Questions tagged [self-study]

A routine exercise from a textbook, course, or test used for a class or self-study. This community's policy is to "provide helpful hints" for such questions rather than complete answers.

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11 views

How do I build a probability distribution from a $P(A_k| \cap_{i=1}^{k-1}A_i)=\theta^k$?

A car windshield gets progressively weaker as it suffers from repeated debris strikes. Let $A_i$ be the event that the windshield survives the $i$th strike. For $0 < \theta < 1,$ let $P(A_1)=\...
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What is the conditional expectation of some random variable?

I have the following problem: Consider the function $g(\tilde{w})=-e^{-\delta\tilde{w}}$, where $\delta>0$ some constant and $\tilde{w}$ is some random variable s.t.$\tilde{w}=\alpha-\beta\tilde{x}...
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62 views

What is the moment generating function of $P(X=x)=\alpha\theta^x$?

Let $X$ be a discrete random variable such that $P(X=x)=\alpha\theta^x$, $x=1,2,\ldots,$ and $0 < \theta < 1$ and $P(X=0)=1-\sum^\infty_{x=1}\alpha\theta^x$, where $\alpha$ is a constant. Find ...
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How do I find all even moments (and odd moments) for $f_X(x)=\frac{1}{2}e^{-|x|}$?

I was asked to find a formula for all even moments of the form $E(X^{2n})$ and all odd moments of the form $E(X^{2n+1})$ using a mgf. Can you help me find the even moments? I will attempt to solve for ...
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1answer
79 views
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Why do output coefficients not resemble true coefficients in a linear model?

How to estimate multiplicative model of spice harvesting? Why do output coefficients not resemble true coefficients in a linear model? This is a story about generating values by function, noising it ...
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143 views

How can one implement PCA using gradient descent?

I have to implement PCA using gradient descent and stop at convergence. I am not able to find the objective function. I know that the aim of PCA is to reduce the $n$-dimensional matrix to $k$ ...
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29 views

Variance of quadratic form

This is a homework problem I’m trying to solve but I can’t seem to solve Q1b without using the theorem. I am also given the fact that $$E(y’Ay)=tr(A\Sigma)+\mu’A\mu$$ I’ve tried using the trace-...
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20 views

Asymptotic Distribution of Covariance

I've seen a lot of questions revolving around the asymptotic distribution regarding the sample variance, such as $\sqrt{n} (s^2 - \sigma^2) \xrightarrow{d} N(0, \mu_4 - \sigma^4)$, however, what would ...
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Argue that MLE of GEO(p) is biased

Question: consider the case of having a random sample $X_1,\ldots,X_n$ from a $GEO(p)$ distribution adn recall that the MLE is given by $\hat{p}=1/\overline{X}$. Argue that the estimator $\hat{p}$ is ...
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64 views

The frog problem with negative steps

In this question The Frog Problem (puzzle in YouTube video) a frog has to jump from leaf to leaf on a row of leaves. And the question is how long it takes on average to reach the end. In that ...
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Asymptotic Distribution of Minimum Uniform Random Variables

I've been working on this problem for a while, and I've made some progress, but I'm still stuck on some parts. I was hoping to get some assistance with this! Let $M_n = \min(X_1, ..., X_n)$ where $...
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52 views

Multivariate Gaussian distribution explanation needed [duplicate]

I am pretty new in statistics. I Googled the multivariate Gaussian distribution, but still have no idea how to solve this. I tried to make $\mu_{x} \rightarrow a\mu_{x} \ and\, \mu_{y} \rightarrow ...
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What is the PDF sum of N random variables

I have N random variables: $X_1,...,X_N$ which are all independent. The PDF (probability density function) of each random variable $f_{X_i}=e^{-a/(x^{2/b})}x^{-(4+b)/b}$. What is the PDF $f_S(x)$ ...
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sampling from joint distribution to recover marginal distribution

I'm going through Bayesian Core and have gotten stuck at this remark on page 233: " A first remark that motivates the use of the Gibbs sampler is that, within structures such as $$ \pi(x_1) = \int \...
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Posterior distribution of Bernoulli distribution

The pdf of X | $\theta$ is given by $\theta^x (1- \theta)^{1-x}$ and its prior distribution is given by $p(\theta) \frac {1} {B(\alpha, \beta)} \theta^{\alpha - 1} (1 - \theta)^{\beta - 1}$ where $...
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Likelihood and minimal statistic [closed]

Suppose that (x1,..., xn) is a sample from the Bernoulli(θ) distribution, where θ ∈ [0, 1] is unknown. Determine the likelihood function and a minimal sufficient statistic for this model. (Hint: Use ...
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signification of average partial effect in probit regression

