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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Yule Walker equations

The Yule-Walker equations relate the auto covariance of a random signal to the autoregressive (AR) model parameters. They can be used to estimate AR models from data by first estimating the auto ...
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214 views

Hypothesis testing question

The article “Statistical evidence of discrimination” (J. Ameri. Stat. Assoc., 1982, 773-83) discussed the court case of Swain v Alabama (1965), in which it was alleged that there was discrimination ...
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310 views

Largest Number Drawn - Why are These Approaches Not Equivalent?

Here's the question: Four numbers are drawn at random from a box of ten numbers 0, 1, ..., 9. Find the probability that the largest number drawn is a six if the draws are made with replacement. The ...
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Calculating lambda in statistics

I am trying to figure out how to find lambda for a question: Compute the probability of taking at least 7 minutes to receive a call. The exponential mean is 5.
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Prove $\sum_i \frac{\bar{x}(x_i-\bar{x})}{nS_{xx}} =0$

Prove $$\sum_i \frac{\bar{x}(x_i-\bar{x})}{nS_{xx}} =0$$ That is one detail in a proof of the variance of the intercept $\alpha$ in the simple linear regression $Y_i=\alpha+\beta x_i$.
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How would I solve this poker statistics question with an infinite deck?

My professor threw out this question for the class as an extra credit question. I'm not asking for an answer, but rather, a nudge in the right direction to solve the problem. I'm currently stuck ...
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Prove $SST=SSE+SSR$

Prove $$SST=SSE+SSR$$ I start with $$SST= \Sigma (y_i-\bar{y})^2=...=SSE+SSR+ \Sigma 2( y_i-y_i^*)(y_i^*-\bar{y} )$$ and I don't know how to prove that $\Sigma 2( y_i-y_i^*)(y_i^*-\bar{y} )=0$ a ...
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120 views

regression on circular data

How would one design a regression where the dependent variable is measured in degrees on a circle? The dependent variable is on the range [0, 360), and the independent variables are demographic ...
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how to calculate p value of 2 sample

James and Margaret grow watermelons. Margaret claims her watermelons are heavier than James's watermelons. $40$ sample has been taken from both. James: mean $7.3$ variance $0.4$ Margaret mean $11.3$ ...
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522 views

How many strings in the letters a, b, and c have length 10 and exactly four a's?

part a) How many strings in the letters a, b, and c have length 10 and exactly four a's? I did $\binom{10}{4} = 210$ different ways for the strings of length $10$ in part a to be arranged, but I'...
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Same mean, different standard deviation in data sets

How would a data set containing the values of a variable with a mean of 50 and a standard deviation of 3 compare with another data set containing the same variable, but a mean of 50 and a standard ...
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196 views

Find the probability that a person has a headache given that the person has a fever.

For the problem below are we going to use the equiprobable space. Which is in particular, if $S$ contains n points then the probability of each point is $1/n$. For an event $A$, $P(A) =$ (number of ...
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2k views

Uncertainty in parallel resistors.

I need help with one of my study guide questions. We learned about uncertainty in class but am not sure how to attack this problem: Could someone walk me through this example? Any help will be ...
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1answer
208 views

Calculate variance

I was given this question as HW: A survey reveals that on average 1 computer out of 800 computers crashes during a severe thunderstorm. A certain company had 4,000 working computers when the ...
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73 views

Probability density function of random variables.

I need help with one of my study guide questions. We learned a bit about PDF but I am not sure how to combine them. Here is the problem: Could anyone walk me through how to do this? Any help will be ...
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1answer
720 views

Probability and proportion from standard deviation and mean [duplicate]

I do not know how to start this problem-help needed-kindly help me The lifetimes of a colour TV picture tube is normally distributed, with a mean of 8 yrs and a standard deviation of 2 years. ...
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1answer
98 views

Allocating observations to test whether two expected values are equal$\mu_{1}-\mu_{2}=\Delta\neq 0?$

Question: Suppose that you wish to test the hypothesis $H_{0}:\mu_{1}=\mu_{2}$ versus $H_{1}:\mu_{1}\neq \mu_{2}$,where both variances $\sigma^2_{1}$ and $\sigma^2_{2}$ are know ,A tatal of $n_{1}+...
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101 views

Showing independence of random variables

When proving $\bar x$ and $S^2$ are independent in my noted it says that "functions of independent quantities are independent ". Can someone tell me how functions of independent quantities are ...
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1answer
50 views

Beginner question in probability

i'm a beginner in statistics and probability and i need help with a given problem please. We are given a guy who has a machine and a button, the outcames when he presses the button are: a)Music ...
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302 views

Roll a 6-sided fair die until a 6 appears. Let X = the number of 1's that are rolled. Find Var(X).

