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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2
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0answers
7 views

Risk in density estimation: grasping the definition

When generalizing estimators to an entire function what is the space in which we perform the integral to obtain the expected value (with respect to this function)? For example, when estimating ...
0
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1answer
14 views

intuition behind convolution of sum of two densities

I'm trying to understand the intuition behind convolutions for densities Suppose X,Y are RVs and Z = X + Y then $f_Z(z) = \int_\infty^\infty f(z-y,y)dy$ The lower bound should be negative infinity ...
0
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0answers
11 views

convergence for dependent random variables

Let $X_1,X_2,\ldots,X_n$ be a sample of iid rvs. Suppose their distribution has a parameter $0<\theta<\infty$ and support $(0,\infty)$. Let $\theta_n=\theta_n(X_1,\ldots,X_n)$ be an estimator of ...
0
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1answer
14 views

What is the joint distribution of sample mean and sample variance of normal distribution?

${X_i} \sim N\left( {\mu ,{\sigma ^2}} \right)$, define $\overline X = \sum\limits_{i = 1}^n {{X_i}} $, ${S^2} = \dfrac{1}{{n - 1}}\sum\limits_{n = 1}^n {{{\left( {{X_i} - \overline X} \right)}^2}}$. ...
0
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0answers
18 views

Meaningful statistic measure of data pairs

I have a dilemma. I have pairwise data, (a,b), that represents some form of speed, whether it's miles/hour or megabits/second. Let's say that we have the following set of data from measuring the ...
0
votes
1answer
4 views

Distribution under null-hypothesis and type 1 error

Given random variables $X_1,...,X_n \overset{i.i.d.}{\sim} N(\mu, \sigma^2)$ where the variance $\sigma^2$ is known let the null hypothesis be $H_0: \mu = \mu_0$ For the statistic $T=\sum_{i=0}^nX_i$ ...
1
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1answer
13 views

Discrete Bivariate Distributions, find constant $c$

I am given $f(x,y) = c(x + y)$, and I have to find constant $c$ such that $f(x,y)$ satisfies the conditions of being a joint pmf for two discrete random variables $X$ and $Y$. $x = 1$, 2, 3, and $y ...
0
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0answers
7 views

Mix of two bivariate distributions (two correlations hidden in data)

We have two sample vectors $X$ and $Y$ which are realizations (observations) of metric (continuous) random variables, and are interested in a sample correlation between them. Actually, a correlation ...
-1
votes
0answers
13 views

for one study , research sampled over 100,000 first-time [on hold]

I have already answered the questions but I just need to make sure if my answers are correct or no thanks in advance :)
1
vote
2answers
18 views

Finding simplest function to distinguish 2 sets

I wish to find a function that distinguishes $2$ sets. I have m data values in form of n-tuples out of which k are supposed to be mapped to a value less than $0$ and other m-k are supposed to be ...
1
vote
1answer
26 views

How do I find if this estimator is unbiased and also its variance?

I need to find if the estimator $\tilde{\beta } _{2} = \frac{(y_{n}-y_{1})}{(x_{n}-x_{1})} $is unbiased given that i) $E(u_{i}\mid x)=0$ ii) $E(u_{i}\mid x_{i})=0$? I also need to calculate its ...
-1
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0answers
7 views

What's std Formulae for Adding Tax in base price then deducting same percent from gross to get 100% base price?

So, assuming you folks have read question, here is the case; **If my selling price is 750 for one item and I need to add 5.5 % TAX on it which will make it: 791.25 but my buyer has to deduct the ...
0
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0answers
17 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 ...
-3
votes
2answers
19 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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0answers
20 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 > ...
-1
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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 ...
1
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3answers
17 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 ...
0
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2answers
22 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 ...
0
votes
1answer
56 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 ...
0
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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 ...
0
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0answers
9 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 ...
0
votes
2answers
52 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 ...
1
vote
1answer
12 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. ...
0
votes
0answers
15 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 ...
0
votes
1answer
40 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. ...
1
vote
2answers
40 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 ...
0
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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 ...
0
votes
0answers
23 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 ...
3
votes
2answers
27 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
votes
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 ...
0
votes
1answer
15 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 ...
0
votes
1answer
36 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 ...
1
vote
1answer
18 views

Bivariate Continuous Distributions

What is the marginal density of $X$ and $Y$ given the probability density function, ${f(x,y)= \lbrace3x ,\;\;0\le y\le x\le1}$
0
votes
1answer
41 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 ...
1
vote
1answer
24 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 ...
-1
votes
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 ...
2
votes
1answer
30 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 ...
0
votes
0answers
15 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 ...
1
vote
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 ...
-2
votes
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?
0
votes
1answer
26 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= ...
0
votes
0answers
19 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 ...
1
vote
2answers
36 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. ...
0
votes
2answers
25 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 ...
0
votes
1answer
20 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
vote
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
votes
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 ...
0
votes
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 ...
0
votes
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 ...