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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Average number of distinct values

Let $q$ such that $q < n$. I pick at random $n$ values in $\mathbb{F}_q$. What is the average number of distinct values ? Thank you
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39 views

Standard deviation shouldn't reduce?

I need to do a algorithm to calculate an integral via Monte Carlo Method, and for a purpose of simulation, I need to calculate the standard deviation of a sample generated in my program. My problem is ...
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1answer
430 views

Most Conservative Estimate : $p = 0.5$

Question : An Electronics Company has just installed a new machine that makes a part that is used in clocks. The company wants to estimate the proportion of these parts produced by this ...
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2answers
578 views

Getting the upper and lower quartiles in data with an even number of observations, or where the quartile lands on a decimal number

I want to draw a box plot, which requires that I know the median, the lower and upper quartiles, and the minimum and maximum values of my data. I understand that the quartiles are simply the value on ...
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1answer
66 views

Find unbiased for $\theta$ of $\hat{\theta_1} = (n+1) ~y_{(1)}$given a uniform distribution on the interval $[0, \theta]$.

Show that $\hat{\theta_1} = (n+1) ~y_{(1)}$ is unbiased for $\theta$. For $$P[Y_i \le Y] = 1 - y/\theta$$ Then for $ P[Y_{(1)} < y] = 1 - [1-F_{(Y_i)} (y)]^n$ which should equal to $1 - [1-(1-y/\...
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1answer
226 views

95% Confidence Interval for $σ^2$ & Determining the Reliability of the Claim

Question : A manufacturer of car batteries claims that his batteries will last, on average, 3 years with a variance of 1 year. If 5 of this batteries have lifetimes of 1.9, 2.4, 3.0, 3.5 and 4....
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1answer
38 views

Use $2\sum_{i=1}^n Y_i/\beta$ which is a pivotal quantity to derive a 95% confidence interval for $\beta$

Suppose $Y_1$,...$Y_n$ is a random sample from a gamma distribution with $\alpha = 2$ and unknown $\beta$. GOAL: Use $2\sum_{i=1}^n Y_i/\beta$ which is a pivotal quantity to derive a 95% confidence ...
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61 views

$Pr(X+Y \geq \frac{\pi}{2})$

I want to find $Pr(X+Y \geq \frac{\pi}{2})$ for joint pdf $f_{X,Y}(x,y) = x \cos y, 0 \lt x \lt \frac{\pi}{2}, 0 \lt y \lt x, 0$ otherwise. I believe I have found conditional pdf of $Y$ given $X=x$ ...
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1answer
46 views

Likelihood across multiple averages… Sorta

Hopefully my title isn't misleading... i'm not sure what this math problem is called, making searching for the answer quite difficult. Maybe someone can tag this question appropriately if you know the ...
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1answer
97 views

Finding the conditional pdf of $Y$ given $X=x$ from a joint pdf. Answer confirmation!

I have a continuous joint pdf, and I am working out the conditional pdf of Y given X=x. Is my method correct? I am given: $f_{X,Y}(x,y) = x\cos y , 0 \lt x \lt \frac{\pi}{2}, 0 \lt y \lt x$ $$\...
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24 views

Mean Value of a Non Linear Function

If I have the following function $f(x)=1-x^c$ with $x$ being periodic and can be written in the form of $x(t)=Asin(wt)+B$. $c$ is real (most of the time not an integer) and $B$ is chosen in a way ...
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0answers
72 views

Joint probability table converted to a conditional probability table

What is being asked of me: Draw up similar tables defining the conditional probability function of $Y$ given $X = x$, and $X$ given $Y = y$. Of: This table giving the joint probability function of $...
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1answer
627 views

Finding a 94% Confidence Interval for μ1 - μ2

Question : A random sample of size n1 = 25, taken from a normal population with a standard deviation σ1 = 5, has a mean1 = 80. A second random sample of size n2 = 36, taken from a ...
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52 views

ML vs MAP estimation, when to use which?

