at the average concentration of one per twenty cubic centimeters of water. Part (c) In testing for the concentration ...
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0answers
10 views

How can I minimise the MSE in binomial distribution

$X_1$-Bin(n,p) and $X_2$-Bin(n-$x_1$,p). n is the unknown total population. I am given that $T_b$ is the estimator of n where $T_b$=a$X_1$+b$X_2$. Also, the ...
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0answers
17 views

The Hessian Matrix I calculate is twice as much as it should be. Why?

I have a function "fkt." In this example, let it be as simple as $y=a \cdot x+b$. I have a real dataset with values obeying to the model. After regression of the points to the model, I find the ...
1
vote
1answer
21 views

Apples Binomial [on hold]

"Apples are packaged automatically in 3-pound bags. Suppose that 4% of the time the bag of apples weighs less than 3 pounds. If you select bags randomly and weigh them in order to discover one ...
1
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1answer
29 views

When to use Central Limit Theorem or Cramers Theorem

In for example this paper the authors say The central limit theorem provides an estimate of the probability \begin{align} P\left( \frac{\sum_{i=1}^n X_i - n\mu}{\sigma \sqrt{n}} > x \right) ...
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0answers
21 views

Questions dealing with Poisson distribution technique

"A waste disposal company averages 6.5 spills of toxic waste per month. Assume spills occur randomly at a uniform rate, and independently of each other, with a negligible chance of 2 or more occurring ...
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0answers
33 views

Joint multivariate normal distribution

If $\mathbf{y}$ has a multivariate normal distribution $N_n(\mathbf{a},\sigma^2\mathbf{I})$, can I say that $(\mathbf{y},\mathbf{a}^\intercal\mathbf{y})$ also has a multivariate normal distribution? ...
1
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1answer
25 views

What is Neyman-Pearson lemma? Why is this proof of Neyman-Pearson's lemma look so diffcult?

What is Neyman-Pearson lemma? Why is this proof of Neyman-Pearson's lemma look so diffcult? I am consider taking a undergraduate course in my college called mathematics of statistics and in the ...
3
votes
3answers
36 views

What is the correct statistical language to conclude using type II error?

Update: This question arises when I read a neuroscience paper. In the natural science community, people are generally less careful about the correctness of statistical language. I am clearly aware of ...
1
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1answer
29 views

50% plus or minus 13% is good or bad guessing. How was this reached?

I watched a documentary some years back and I cant find it anymore. It was either about genetics or twins (specifically identical twins). But I recall a test was done to see statistically if identical ...
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votes
3answers
29 views

Suppose U is a uniform random variable over [0, 1] [on hold]

This is one of two HW problems that I'm positing that I have no clue how to go about. Suppose U is a uniform random variable over [0, 1]. Show that $Y = (b - a)U + a$ is uniform over [a, b] Use ...