Let X = the number of 1's that are rolled. Find E[X] and Var(X). I can't seem to calculate Var(X). I've calculated E[X] = 1. I let R = the number of non-6 rolls, and I let Y = the number of rolls ....
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Unbiased estimators in an exponential distribution

We have $Y_{1}, Y_{2}, Y_{3}$ a random sample from an exponential distribution with the density function $ f(y) = \left\{ \begin{array}{ll} (1/\theta)\mathrm{e}^{-y/\theta} & y \gt 0 \\ 0 &...
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Why do statisticians like “$n-1$” instead of “$n$”?

Does anyone have an intuitive explanation (no formulas, just words! :D) about the "$n-1$" instead of "$n$" in the unbiased variance estimator $$S_n^2 = \dfrac{\sum\limits_{i = 1}^n \left(X_i-\bar{X}\...
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1answer
159 views

Power function of a hypothesis test

Let $X$ ~ Exp($\lambda$) Consider testing $H_0 : \lambda \geq \lambda_0$ vs. $H_1 : \lambda < \lambda_0$ I'm trying to find the power function of the test that rejects the null hypothesis iff $X$...
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1answer
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Confidence Interval and margin of error

I had the following question on a study guide and was wondering if I did it correctly. I was confused because of the way the question is worded. Here is the question: And here is how I attempted it: ...
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What is the log derivative of a gamma function with a shift in its parameters?

Forgive my ignorance here. I am going through "Latent Dirichlet Allocation" (pdf) where they state the following (just assuming common knowledge about the Dirichlet distribution): $\frac{1}{\delta\...
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1answer
36 views

Fidelity of measurement using conditional probabilities

Let me begin by saying that I'm not entirely sure if this is the correct forum, or if Cross Validated would be more suitable. The problem I'm about to describe is statistical in nature, but I believe ...
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1answer
148 views

calculate Limiting distribution $\displaystyle\frac{\sum_{i=1}^n X_i}{\sum_{i=1}^n Y_i}$

let $X_1,X_2,\ldots,X_n$ are random sample of bernoulli distribution with parameter of $\displaystyle\frac{\theta_1}{\theta_1+\theta_2}$ and $Y_1,Y_2,\ldots,Y_n$ are random sample of geometric ...
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48 views

Weighted averages with fractions instead of frequencies?

So I have a bunch of scores 0-100 and a two sets of weights, the frequency which is the number of people in the sample with an average score among those people. Then I have another value that is the ...
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1answer
191 views

Expected number of bridges

There are 10 islands in a row. Each island is linked to the successive one by 2 bridges. One of these two is the correct bridge i.e. it will take you to the other side. The other is a bad bridge i.e. ...
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1answer
59 views

Error In Proving Variance Of Sample Mean

So the variance of the sample mean $\bar X$ is $\sigma^2/n$, and I understand how to prove it through the formula $Var(nX) = n^2Var(X)$. However, I was approaching it through another method, and seem ...
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Can we prove this after removing one condition?

mathematicians! I want to ask about redundant condition in the below proof. Suppose there exists a partition $A_{0}, A_{1}, \ldots, A_{k}$ of $\chi$ such that $P(X \in A_{0}) = 0$ and $f_{X}(x)$ is ...
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There are 5 servers. Each server has 1% downtime. What's the probability that at at least three servers are down?