ML = Maximum Liklihood MAP = Maximum a-posteriori ML is intuitive/naive in that it starts only with the probability of observation given the parameter (i.e. the likelihood function) and tries to ...
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1answer
58 views

Bringing a density in a normal distribution form

Because I do not want to exaggerate this thread Show that $E(Y\mid X=x)$ is a linear function in $x$ I continue my special problem here. In order to make the setting clear I'll give some information. ...
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2answers
34 views

Ball Probability help

A bowl contains 5 red balls, 3 white balls and 2 blue balls. Two balls are seleceted at random from the bowl (without replacement). A) What is the probability that both are red? which is 2/9 B) What ...
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1answer
17 views

Stats question to compound drop off

Let say we have a population of 1,000 people. Let say we wanted to know the death rate out of the population in a period of 100 years. The thing I want is something to describe probability of someone ...
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1answer
20 views

Who do the Expected Values have this bound

Let $E_i$ be the expected frequency and $O_i$ the observed frequency. For example, if we role a six-sided die $120$ times then $E_i = 20$ for all $1 \le i \le 6$, while the $O_i$ will be however ...
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1answer
20 views

Question on Averages

This may be very simple to answer. But I have an excel spreadsheet. I am trying to figure out the average rate of down payments on a car. I have 70 examples. I have divided the down payment by the ...
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2answers
74 views

Expected value of SinhX

A random variable X is distributed according to a normal distribution with mean u and variance d.How to evaluate the expected value of SinhX?Thanks
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159 views

Show that $E(Y\mid X=x)$ is a linear function in $x$

Let $Y$ and $X$ be bivariate normal distributed with expectationvector $\mu=(\mu_Y,\mu_X)^T$ and covariance matrix $\Sigma=\begin{pmatrix}\sigma_Y^2 & p_{XY}\\p_{XY} & \sigma_X^2\end{pmatrix}$....
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1answer
26 views

Uncertainty on an estimator

Well first I want to say that I have a very weak background in stats/probability but I still hope that the question I want to share with you makes sense. I have a set of binary (coming from a set of ...
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2answers
50 views

The Deck of cards!

A standard deck of cards has 26 red cards and 26 black cards. 2 cards are drawn out of the deck. What is the probability the second card is red, given the first card was black (The first card was not ...
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1answer
32 views

Variance algebra

This might seem very simple but I'm having some trouble getting to the answer. If I have a random variable that's normally distributed $$X\sim N(30, 3^2)$$ and another random var. $$Y \sim N(20, (2.5)^...
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32 views

Colorful marbles

Ok so I'm having some work samples at school soon and I'm trying to practice my statistics and probability. I was given a problem with marbles. Ok.. So there are 3 red marbles, 5 greens marbles, and 2 ...
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263 views

Find the standard deviation of X, a biased coin

When tossed, a biased coin yields heads with probability 20% and tails with probability 80%. Let X denote the number of heads obtained when the coin is tossed 36 times. Find the standard deviation of ...
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1answer
60 views

Exponential Random Variable representation of criminal trial

Assume the amount of evidence against a defendant in a criminal trial is an exponential random variable $X$. If the defendant is innocent, then $X$ has mean $1$, and if the defendant is guilty, then $...
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1answer
24 views

Statistics Sample requirements

In statistics , a sample should be random, have a sample size less than one-tenth of the population and be normally distributed for it to be viable for simple statistical analysis. I understand fully ...
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1answer
113 views

What's the expected score for guessing on a word bank/“matching” test?

Some quizzes/tests have a "matching" or "word bank" section, which is set up as follows: There are $n$ questions that the student must answer. The answer to each of the $n$ questions is one of $k \...
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1answer
26 views

Probability a Random Variable will assume a value

In a fictional town of ABC, the weather patterns on different days are all independent of each other. Assume that each given day in ABC is sunny with probability 30% and rainy with probability 70%. ...
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164 views

Finding the probability density function for IID rv [duplicate]

The question is as follows: Suppose that X1 and X2 are independent, identically distributed exponential random variables. Determine the PDF for for X1 - X2. I understand that because X1 and X2 are ...
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81 views

Lognormal Distribution

I have a question about mle's and their regarding to a certain distribution: the lognormal distribution. And, I have that the expected value for the distribution is: The mle for the expected ...
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0answers
47 views

Moment Generating function proof

Let X have moment generating function M(t), and let v(t)=lnM(t). show that v'(0)=E(X) and v''(0)=Var(X). I know the formula's for E(X) and Var(X), but don't I need an original pdf of X to compute ...
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1answer
18 views

Consider the sample space of $2$ die. If $(1,2)$ does not mean the same as $(2,1)$, why is there not two sets of $(1,1)$.. etc?