Consider the probit regression: where "yi" and "di" are dummy variables and "xi" is a continuous scalar regressor. Is the following statement true? Even if the coefficient estimates "B1" and "B3" are ...
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33 views

Whats the Probability of pulling 5 aces in 10 pulls with replacement

This is not homework. I am just going through the Bayesian Statistics the Fun Way book ( https://nostarch.com/learnbayes ) and I can't figure out how they come up with inputs for Chapter 4 question ...
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1answer
51 views

Prove that the variance of the sample mean is smaller than that of the mean of a simple random sample of the same size n drawn with replacement

In statistics, a simple random sample is a subset of individuals chosen (one by one) from a population. Each individual is chosen randomly such that each individual has the same probability of being ...
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56 views

Normal V Binomial Distribution (Elementary Question)

CONTEXT: First year university statistics course exam question Suppose couples decide to have children until they either have a child of each sex, or they have three children. Assume that births are ...
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From what type of distribution does observed z score come from, and based on this picture, what is the variance and mean?

I am trying to understand this conceptually and mathematically but somehow these formulas and variable expressions are confusing. If someone can explain it in simpler terms, it would be appreciated. I ...
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Derivation of posterior predictive distribution for autoregressive model

[Bayesian Choice ex. 1.37] If $x \sim N(\theta, \sigma^2)$ and $y \sim N(\rho x, \sigma^2)$, as in an autoregressive mode, with $\rho$ known, and $\pi(\theta, \sigma^2) = 1/\sigma^2$, give the ...
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Binomial distributions?

Research has shown that people are likely to bypass tomatoes that weigh less than 70 grams. a produce company produces tomatoes that average 78 g with a standard deviation of 5.2 g. The company ...
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24 views

Maximum likelihood: Bernoulli

I would really appreciate if anyone you can explain how it went from step 1 to the answer provided below. This is from the book Doing Data Science by Cathy O'Neil and Rachel Schutt pages 101 to 102. ...
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1answer
42 views

p-value and hypothesis testing [closed]

(The p value) Which one of the statements about the p value is correct? a) The p value is the predefined probability of making a mistake when the null hypothesis is false. b) The p value is the ...
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32 views

Clarification on the concept of a cumulative distribution function of a measure (measure theory)

I was asked to show that $g_f(x)=\mu(f\leq x)$ defines a cumulative distribution function for any measurable function $f$. Let $(\Omega,\mathcal{F},\mu)$ be a measure space and $(\mathbb{R},\...
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32 views

Probability and Normal Distribution

(sampling distribution) We randomly draw another sample of size $10$. What is the probability that the mean word count of these $10$ songs is greater than $436.531 = 374.9149 + \dfrac{2∗97.42498}{\...
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Given that two diagnostic tests are positive when administered, what is the probability that this person has the disease in question?

I have seen many similar questions on this forum; however, this question has two diagnostic tests, which appear to complicate the problem a bit. Two tests (Test A and Test B) for a disease exist. ...
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40 views

Mean and variance after transformation

A psychology teacher has given an exam in which the highest score possible is 200 points. the mean score for the 30 students who took the exam was 156, and the standard deviation was 24. Because there ...
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1answer
61 views

$P(h\leq k)=P(t \leq k),\forall k\in \mathbb{R}$ implies $h=t$

Consider the measurable space $(A,\mathcal{A})$. Let $h,t:A\rightarrow \mathbb{R}$ be mesurable functions. Show that If $P(h\leq k)=P(t \leq k),\forall k\in \mathbb{R}, \forall P$ probability on $(A,\...
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735 views

What is the probability that Person A will require more tosses of a particular coin than Person B to obtain the first head?

Person A and Person B each independently toss the same unbalanced coin and count the number of tosses it takes each of them to obtain the first head. Assume that the probability of obtaining a head ...
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1answer
31 views

What is the probability that none of the people call the same office and the probability that all of them call the same office?

Suppose there are $n (n \ge 2)$ offices in a city. Suppose that $k (2 \le k \le n)$ people each independently and randomly call one these $n$ offices for an appointment. What is the probability that (...
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1answer
69 views

Is either $P(A)=\sum_{x \in A} p(1-p)^x$ or $P(A)=1$ if $A$ has a finite number of elements a probability on (S, B)?

