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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.

2
votes
2answers
54 views

Expected value of $e^{sP}$ where s is a complex number and P is a Poisson rv

For each positive integer $N$, let $ B_N$ be a binomial $(N,1/3)$ random variable and $P$ be a Poisson(5) random variable. I am trying to understand the statistics of $B_P$. Could someone please hint ...
5
votes
1answer
84 views

Find the values of $ a$ so that A is positive definite (p.d)

Let $A=(1-a)I_n + a J_n$ Find the values of $~a~$ so the Matrix is p.d ? Note:$~I_n~$ is the identity matrix and $~J_n$ is the $1's$ matrix. I know that $~~A$ is p.d $~iff~λ_i >0$ so, I need to ...
3
votes
1answer
50 views

Consistency in uniform distribution

I know what consistency is but in options C and D both U and V are given whose covariance is quite difficult to find.
1
vote
0answers
40 views

Finding a critical region for a simple test, with PDF $f(x) = (\frac{x}{\theta} + \frac{1}{2}) \space \mathbb{1}_{(-1,1)} (x)$

I'm dealing with a simple inference problem which involve a PDF I've never dealt with before. We have $X_1, ..., X_n$ iid variables where $X_i$ has a PDF $$f(x) = (\frac{x}{\theta} + \frac{1}{2}) \...
3
votes
1answer
16 views

Expected time to visit all countries by random flight paths [closed]

Say there are $n$ different countries, the flight starts from some initial country. At each step, the flight can go to a random country other than the one where it currently is. The probability of ...
0
votes
0answers
25 views

Calculating the true error comprised of two probability distributions

Let $X$ = {0,1,2,3,4} and $Y$ = {0,1}. A probability distribution $D$ defined on $X\times Y$ such that $D_x$ = Binomial(4, 0.5) and $D_{y\mid x}$ = Bernoulli(0.5). Given the predictor: $h(x)$ = $0$ ...
1
vote
0answers
30 views

F-test in multiple linear regression

I'm currently reading Introduction to statistical learning. When trying to prove the collective significance of a regression linear model, we use the F-test with the following formula. $$F=\frac{(TSS-...
0
votes
0answers
20 views

A question about selection of appropriate test procedure in Testing of Hypothesis

Q. Let (X, Y) be a paired response measured in continuous scale and obtained by administering a a certain drug on a patient. Let 'm1' and 'm2' denote the expected responses. Under suitable assumptions,...
0
votes
1answer
30 views

Explain coefficients in a multiple regression are the same as in simple regressions

Given the matrix of covariances, $M$ (below), three variables $X, Y, Z$, and a multiple regression $\hat{Z} = \frac{5}{4}X + \frac{4}{5}Y$: $$M=\begin{bmatrix}16 & 0 & 20\\0 & 25 &...
1
vote
0answers
60 views

Estimation of parameter $\widehat\beta$ in the linear model [closed]

Consider the simple linear model $Y=X\beta+\varepsilon$ where $\varepsilon\sim N_n(0,\sigma^2I).$ It known that $\widehat\beta=(X^tX)^{-1}X^tY$. Also, $$\pi(\beta\mid Y)\propto\Gamma\left[\frac{1}{...
0
votes
2answers
68 views

Showing a Normal and a Chi square are independent

Student's t distribution is defined as the ratio of a standard normally distributed random variable and the square root of a Chi-square distributed random variable divided by its degrees of freedom, ...
0
votes
0answers
22 views

Calculating probability of two random variables [duplicate]

I'm having trouble calculating the probability of two random variables. A question on an exam review is P(X+Y>38). I know how to do this problem with just one variable, using z scores, but i'm not ...
3
votes
1answer
55 views

What is the expected number of times you need to flip a coin before you see 2 heads? The heads do not need to be in a row

My attempt: The 2nd head must appear last in the sequence of k flips. Therefore the first head can appear in any of the first k-1 flips. The number of ways the first head appearing in the first k-1 ...
0
votes
1answer
40 views

A question about Testing of Hypothesis

Q. Suppose for an experiment, the test for normality of an underlying population is accepted on the basis of a data set. To cross check the result, the experimener repeated the test with another ...
1
vote
0answers
25 views

A 2 component mixture is symmetric if and only if $\lambda\in \{0,1,\frac{1}{2}\}$

Consider the following mixture of two densities $$ f(x)=\lambda g(x-\mu_1)+(1-\lambda)g(x-\mu_2) $$ with $\lambda\in [0,1]$, $g(\cdot)$ symmetric around zero, $\mu_1<\mu_2$. Claim: the mixture is ...
2
votes
0answers
148 views

MLE of $\theta$ when $X_1,\ldots,X_n$ are i.i.d with pdf $f(x)=\frac{2(\theta-x)}{\theta^2}\mathbf1_{0<x<\theta}$

