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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Acceptance Sampling - Random Access Memory Chips

Random access memory chips are packed in batches of $1000$. A sample of size $15$ is randomly selected from each batch and subjected to tests of reliability. The batch is accepted if the sample ...
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1answer
23 views

What is the standard deviation for 1000000 spins if we know the standard deviation for 1000 spins ?

I have to estimate the standard deviation for 1000000 spins . I know the standard deviation for 1000 spins. I found somewhere that : New sd = old sd * sqrt (1000/1000000) but I didn't understand ...
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1answer
18 views

How do I calculate the number of trials required to verify that a failed intermittent test is fixed?

Say I've got a software test that fails randomly one out of ten times. I make a change to the code which I hope will fix it. I know ten trials is not sufficient to verify the fix. How many trials do I ...
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30 views

MIT Statistics 1.1

I'm solving problems from this MIT Statistics class. I will post here my solutions and it will be nice if you will check it and comment my faults. Thanks! Today I'm starting with the first unit: ...
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7 views

linear regression - ANCOVA

Is it possible to have more than one independent continuous variable in ANCOVA with one categorical variable? That is to say is it valid to write: ...
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0answers
11 views

Where does the probability distribution of t-statistics come from?

How was the t-table created? Was it simply based on empirical estimates? For example, why does a t value of 7.5938, with 31 degrees of freedom lead to a p-value = 1.463e-08?
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16 views

How to isolate and solve for k in a Sigma notation probability mass function equation?

"isolate and solve for k:" $$P(X = k) = \sum_{k=0}^n {{{K \choose k} {{N-K} \choose {n-k}}}\over {N \choose n}}$$ If the above equation is a function of P, how would the equation be stated as a ...
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1answer
21 views

Power function exponential distribution

I am trying to find the power function for a test. I know that the power function is calculated by $\beta(0) = P_0(x \in R)$ where $R$ is the rejection region. What I know about this test is that $X ...
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6 views

Use past and future data to predict or estimate missing values

I have a huge data set for a single variable $z$ , say WEATHER, not necessarily complete one. That is it has many holes in it(missing data) $z \hspace{3mm} is \hspace{3mm} a \hspace{3mm} 6000\times1$ ...
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2answers
46 views

Find the mathematical expectation [on hold]

Find the expectation of $$f(x) = a(1+x)^{-(1+a)}, \quad x>0.$$ The answer given is $\frac{1}{a-1}$. I am not getting the answer. Please help.
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1answer
10 views

Question about the support of a joint distribution

Let X and Y be continuous random variables having the joint pdf $$f(x,y) = 8xy , 0\leq{y}\leq{x}\leq{1}$$ Find $g(x|y=\frac{1}{2})$ the conditional pdf of $X$ given $Y = \frac{1}{2}$ I found that ...
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11 views

Variance reduction by conditioning [on hold]

Say that given Y=y, N have a probability function of $f_N$ and Y have a distribution function $F_Y\in(0,1)$. We want to estimate the the probability p for when N is larger than some constant number c. ...
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1answer
25 views

Linear regression proof

My teacher wanted us to try to attempt to prove this. So I noticed the summation on the left represents SST (total sum of squares) and on the right I noticed the second summation was the measure in ...
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0answers
12 views

Negative Binomial convolution

I've seen a couple of questions where some users provide some help on how to calculate the convolution of two independent variables $X\sim NB(r,p)$ and $Y\sim NB(s,p)$ link 1, link 2. However they ...
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0answers
12 views

How to determine 2 pairs of random variables if they share the same dependency ?

For example, I have 2 pairs of random variables $X_1,Y_1$ and $X_2, Y_2$, and I want to know if the dependency between $X_1$ and $Y_1$ is the same with $X_2$ and $Y_2$. You can think $X_i ;i=1,2$ is ...
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36 views

Exponential is to Poisson as Normal is to ???

