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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Narrowing down a location on a grid based on multiple data points

I'm working on a program for triangulating wireless device locations on a map. So far I've cooked up the triangulation algorithm, but the problem is that wireless signals can bounce around depending ...
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18 views

Some Cumulative Distribution and Density Function Question.

Suppose that x is normally distributed N(5.2,0.64), If Y=e^X; a.find the cumulative distribution function for Y. b.find the probability density function for Y. c.calculate the probability P(200 ...
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1answer
11 views

What is $\text{cov}$$(\hat{y_i}, \hat{y_i}^*)$?

Using the simple linear regression model: $ {y_i}= {\beta_0} + {\beta_1}x_i + \epsilon_i$, where E[$\epsilon_i$]=0 and var[$\epsilon_i] = \sigma^2$... If $ \hat{y_i}= \hat{\beta_0} + ...
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3answers
18 views

Finding Survival Function given hazard rate

Let X be a random variable defined for 0 < x < 4 with hazard rate $$lambda(t)=1/(4-t)$$ for 0 <= t <= 4. find the survival function, S(x) = P(X>x). Using the formula S(x) e^-integral from ...
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What is ${\rm cov}(e_i, \hat y_i)$ in simple linear regression?

The model is $y_i = \beta_0 + \beta_1x_i + \epsilon_i$ What is ${\rm cov}(e_i, \hat y_i)$? What is ${\rm cov}(\epsilon_i, \hat \beta_1)$? What is ${\rm cov}(e_i, \epsilon_i)$? For 1, I am writing ...
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23 views

what is the solution of mathematical statistics and data analysis chapter 7 problem 26? [on hold]

I think I got most of questions, but I did not get part e and f.
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1answer
14 views

Calcuate the test statistic

I am facing a little problem is this question. Can somebody please help e here A sample of 500 drivers was asked whether or not they speed while driving. The following table gives a two-way ...
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18 views

Uniform distribution parameter estimation

How do I estimate the parameter, when the distribution is $\mathrm{U}(-a,a)$? I get $a=\min(X_i)$, $b=\max(X_i)$, when it's $\mathrm{U}(a,b)$, but these do not work here.
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24 views

The critical value

I am facing a little problem is this question. Can somebody please help e here A sample of 500 drivers was asked whether or not they speed while driving. The following table gives a two-way ...
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1answer
25 views

Probability math problem. [on hold]

Certain data obtained from a study of a group of 1000 subscribers to a certain magazine relating to their sex, marital status, and education were reported as follows: 312 males, 470 married, 525 ...
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3answers
35 views

The Objectivity of Statistical Testing

I have a very generic question about applied statistics. Suppose, to make things simple, we have a biased coin with probability $p$ of landing heads. We want to determine if our coin is truly fair - ...
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1answer
23 views

Calculating the critical value

I am facing a little problem is this question. Can somebody please help e here A sample of 500 drivers was asked whether or not they speed while driving. The following table gives a two-way ...
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1answer
31 views

Calculate Critical Value

Experience in investigating insurance claims shows that the average cost to process a claim is approximately normally distributed with a mean of 80 dollars. New cost-cutting measures were started and ...
2
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1answer
18 views

Confidence interval multiplication

The question looks pretty simple but I can't get my hands on it: Say I have a probability which is the product of two other independent probabilities $p = p_1p_2$. I have estimated each probability ...
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1answer
15 views

Hypothesis testing on minimum of exponentially distributed random variables

I am completely stuck with the following problem, because I do not know how to start: Let $X_1,...,X_n$ be independent and exponentially distributed with unknown parameter , and let ...
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0answers
27 views

Help with finding $E((\Phi(aX+b)^2))$ when $X$ is standard normal

I need help with calculating $$ E((\Phi(aX+b))^2)=\int_{-\infty}^\infty (\Phi(ax+b))^2\phi(x)dx $$ where $X$ is a standard normal random variable and $a$ and $b$ are constants. $\Phi$ and $\phi$ ...
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1answer
29 views

Source needed: Does asymptotic normality yield asymptotic unbiasedness and consistency?