My GA proposed the following question in class. I am not too sure which textbook this question came from, so if you can identify which textbook she is sampling, I would enjoy doing some additional ...
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8 views

Convergence of Modified Expectation-Maximisation Algorithm - interpreting language of question

We're going to consider a modified E-M algorithm and its convergence properties. To do so, we will first need to review the convergence of the standard E-M algorithm as I'll need to refer back to it. ...
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30 views

Convergence of a Expectation Maximisation Algorithm

Consider using standard expectation maximisation to learn the parameters of a Hidden Markov Model. We can show the effect of standard expectation-maximisation on the log-likelihood by first writing ...
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1answer
13 views

Representation of $l_x, d_x, L_x, T_x$

There are some problems in my textbook where a life table is given, with some of the entries missing. Based on the given entries, I have to fill up the complete life table. Suppose it is given $l_0 = ...
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43 views

Expected value of ratio of random variables

Let X and Y be independent random variables with $$E(X) = 0\ and\ Y > 0$$ Find the mean value of $$ X/Y$$ My attempt: We have for independent random variables $$E(XY)=E(X)\times E(Y)$$ Hence, $...
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10 views

Equivalent definitions of cointegration rank for VAR(2)

Let us have Vector Autoregression $$\begin{cases}I: y_t = 2 + 1.3 y_{t-1} + e_{1t} \\ J: x_t = 3 - 0.3 y_{t-1} +1.3 x_{t-1} + e_{2t} \end{cases}$$ I am asked to find the cointegration rank and ...
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1answer
21 views

Interpretation of short and long run reactions in ADL(1,1)

This is a task from my statistical minor. The question proposes that this equation is a model, where $x$ are health expenditures and $y$ is life expectancy in some country. If so, why is $\phi(0)$ ...
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2answers
73 views

Is there a fundamental difference between artificial neural networks and “other” supervised machine learning models

I would like to link three of these resources, present my understanding of what I read, and pose the question if my understanding is approximately correct Is Machine Learning glorified curve fitting ...
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2answers
27 views

Revert AR process with constant

I have got this task at the Time Series course as a part of Statistical minor. I am math major, and have gone through basic Probability (read:measure theory) course. Let us have $y_t = 0.4y_{t-1}+2+\...
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1answer
50 views

Bayesian Data Analysis 2.7a

I'm self-studying Bayesian Data Analysis by Gelman et al. and I'm struggling to understand the solution to exercise 2.7a. The question: For the binomial likelihood, $y\sim Bin(n, \theta)$, show that $...
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12 views

Graphical Models Showing Independence Relations

I am attempting an old assignment question on graphical models. I am given a paragraph of information and asked to draw a directed graphical model showing the relationships between the variables and ...
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30 views

Maximum Likelihood to estimate function parameters from experimental data

Problem I have a set of $n$ measurements $V_{i}^* = V^*(t_{i})$ of the observable $V$ in function of time $t$. I also have the associated sensibility errors (determined by the measuring device's ...
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L sets and Probability theory

I am required to prove the following: Let $L_1$ be the space of real-valued random variables on $(\Omega,\mathscr{A},\mathbb{P})$ which have finite expectation $\mathbb{E}(|X|)<\infty$ and let $...
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1answer
39 views

Type 1 and Type 2 Errors with Costs

I know the ex-ante cost of type 1 and type 2 errors in my study. How do I select my alpha, given that I know that alpha governs both type 1 and type 2 errors? Suppose my null is that $\beta=0$ and my ...
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57 views

Pre PhD preparation: What should I focus on? [closed]

I am going to start my PhD in Statistics next fall and I am currently studying some math (in part because I don't want to stay too much time without studying something and in part to be sure that I ...
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1answer
174 views

Limiting distribution of $\frac1n \sum_{k=1}^{n}|S_{k-1}|(X_k^2 - 1)$ where $X_k$ are i.i.d standard normal

Let $(X_n)$ be a sequence of i.i.d $\mathcal N(0,1)$ random variables. Define $S_0=0$ and $S_n=\sum_{k=1}^n X_k$ for $n\geq 1$. Find the limiting distribution of $$\frac1n \sum_{k=1}^{n}|S_{k-1}|(X_k^...
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1answer
91 views

How to find Expectation?

If $X \sim N(0, \sigma^2)$ and $Y \sim N(0, \sigma^2)$ are independent, how can we find the expectation $$E \left(\frac{X }{\sqrt{X^2+Y^2} }\right)\,?$$
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55 views

How to show an alternate data processing inequality concerning KL divergence between conditionals?

Suppose $(Y,X) \sim F \in \mathcal{P(\mathbb{R^d})}$. Consider an arbitrary transformation $f$ that acts on $X$. My intuition is that the following should be a result in information theory: $$ \...