$(1,2)$ does not equal $(2,1)$ So why does the sample space not include $(1,1)$ twice?
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1answer
2k views

Intuition and the math behind normalization

What exactly is the purpose of normalization. From what I read, it is to adjust two different sets of values so you can compare them, but I don't understand why, nor the math behind it. Could anyone ...
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1answer
51 views

Probability distribution of count of factors for all numbers

Is the following known? Define "factor count" as the number of prime factors of the number, minus 1. For example: Prime numbers have a factor count of 1-1 = 0 4 has a factor count of (2 and 2)-1 = ...
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103 views

What is the math symbol ~ with ind over it?

The symbol I'm talking about is from a statistics article here:
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46 views

Central Limit Theorem with no mean given

Samples are to be taken from a random variable $X$ which is normally distributed with standard deviation equal to $5$. Find the size of the sample needed so that the sample mean is within $1$ unit of ...
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964 views

Deck Building Probability Stats for Fun/Beginners (HEX tcg)

EDIT Given the amount of time spent on the problem I eventually solved it, so below I leave the structure of the original problem and the answer I came up with. I will be writing it up over the next ...
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1answer
130 views

OLS slope estimate of AR(1) with autocorrelation

I've stumbled upon this question in my econometric textbook and can't work out the right answer. The question: Consider $$Y_t = B_2Y_{t-1} + u_t\\ \\ u_t = pu_{t-1} + \epsilon_t\\ $$ where $ \...
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0answers
20 views

Compare Unequal Entities

This is more of like an analytic question. It could be vague but let me try to narrow it down a little. Assuming, i am working on an analysis in primary education which include factors like ...
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1answer
146 views

An Application of the Neyman-Pearson Lemma.

Problem: Using the Neyman-Pearson Lemma, determine the most powerful test of size $ 5 \% $. I know the Neyman-Pearson Lemma says that the test with the critical region $$ \left\{ x \in \{ 1,2,3,4 \} ...
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2answers
143 views

How to calculate the right size for error bars

I am estimating $P(A|B)$ by random sampling. I do $N$ samples and event "$A$ and $B$" occurs $x$ times and event $B$ occurs $y$ times so my estimate of $P(A|B)$ is $x/y$. I would like to plot ...
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1answer
38 views

Determining hypothesis test and application

I'm running into issues trying to figure out what type of test to use and whether two tailed or one tailed in different scenarios. How do you determine what to do in the scenarios presented? For ...
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1answer
51 views

Is extra sum of squares $SSR(X_p|X_1,…X_{p-1})$ in multiple regression always positive?

An extra sum of squares $SSR(X_p|X_1,...X_{p-1})$, assuming that no pair of predictor variables are perfectly correlated, measures the marginal reduction in the error sum of squares. Eventually one ...
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284 views

Combinatorics question - seats

In how many options can we arrange random number of men (identical men) in line of 15 chairs but: 1. 2 men can't sit next to each other. 2. next to each empty chair - there is at least one men. i can'...
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1answer
367 views

Testing hypothesis using significance levels

A random sample of n=26 observations on escape time (sec) for oil workers in a test resulted in sample mean 370.69 and sample standard deviation 24.36. Suppose the investigators had believed a priori ...
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1answer
86 views

Probability, Uniform Distribution.

Suppose $A$ is a random number chosen uniformly from the interval $(-5,6)$,what is the probability that the quadratic equation $x^2 + Ax + 1 = 0$, has no real root? This is my approach, Using the ...
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1answer
124 views

When tossing a coin, what's the likelihood of coming up heads when it came up 999 tails consecutively? [duplicate]

Consider that you're tossing a coin $1000$ times. First $999$ times, it came up heads. Is it $50\%$ chance that the coin tails after $1000^{th}$ trial? Are these discrete events? Since the mean of ...
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1answer
100 views

How to calculate Critical Values of the F-Distribution

I need to program a function to calculate the critical values for the F-Distribution, I mean the values from the tables, for example $F_{1,1}(0.95) = 161.45$. I was looking in the Abramowitz & ...