Assume that $$\sqrt{n}(\hat g - g(\theta)) \xrightarrow{d} Z, $$ where $Z$ is $N(0,\sigma^2)$. Does this already imply asymptotic unbiasedness and/or consistency, i.e., $$ E[\hat g] \rightarrow ...
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1answer
20 views

$MLE$ of $\theta$ when $X_1 , X_2 , …, X_n$ is a sample with pdf $f(x, \theta) = e^{\theta - x}; x \ge \theta$

How can we find the $MLE$ of $\theta$ when $X_1 , X_2 , ..., X_n$ is a sample with pdf $f(x, \theta) = e^{\theta - x}; x \ge \theta$ ? $L(\theta) = \prod_{i = 1}^{n} e^{\theta - x_{i}}$ $L(\theta) = ...
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1answer
13 views

Paired t test, finding mean and variances

Paired t-test: Let x1, x2,...,xn be iid ∼ N(mu1, sigma^2). Let y1, y2,...,yn be iid ∼ N(mu2, sigma^2). Suppose also that the pair of observations within subject i, (xi,yi), has correlation rho. This ...
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0answers
13 views

Type I Error Statistics [on hold]

Consider a random variable $X$, which is distributed as follows: $$f(x)=\begin{cases}\frac{1}{k}& 0<x<k\\0& \text{otherwise}\end{cases}$$ We observe only one observation $X=x$ from this ...
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1answer
20 views

How to know when to use which test statistic?

I'm studying for my stats exam and I seem to always get stuck on which test statistic to use. This is even after many examples that I have done. For example: With this question: I attempted to ...
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10 views

How large a sample will need to be taken to meet the administrator's needs?

A hospital administrator wants to measure average time per hip replacement procedure performed in the institution's operating suite. He is willing to be within 10 minutes of the true value with a ...
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13 views

Find unbiased estimate for mean

These data are obtained: 42.65 45.15 39.32 44.44 41.63 41.54 41.59 45.68 46.50 41.35 44.37 40.27 43.87 43.79 43.28 40.70 The sample is normally distributed with mean "mu" standard deviation 2. The ...
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6 views

How to get F-test p-value

Suppose we can choose from two differents catalysers. 10 substances are taken from the first one and 12 from the other one. $S1 = 0:14$ and $S2 = 0:28$, can we reject at $\alpha = 5%$ the hypothesis ...
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1answer
21 views

If I have a random variable X with a given Probability Density Function, How do I find the PDF of the area of a circle with radius X?

To find the PDF of the area of the circle, do I just substitute the PDF of the random variable X in for the radius in the circle area equation?
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16 views

Expectations of sum of indicator rvs

Question: Suppose that 2 tomatoes, 2 apples, and 2 oranges are randomly divided into three lunchboxes, with each lunchbox having 2 fruits, with all possible ways of forming the pairs of fruit being ...
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1answer
17 views

Calculating confidence interval - formula

I have the following problem that I get the feeling I'm mixing formulas. ...
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1answer
13 views

Please explain uniform distribution to me [duplicate]

I am confused about Uniform Distribution why does $$P(v < 2b1)$$ equal 2b1 ?
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2answers
29 views

What's the densitiy of the product of two independent Gaussian random variables?

Suppose that $X,Y$ are two scalar independent normal random variables, $X \sim N(\mu_X,\sigma_X^2)$, $Y \sim N(\mu_Y,\sigma_Y^2)$. I'm particularly interested about the case where we don't assume ...
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1answer
31 views

Game Theory and Uniform Distribution question?

In an Auction , two players are bidding. Their bids will be a unknown fraction of their valuations. The valuations come from a uniform distribution $$[0,1] $$ If Player 2 bids $$ v/2 $$ and Player ...
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1answer
9 views

class conditionals and priors

Let's say we have an event A that indicates whether an image contains a person or not. A = 1 indicates that the the image contains a person and z = 0 means that it does not. Assume that L which can ...
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17 views

Paired T-test Question

My study group is having a hard time coming to a consensus over how we would solve this problem because we've never dealt with paired data without a mean. We have a paired t-test with two ~N(mu, ...
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Addition of two box plots

