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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1answer
42 views

Subgame perfect Nash equilibrium & perfect Bayesian Nash equilibrium - Game theory

For a week or so I have been struggling with the topics around the concept of subgame perfect Nash equilibrium (SPNE) and the perfect Bayesian Nash equilibrium (BNE). Namely: Is it possible to apply ...
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2answers
43 views

Find the mean, variance and $f(21)$ of this if it is a p.g.f.

$$G(t) = t^{11}e^{9t-9}$$ Find the mean, variance and $f(21)$ Please help me with find $f(21)$ as easily as possible. For the mean and variance: $$G'(t) = 11t^{10}e^{9t-9}+9t^{11}e^{9t-9}$$ ...
2
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1answer
55 views

Probability of Bit Errors Poisson Question

I'm not quite sure how to get the correct probability for this question. Q: The probability of error in the transmission of a binary digit over a communication channel is 1/10^3. Write an expression ...
0
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1answer
42 views

Expected Value / Variance Problem - X is equally likely to be four values

Q: Suppose that $X$ is equally likely to take on any of the values 1, 2, 3, 4. Compute a) $E[X]$ and b) $\mathrm{var}(X)$ For a) $E[X] = \int_{-\infty}^\infty x ~ f(x) ~ \mathrm dx$, but I'm not ...
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1answer
16 views

Understanding continuity of probabilities

I am reading a book called All of Statistics by Larry Wasserman that includes a theorem called "continuity of probabilities." It says that if $A_{n} \rightarrow A$ then $P(A_{n}) \rightarrow P(A)$ as ...
2
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1answer
18 views

How to determine the expectation of the square of a binomial collection

I'm trying to find how to express the expectation of the square of a collection of binomial measurements. If we have a collection: $$A = a_1 + a_2 + \cdots + a_n$$ The expectation of $A$ is the sum ...
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0answers
21 views

Needle game with 3 payouts and fair game. Have two answers, but not sure which one.

The question is below, followed by two answers. Both look like reasonable ones but I am not sure which one is right. Answer 1: Answer 2:
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0answers
19 views

Why are the eigenvalues of $I_R + \beta' \alpha$ bounded by one?

Trying to understand the Granger-Johansen Representation Thm (see p. 7 here; we are assuming Condition 5). We have $(p \times r)$ matrices $\alpha, \beta$. We know that $|eig(a \beta')| \leq 1$ and ...
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0answers
11 views

How to choose assymetry for KL divergence?

I have two 2D probability distributions of eye movements of two different images. Suppose I call the first distribution of Image 1: $P$, and the second distribution of image 2: $Q$. Since ...
2
votes
2answers
31 views

Deriving the variance of a binomial distribution

I know that the variance of a binomial distribution is the number of trials multiplied by the variance of each trial, but I'm not seeing the derivation of this. Here's my logic so far: For each trial ...
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0answers
10 views

Minimum Variance Unbiased Estimator question

Let x_1, … , x_m be independent Poisson (λ_x) and independently Y_1,…,Y_n be Poisson (λ_y). Find the MLEs of λ_x-λ_y and λ_x+λ_y. Are these estimators MVUEs? I'm having a very hard time understanding ...
2
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0answers
19 views

uncertainty of slope.

i have a graph that i fitted a line on it using least squares fit. Now i want to calculate the uncertainty of slope. i calculated the standard error of slope and now i have this question: the ...
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votes
1answer
42 views

Describe steps omitted in this central limit theorem problem. [on hold]

I'm having a hard time seeing the steps they skipped in this problem. Question gives mean of 75. How many students would have to take the examination to ensure with probability at least .9 that ...
0
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1answer
17 views

Signals send over a channel

Signals are being send over a channel. A signal is either a 0 or a 1. An error can occur during transmission, and a 1 could be received while a 0 was sent, and a 0 could be received while a 1 was ...
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1answer
48 views

What $\mathsf {Cov}(\tilde X, \hat X)$ is in this question? [closed]

$\{X_1, X_2, \ldots, X_n\}$ for n>30, is a random sample from probability distribution $X$ with mean $100$ and variance $180$.   Let $\tilde X=\tfrac 1{10}\sum_{i=11}^{20}X_i$ and $\hat X ...
0
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1answer
33 views

Expected value problem… Infinite series with factorial

In an exercise I have to calculate the following expected value: $$ \sum_{i=4}^\infty500\frac{e^{-4}4^i}{i!} $$ Leaving $500e^{-4}$ outside and using D'Alambert's criterion (with $a_n = ...
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2answers
15 views