In a time series, if the gap between successive events follows an exponential distribution with PDF $\lambda e^{-\lambda}$, then a Poisson distribution with parameter $\lambda$ tells you the ...
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0answers
42 views

How to assess how much time it will take for my YouTube channel start to make 1000 dolars a month? [on hold]

I have a YouTube channel which is making some cents every day, but I need to assess this channel profitability so that I can decide if it is worth to keep working on this channel or if it will take ...
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0answers
9 views

How to compare two tests according to the power of the test?

enter image description here Can the rejection region calculated from the problem? I'm a little bit confused by all this staff.
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1answer
23 views

Find the conditional pmf of $Y$ given $X = 0$

Let $X$ and $Y$ have the joint pmf defined by $f(0, 0) = f(1, 2) = 0.3$, $f(0, 1) = f(1, 1) =0.2$ $(a)$ Tabulate the conditional pmf of $Y$ given $X=0$ $(b)$ Tabulate the conditional pmf of $X$ ...
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1answer
21 views

Finding correlation Using only Expected values and Variance

I am doing an assignment and arrived at a question that I could not figure out and was hoping for some hints. Let X and Y be two random variables with common variance $a^2$ (a > 0). Suppose that $E ...
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1answer
20 views

Finding the Probability of a Normal Distribution

The mean IQ scores of 30 primary school students is 108.56 and the Standard deviation is 12.33. Assume that IQ scores for primary school students that have been kept for 50 years illustrate a normal ...
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1answer
31 views

Calculating the mean and variance of continuous distribution

The main question was "A machine produces 2mm to 12mm usb sticks. Any usb greater than 10mm in size will need to be thrown away." Part A) Calculate the portion that needs to be thrown away, and I got ...
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1answer
14 views

Showing that $\mathbb{E}[ \frac{S'_n}{n \log_2 n}]$ converges to 1 for a problem related to geometric distribution

We define independent random variables $X_i$ which follow the law $P(X_i = 2^k)=\frac{1}{2^k}$. We set $S_n = X_1+ \cdots +X_n$. Since we cannot apply the law of large numbers to $S_n$, we define ...
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1answer
22 views

How would you go about isolating and solving for k in a probability mass function equation? [on hold]

"isolate and solve for k:" $$P(X = k) = {{{K \choose k} {{N-K} \choose {n-k}}}\over {N \choose n}}$$ If the above equation is a function of P, how would the equation be stated as a function of k? ...
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29 views

Central limit theorem and the sequence with general term $e^{-n} ( 1+n+ \cdots + n^n/n!)$ [Proof check] [duplicate]

As an exercise I need to find the limit of the said sequence $$e^{-n} ( 1+n+ \cdots + n^n/n!)$$ using the toolkit of probability theory. Since no solution (only hints) is provided, I would appreciate ...
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0answers
20 views

How to show that a test statistics can't reject H0 with a small change in the hypothesized value

In the homework, suppose the null hypothesis is given by $H_{0\text{ }}: \alpha =\alpha _{0}$, and a test statistics $T\left( x\right) $ is proposed. The question is to show that $T\left( x\right) $ ...
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1answer
33 views

How do can i solve the integral, finding cdf [on hold]

Let $X$ be an exponential random variable with mean 1 and Y a uniform random variable between $0$ and $1$. Assume X and Y are independent and let $Z =e^{X/2}$ Find the joint cumulative ...
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31 views

Uniform probability bound involving two binomial random variables

Fix $c>1$. Does there exist number $m$ and function $f(\epsilon)$ such that for every $0<\epsilon<1$, $0<p<\epsilon$, and $n > f(\epsilon)$, we get ...
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1answer
23 views

Snakes and Ladders and Sample Space

for my Data class project we had to play a board game and do an analysis of it. My group chose rehashed version of Snakes and Ladders. I am almost done the majority of the project, but am stuck on ...
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1answer
35 views

Books on Statistics and Optimization

I'm trying to close gaps in my education especially in Statistics and Optimization theory. I had an awful class on Statistics so I want to learn it by myself. As for Optimization we had a pretty good ...
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2answers
22 views

Distinguishing between unimodal and bimodal normal data

I have a large number of data sets that have either a unimodal normal distribution or a bimodal normal distribution. I'm not a statistician by any means, so I'm quite limited in my experience. For ...
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1answer
19 views

t distribution : formula for the degrees of freedom

I understood why we are using a t distribution in this case , because the sample isn't big enough to approximate the true standard deviation of the population by the sample's . But what I can't find ...
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0answers
15 views

Finding the probaility of y for standard deviation and mean [on hold]

Write down the value of y where y is 1 standard deviation smaller than the population mean and estimate the probability that the IQ score of a randomly chosen primary school student is greater than y. ...
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1answer
53 views

How many palindrome numbers between 1 and 1,000,000? [on hold]

Though trivial since a complete response will include a valid proof, single-digit numbers will be considered to count towards the total.
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7 views

Continuously go from a lognormal distribution to a power law

Do you know any phenomena that are described by a continuous mappings between a lognormal and a power law distribution? Of course, one could give a simple linear combination of the two ...
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0answers
22 views

What should I be studying if I want to calculate correlation between data sets?