I have two box plots that I'd like to add together to form one box plot representing both plots in one. Here's an example of two box plots: At a school 200 boys and 200 girls participated in a test. ...
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41 views

Hironaka 1964 theorem in the context of S. Watanabe 2009 book

I am trying to read the following book of S. Watanabe: "Algebraic Geometry and Statistical Learning Theory". More particularly, I am currently interested in chapter 2 and Hironaka (1964) theorem on ...
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23 views

Averages; mean, median, modes and the range [on hold]

If two sets of data have the same range and median but the mean in one set of data is higher than the other mean, in the other set of data. What can you say about the two sets of data?
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1answer
65 views

Card probability

There are two 10-card decks, consisting of 5 red cards and 5 blue cards each. Both are shuffled separately. One card is then dealt from each deck and compared. This is repeated for all 10 pairs of ...
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27 views

How to find the MLE of the mean of Gamma distribution

If I parameterize Gamma distribution in the way as $\Gamma(\alpha,\frac{\mu}{\alpha})$, am I able to find the maximum likelihood estimator of $\mu$. Here, $\alpha$ is the shape parameter, ...
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0answers
7 views

is UMP test a unbiased test?

as shown in the title, is that statement true? Since UMP is defined in $(H)^c$ space, and unbiased test is the comparison between $(H)^c$ and $(H)$ space, How can we decide if this is true or not?
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2answers
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Relationship between Fisher information conditional on MLE and Fisher Information conditional on True Value

As described in most textbook, the formal definition of Fisher Information is a function conditional on true value of parameter $\theta$, i.e. $I(\theta)$. Today I came across this phrase: ...
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1answer
18 views

Alternative Hypothesis

I am facing a little problem is this question. Can somebody please help e here A sample of 500 drivers was asked whether or not they speed while driving. The following table gives a two-way ...
1
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1answer
10 views

Self-Selection Bias Intrinsic to Survey Samples?

I'm new to stats so bear with me in asking this question. I'm sure my novice will shine through. I've noticed that with any survey there is an intrinsic opportunity for self-selection bias (There is ...
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1answer
34 views

Null Hypothesis

Experience in investigating insurance claims shows that the average cost to process a claim is approximately normally distributed with a mean of 80 dollars. New cost-cutting measures were started and ...
1
vote
1answer
11 views

is UMVUE unique? is the best unbiased estimator unique?

guys Here is the question: is the best unbiased estimator unique? My understanding is that the best unbiased estimator must be the UMVUE, so the original question turns into the uniqueness of UMVUE. ...
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1answer
29 views

Normal Ratio Distribution with CDF Method

I think I'm missing something glaringly obvious here that's causing problems for me in the entire subject. I have two independent standard normal random variables, X and Y ~N(0,1), and I need to find ...
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1answer
34 views

Lickly hood estimators and discrete random var

If have discrete random variable X and have the observations that X=1, twice, X=3, once and X=4, six times is the likely hood of these observations, L = P(X=1)^2 . P(X=3)^1 . P(X=4)^6 or L = ...
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1answer
19 views

Approximating the expectation of a function with sampling

I'm reading a paper (section 5.1) that approximated the expected value of a function $f(X,Y)$ of two random variables using Gibbs sampling. As far as I know the expectation of $f(X,Y)$ is defined to ...
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1answer
6 views

Variences and adding them from independent random variables?

If I have 3 random varibles X, Y and Z and X=Y+Z then var(X)=var(Y)+var(Z), but Y=X-Z therefore var(Y)=var(X)+var(Z), it is clear that these two contridict, so what makes one of them right and the ...
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14 views

Proving a statistic is sufficient and NOT complete

Let X be a single observation from the distribution , $P(X=x)= \begin{cases} \theta, & \text{if $x$ =-1} \\ (1-\theta)^2\theta^x, & \text{if $x =0,1,2,3,...; 0<\theta<1$} \\ ...
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Unbiased estimator exponential distribution [on hold]

Is the reciprocal of the sample mean an unbiased estimator of the exponential distribution parameter? How can we get it unbiased? Hint: use out, that the sum (convolution) of independent exponential ...