Let $X_1,X_2,\ldots,X_n$ be i.i.d random variables with pdf $$f(x\mid\theta)=\begin{cases}\frac{2(\theta-x)}{\theta^2}&,\text{ if }0<x<\theta \\ 0 &,\text{ otherwise }\end{cases}$$ ...
1
vote
1answer
32 views

Number of parameters in Bayesian Classifier

Problem Assume we have a Bayesian classifier with the three following features to determine whether a software user is a student, an ...
3
votes
1answer
53 views

Probability question about panda births and statistical tests

I am self-learning statistics, and I have a question about how to do the following problem: There are two species of panda bear, A and B. Both are equally common in the wild and live in the same ...
2
votes
4answers
62 views

CLT for uniform distribution

I don't understand how the CLT can hold for a uniform distribution. Say I have U[0;1], then whatever value I will be able to sample from the population will always be 1. Therefore, every sample mean I ...
0
votes
1answer
69 views

Hierarchical Bayesian Negative Binomial model with Gamma prior on mean

I am interested in deriving the full conditional for the mean parameter in a Neg-Binomial model with a Gamma prior on the mean, as such: \begin{align*} Y|\lambda,\phi\sim & NB(\lambda,\phi)\\ \...
1
vote
2answers
121 views

How to compute Integrated Squared Error for kernel density estimation in R

I am working on R for kernel density estimation. I am testing different kernels and I need to evaluate them. I use next code for density estimation: ...
1
vote
0answers
34 views

Joint Probability Distribution and covariance

If $$f(x,y)=1/4 $$ $x=-3,y=-5; x=-1,y=-1; x=1,y=1; x=3,y=5. $Find cov (x,y). I know the formula for cov (X,Y) but I'm stuck at finding E (x) and E (y).
1
vote
1answer
21 views

Using independence_test in R with unequal sample sizes [closed]

This is a homework problem that I have where I'm testing the means of sampled high temperatures in two cities: Des Moines and Chicago. The first part of the question required me to run an unpaired t-...
1
vote
2answers
58 views

Distribution of a Random Sum [duplicate]

The following experiment is performed: An observation is made of a Poisson random variable $N$ with parameter $ \lambda $. Then $N$ independent Bernoulli trials are performed, each with probability $p$...
2
votes
0answers
18 views

Conditional probability given only the converse conditional probability, and the average of one variable

I’ve been working on this question for a few days now. Full disclosure: this is from a homework problem set. This is one of the exercises of Barnett's book on quantum information. A particle counter ...
0
votes
1answer
9 views

Calculate reliability of a system with probabilites

I have a system, that has several components. These components are working with p probability. In order for the system to work there must be a continuous line active. What I have done so far: P(...
0
votes
0answers
23 views

Estimating the mean and the error of estimator for Poisson random variables

$Y_1,...,Y_n$ ~ $P(\lambda)$, $V(Y_i)=\lambda$ $E[\bar Y] = \lambda$, $V[\bar Y]=\lambda/n$ Question is how would I employ $Y_1,...,Y_n$ to estimate $\lambda$, and how would I estimate the standard ...
1
vote
1answer
41 views

Distribution function of a biased estimator

$f(y) = ay^{a-1}/θ^a, 0<y<θ$ $ \hat{\Theta} = max(Y_1, Y_2, . . . , Y_n).$ How do I find the $E[\hat{\Theta}]$ ? I'm trying to show that it's a biased estimator, then I'm going to find ...
2
votes
1answer
95 views

bias variance tradeoff — properties that do not follow

Going through this lecture note on bias-variance trade-off, I didn't follow the latter part of this paragraph. It shows the common situation in practice that (1) for simple models, the bias ...
3
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0answers
36 views

Require understanding regarding the concept of restricted estimators

I was reading "The Elements of Statistical Learning Book by Jerome H. Friedman, Robert Tibshirani, and Trevor Hastie" where I encountered the following: The part tells us that the RSS criterion will ...
0
votes
0answers
18 views

$R^2$ associated with a restricted LS estimator is never larger that that of the unrestricted LS estimator

Prove that the $R^2$ associated with a restricted least squares estimator is never larger than that associated with the unrestricted least square estimator. So, I tried doing this question but I can'...
1
vote
0answers
23 views

Error with polr . Error in svd(X) : valori infiniti o mancanti in 'x' [closed]

Good evening, I'm a university student. For the thesis work I am processing an ordinal logistic regression. All Variables are rasters: Aspect,Basin,Curvature,Elevation, Geology,Plancurv,Profcurv,...
2
votes
1answer
64 views

Finding variance of infimum of a set

Let $X_1, X_2, \cdots,X_n$ be independent and identically distributed random variables having an exponential distribution with mean $\frac{1}{\lambda}$. Let $S_n=X_1+X_2+\cdots+X_n$ and $N = \text{...
0
votes
0answers
10 views

In a LGSSM how do we know that the prediction distribution is Gaussian?