There are 5 servers. Each server has $1$% downtime. What's the probability that at at least three servers are down? My reasoning is the following: A) There is $(1-0.01)^5$ probability that $5$ from $...
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1answer
31 views

( X ~ $B(n_1, p) $, Y ~ $B(n_2,p)$, (X,Y) independent ) $\implies Z = X + Y$ ~$ B(n_1 + n_2, p)$

Hello I have some exercise from probability and statistics, but I don't know what they are really asking about: Random variable X has Binominal distribution $B(n_1, p)$. Random variable Y has also ...
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39 views

Bounded function of geometric random variable

if X~ Geometric(p), with q=1-p, then show that for any bounded function f with f(0)=0, we have E(f(x)-qf(x)+1)]=0. Our professor asked us to try solving this problem as a good practice but I have no ...
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135 views

Accuracy of a Normal Approximation for a Poisson random variable.

compute bound on accuracy of a normal approximation for a poisson random variable with mean 100? I understand what the question is trying to ask me but I have no idea how to approach it and solve it. ...
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Confidence interval for a binomially distributed observation with few trials?

If there are few trials and you want to get the confidence interval of a binomially distributed observation, is it still okay to use the normal approximation interval, or is that only accurate for a ...
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Derive a model for the variances $\sigma_i^2$ for which $b_1$ is the best linear unbiased estimator (BLUE) of $\beta$

Consider the model $y_i=\beta x_i + \epsilon_i$ (without a constant term and with $k=1$), where $\mathbb{E}[\epsilon_i]=0, \mathbb{E}[\epsilon_i \epsilon_j]=0, \forall i \neq j$, and $\mathbb{E}[\...
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1answer
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Determining total number of tickets sold

Let's say there is a lottery where ticket numbers 58,145,350, and 400 won. Is there a way to determine the most likely number of tickets sold? We are assuming non-replacement and fair draws.
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normalizing a non gaussian distribution

I have a set of empirical data that I have collected from different people with several attributes. For privacy reasons I will refer to these attributes as a,b,c,d,... etc. Since the data was ...
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243 views

Clarify my understanding for central limit theorem from a statement

Asked what the central limit theorem says, a student replies, "as you take larger and larger samples from a population, the histogram of the sample values looks more and more Normal". Is the student ...
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1answer
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What type of distriburtion?

A comparative study of responses to a questionnaire addressed to consumers, showed that the lifetime of televisions usually distributed with an average of 8.2 years and a standard deviation of 1.3 ...
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2answers
3k views

How to find (and plot) a probability distribution function?

I'm working on my biometrics course, and I have to plot a pdf (I think it means probability density / distribution function). Here is a sample pdf graph : Introduction to Biometrics page 5 , figure 2....
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Statistics & Probability -Distribution problem

10% of the chocolate bars that are produced in a factory have unacceptable shape. In a sample of 1000 chocolate bars find the probability that the number of unacceptable shapes is (A) less than 80 (...
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Sum of Random Distributions/ Unusual Results

$$X \sim N(\mu_1,\sigma_1^2)$$ $$Y \sim N(\mu_2,\sigma_2^2)$$ then $$X+Y \sim N(0,\sigma_1^2+\sigma_2^2)$$ One way, I tested this to be true is in excel, I used the ...
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Find a minimum of a variance

Given $Y_1$ and $Y_2$, linearly independent. $Y_1$ has mean $\theta$ and variance $\sigma_1^2$, and $Y_2$ has mean $\theta$ and variance $\sigma_2^2$ Given $Y_3 = a Y_1 + (1-a) Y_2$ So then by ...
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1answer
36 views

How does effect size affect alpha?

As the effect size increases the power of a hypothesis test also increases, but what happens to alpha?
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Why square the result of $x_1 - \bar{x}$ in the standard deviation? [duplicate]

I don't understand the necessity of square the result of $x_1 - \bar{x}$ in $$\sqrt{\frac{\sum_{i=1}^{N} (x_i - \bar{x})^2}{N-1}}$$. In fact I don't understand even why is $N - 1$ on the denominator ...
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1answer
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I have trouble understanding the proof of the Wold decomposition theorem

I'm trying to understand the proof of the Wold decomposition theorem in [1, p.187]. I find a few things about it very irritating. The theorem states: Theorem 5.7.1 (The Wold Decomposition). Let $...
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Probabilty & Statistics Problem

The number of cracks which are present in a part of an international road has an average value of 2 cracks per kilometer. 1)What is the probability that there are no cracks in a section of road ...
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Dummy variable in the probability generating function

I'm struggling to understand what the purpose of the dummy variable $t$ in the probability generating function is? I know it takes a value between 0 and 1, and have heard it described as a 'relative ...