I'm building an app that brings in data from multiple API's (Stripe, Google Analytics, Github). I'd want to be able to analyze the different sets of data against each other if at all possible and draw ...
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13 views

Statistical Methods [on hold]

In the following four problems ( Problem 1- problem 4) you have to use the four following statistical methods: ANOVA, Linear Regression, T-Test, Confidence Intervals. Each method is to be used for ...
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1answer
41 views

A Game of Coin and Die

This game is played with a fair coin and a die. First player flips a coin. If it turns out head(H), the player proceeds with tossing a die. If it turns out tail(T), the player proceeds with flipping a ...
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1answer
20 views

Properties of the solution of a linear system with random equations

$x_i$ is drawn from $\mathrm{unif}(a,b)$, $y_i$ is drawn from $\mathrm{unif}(c,d)$. $x_i$ are independent from each other. $y_i$ are independent from each other. $x_i$ are independent of $y_i$. $i$ ...
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2answers
26 views

How to compute approximate probability with z value?

This is the question: An airplane with room for 100 passengers has a total baggage limit of 6000 lb. Suppose that the total weight of the baggage checked by an individual passenger is a random ...
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0answers
21 views

How can I calculate definite integral of chi-squared pdf with one degree of freedom

enter image description here I need a calculating process of the above definite integral please help me.. (sorry for my poor English)
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2answers
14 views

Is this altered formula for correlation still bounded by $-1$ and $1$?

Recall that $$ ‐1 \le \text{corr}(X,Y) = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} \le 1 $$ The proof for this bound uses the Cauchy Schwarz inequality, and I've been trying to wrap my head around ...
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0answers
30 views

How to find P(Y≥k) for a Poisson distribution

The original question is P(Y=y) for k-1≥y≥0 and P(Y≥k) for x=k. And find the expected value for Y. I know that the expected value of Y is the sum of two separate parts, which is the summation of ...
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0answers
24 views

sum of dependent random variables: [on hold]

Suppose $h=x_1+x_2+..+x_n$ and we have pdf of each $x_i$ (means: $p(x_1 ),p(x_2 ),…,p(x_n )$ )and also $x_{(i )}$ ($i=1,..,n$) are dependent random variables: How can I find the pdf of h?
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18 views

deriving the profit function given probability distributions

I can't seem to get much further in deriving the profit function for part (c). I've attached the question and my attempt, but I'm not sure on what to do next, or if I've done something completely ...
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3answers
33 views

Is it true that $\frac{E[(|X - E[X]|)(|Y - E[Y]|)]}{\sigma_X \sigma_Y} = 1$?

Consider the well-known fact that correlation is bounded between $-1$ and $1$: $$ -1 \le \text{corr}(X,Y) = \frac{E[(X - E[X])(Y - E[Y])]}{\sigma_X \sigma_Y} \le 1. $$ I've been trying to wrap my ...
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0answers
31 views

Odditiies in a StackExchange reputation distribution

As we know from for eg M.SE reputation distribution the reputation distribution is a neat power function, that's to be expected. However, on Travel StackExchange (possibly elsewhere, didn't research ...
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1answer
33 views

If a student is selected at random and is found to be taller than $1.8~\text{m}$, what is the probability that the student is a girl? [on hold]

In a college $4\%$ of the boys and $1\%$ of the girls are taller than $1.8~\text{m}$. Furthermore, $60\%$ of the students are girls. If a student is selected at random and is found to be taller than ...
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33 views
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How to use Expectation Maximization (EM) in Item Response Theory (IRT)?

Could you give a worked example on the steps of Expectation Maximization in Item Response Theory if we use the Two Parameter Rasch Model. The student abilities are unknown and the question parameters ...
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2answers
38 views

Is covariance preserved under transformation?

Let $X_1,X_2$ be normally distributed random variables with $\rho = 0.5$, mean equal to $0$ and variance equal to $1$. Let $U_i = \Phi(X_i)$ where $\Phi$ is the marginal distribution of $X_1,X_2$. We ...