How is this bound being computed in a chebyshev probability question

Problem: X has mean and variance of 20. What can be said about $P(0<X<40)$? Chebyshev formula = $P(|X-\mu|\geq k)\leq\frac{\sigma^2}{k^2}$ The first step has $P(|X-20|\geq ...
0
votes
1answer
21 views

Expectation vs Variance in Economics

When deciding between two projects available, where E[ReturnB] > E[ReturnA] but Var[B] > Var[A], why would this not be a clear case and rather look at the principle of maximization of expected return? ...
2
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3answers
18 views

Are there any limits on Standard Deviation of a data set with given $n$ and mean?

Say a class of 200 students is graded out of 100 marks. The mean of the dataset is 50. Can we put a maximum limit on Standard Deviation for the set ? I thought of putting a number of people onto 100 ...
4
votes
1answer
39 views

Coin Flips and Hypothesis Tests

Here's a problem I thought of that I don't know how to approach: You have a fair coin that you keep on flipping. After every flip, you perform a hypothesis test based on all coin flips thus far, with ...
1
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1answer
23 views

Intuition behind direction of maximum variance?

I'm trying to understand the phrase "direction of maximum variance" which keeps popping up in the context of PCA. For example, in this set of 2D points, it is clear they approximately lie on a line. ...
0
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0answers
18 views

iid sequence of random vectors

If $W_1,...,W_N$ is an iid sequence of random vectors, with $W_i=(X_i,Y_i)^T$, does $W_1,...,W_N$ being an iid sequence imply that $X_i$ will be independent of $Y_i$? Does it imply that $(X_i,Y_i)$ ...
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0answers
20 views

Normal random variable probability

Let X ~ Normal(50,19) and Y ~ Normal(20,9), a assume X and Y are independent. Find the probability X <= 3Y. Hint: -3Y ~ Normal(-60,81) I'm confused about how to do this. Any ideas?
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0answers
12 views

How to test if a tree fit in existed hypothesis?

Saying I have some data, and I build a tree based on the data. Now I want to test if this tree fit in my predefined hypothesis statistically. How can I do it? For example, the null hypothesis is the ...
0
votes
1answer
29 views

Probability of a full house if hand contains at least 2 aces?

What is the probability of having a full house if the hand contains at least two aces? I know this is a conditional probability. I'm not sure how to approach it.
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2answers
29 views

Deriving linear transformation of a probability expectation?

In looking at linear transformations of the expectation of a random varaible $E(X)$, the following is given, but I'm not seeing how it is derived: The linear transform: $Y=aX+b$ The result on ...
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0answers
25 views

Upper bound of coefficient of variation computed with mean average deviation

From $n$ data $x_i$ in $[0,1]$, I am computing a coefficient of variation $C=d/\mu$ where $d$ is the mean average deviation $d=1/n \sum_{i=1}^n |x_i-\mu|$ and $\mu$ is the standard mean $\mu=1/n ...
0
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0answers
16 views

What does it mean when the mean of a sample close to the standard deviation?

I have did a MCMC and get a sample for parameter A in a model. Then the mean of the sample is close to the standard deviation, for example, mean $\mu = 0.01$, standard deviation std = 0.01, then what ...
0
votes
1answer
16 views

Simple Combination Problem with More Spaces to Fill than Items to Place

The problem is to seat 9 people into 3 cars of which car A has 2 seats, car B has 4 seats, and car C has 5 seats. All cars must be driven. I have reasoned that at least one person must be in every ...
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0answers
23 views

Probability of relations in a network

Imagine, i have a predicate $\text{friends}(x_1, x_2)$ and I know that $p(\text{friends}(x_1, x_2)) = p_2$. If I generate a world of $n$ people ($x_1$ to $x_n$), I expect there to be $\binom{n}{2}p_2$ ...
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0answers
9 views

Statistical proof of an inegality

I have a random variable $X : \Omega \rightarrow R \cap \left[ 0,2 \right]$ (it is a measure of the output of an algorithm given a random input). I presume that the actual interval is smaller, let us ...
2
votes
2answers
33 views

is there a concept of asymptotically independent random variables variables?

To prove some results using a standard theorem I need my random variables to be i.i.d. However, my random variables are discrete uniforms emerging from a rank statistics, i.e. not independent: for ...
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0answers
25 views

Probability question, find te expected number we will win

A deck of 2n cards consists of n red and n black cards. The cards are shuffled and then turned over one at a time. Suppose that each time a red card is turned over, we win 1 unit if more red cards ...
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0answers
26 views

Laplace transform for Bernoulli with exponential distribution

If we have, say a power $P_{ri}$, which is received power from node-$i$, distanced-$r$ and distributed exponentially with mean $r_i^{-\alpha}$, and the PDF of that power is equal to ...
0
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1answer
14 views

(Random Sampling + Approximate Probability + Normal Approximation)

Suppose that 16% of the population of the U.S. is left-handed. If a random sample of 130 people from the U.S. is chosen, approximate the probability that at least 20 are left-handed. Use the normal ...
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1answer
28 views

how to check data fits gamma distribution in MATLAB ?

I have two vectors which also contain zeroes. I cannot remove the zeroes as that is also data. I have to see whether my data fits gamma distribution or not. How can I check that?
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0answers
12 views

Calculating Standard Deviation of a Function of Multiple Variables

I am doing an error analysis for my Chemistry project. It involves ethanol. I defined a term called ETA (Ethanol Titer Advantage) to be the difference of ethanol concentration of wild type and that ...
0
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0answers
17 views

Can Poisson distribution be used in my example?

Lets say that I have an infinite population of individuals of finite density. The aim of the individuals is to find shelter. Density of individuals is $x$ and density of shelters is $y$. In a given ...
0
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1answer
68 views

Correctness of a statistical evaluation of a parameter

I have a question about a non-Gaussian distributed parameter that can only take certain values in a defined interval. Knowing that I have to define this parameter starting from a set of its values and ...
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2answers
35 views

I'm confused with one conditional probability problem

everyone! I'm confused with the definition of conditional probability. Assume we have one continuous random variable X and one discrete variable Y. I think $P(X>x|Y=y) = \int_{t=x}^\infty ...
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0answers
14 views

Correlation/Regression for Continuous and Discrete data

I want to correlate a data where one axis is continuous (ranging from 0 to 1), other axis is discrete. Discrete axis scale is 1 to 5 (1 is for Strongly Disagree and 5 is for Strongly agree). How ...
0
votes
1answer
35 views

Probability of batter getting fewer than 200 hits

A certain baseball player has a batting average $p=0.2$ (meaning that the chance of a hit is $0.2$ in each at-bat). Show that the probability of the player getting fewer than $200$ hits in his next ...
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0answers
24 views

How would one generate a set of $x$ random values conforming to a certain criteria?

I basically want to generate a set of $x$ numbers given $x$, a minimum, maximum and average. The set of numbers must conform, yielding the given average, with the smallest number being the given ...
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0answers
6 views

Proving that the Standard Sample Covariance is Biased

I know to be unbiased, the covariance should be $$\frac{1}{n-1} C_{xy}$$ but I'm trying to show how $$\frac{1}{n} C_{xy} = (n-1)C_{xy}$$ so, to do this, I reckon I'd probably go about finding ...
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2answers
39 views

Expected value for number of ties in n coin tosses?

I flip a coin $n$ times, at each toss I count how many heads and how many tails have come out so far. An example with $n=4$ might be: ...
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2answers
22 views

Hypothesis testing rejection question

If a hypothesis is rejected at the 0.025 level of significance, then it may be rejected or not rejected at the 0.01 level. Is this statement true? If it is true can you explain it to me why? I cannot ...
0
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1answer
30 views

Probability that hand contains Ace and King of at least one suit?

Compute the probability that a hand of 13 cards (drawn randomly from a standard deck of 52) contains both the ace and the king from at least one suit. I think I would use the inclusion exclusion ...
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0answers
18 views

How many units in production should be tested to ensure that the batch is healthy?

I have this assigment where I'm supposed to construct a test rig for plastic components due to the fact that a company that manufactures furniture has received lots of warranty issues regarding this ...
0
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0answers
30 views

What is the difference between weak and strong law of large numbers? [duplicate]

Why is the Weak Law and Strong Law always stated separately? From my textbook, weak law is convergence in probability and strong law is about convergence almost surely. But doesn't 'almost surely' ...
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1answer
25 views

Probability of drawing no aces with replacement of cards?

If you choose n cards with replacement (randomly choose a card, note what it is in a list, replace it in the deck, and repeat), then what is the probability that the list of cards contains no aces? I ...