I am trying to follow lecture notes regarding the Kalman Filter from a course taught at Stanford. The lecture notes can be found here. The linear Gaussian state space model (LGSSM) is introduced as ...
1
vote
1answer
33 views

Naive Bayes Classifier Unclear

I read the following sentence regarding the Naive Bayes Classifier: If large number of features have relatively minor effects, taken together, their combined impact could be quite large. Could ...
5
votes
1answer
89 views

Distribution of min(X+Y,Y+Z,X+Z,Z+V,X+V,Y+V)?

Let $X,Y,Z,V$ be i.i.d continuous random variables in an interval $[a,b]$. What will be the distribution of $\min(X+Y,Y+Z,X+Z,Z+V,X+V,Y+V)$? Assume the distribution of the random variables to be $F(.)$...
0
votes
1answer
21 views

Marginal Distribution from Bivariate Distribution Matrix

I am doing some practice problems to prepare for my statistics exam, and I just want to know if my reasoning is correct on one problem, and if not, I want to know how I should reason through this. The ...
0
votes
0answers
21 views

Classification vs regression for binary response variables

I have a breast cancer dataset with ~30 predictors, all of which are continuous/numeric. The response variable is binary (the diagnosis is either malignant or benign). I know classification is good ...
1
vote
0answers
36 views

How to generate a Bernoulli distribution based on a given Bernoulli distribution? [closed]

We have a Bernoulli distribution which outputs 1 with probability C and outputs 0 with probability 1-C. C is unknown. Now we would like to generate a new Bernoulli distribution that outputs 1 with ...
1
vote
0answers
21 views

Concept of Stationary Population

In stationary population, there can be different probabilities for death in a given year for different age groups but these probabilities don't change over time. So the probability a 28 year old will ...
0
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0answers
11 views

What to do when the activation is non-linear when rescaling to compensate for dropout?

At 54:24 of this video, it says that once there is non-linearity in the activation function, the expectation is not exact. So would it post a problem for non-linear activation? Then how would you ...
1
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0answers
19 views

“Number of Persons” instead of “Number of Person-years” in Stationary Population [closed]

I have learnt that the life table function $_{n}L_x$ is the number of person-years lived by the cohort between ages $x$ and $x+n$. But for stationary population, $_{n}L_x$ is defined as the number ...
0
votes
0answers
14 views

definition of integrated- and- exponential autocorrelation time

I understand them (to an extent) both seperately, but i was reviewing my notes from class and my verbal definition is effectively stating the same thing in different words. I have: integrated: ...
0
votes
1answer
40 views

Joint conditional density of two iid exponential random variables conditioned on their sum

I have the following question: Suppose $X_1, X_2$ iid $\sim f_X(x)=\theta e^{-\theta x}1\{x\ge 0\}$ and $S=X_1+X_2$. What is $f_{X_1,X_2}(x_1,x_2|S=s)$? The solution from our exercise class to this ...
0
votes
1answer
15 views

Distinction between “Mid-year population” and “Number of people alive”

Suppose $D_i=$ Observed number of deaths in the age group $(x_i, x_{i+1})$, $P_i=$ Mid-year population for age group $(x_i, x_{i+1})$, $N_i=$ Number of people alive at $x_i$ among whom $d_i$ ...
0
votes
1answer
21 views

Statistics and ML Knowledge Sources

I have a MS in Statistics but I completed my studies 10 years ago. Since then the field of Statistics/Modeling/Machine Learning has continued its rapid advance, and I feel that a number of new ...
1
vote
0answers
32 views

Probability Vs Rate

Suppose $_{n}q_x$ is the $\textbf{probability}$ of dying between age $x$ and $x+n$. And $_{n}M_x$ is the age-specific death $\textbf{rate}$ in the observed population in the interval $(x, x+n)$. The ...
1
vote
0answers
31 views

Variance of bivariate normal distribution plus normal distribution

Problem: $W = -27 + 0.3X + 0.45Y + E$ The pair $\begin{bmatrix} X \\ Y \end{bmatrix}$ behaves like a bivariate normal with vector of averages $\begin{bmatrix} 156 \\ 86 \end{bmatrix}$ and ...
3
votes
1answer
77 views

How to compute a 2D distribution function from its density?

Suppose that we have two random variables $X, Y$ with a joint probability density function $$f(x,y)=1,\ -y\lt x\lt y,\, 0\lt y\lt 1.$$ How can I calculate the cumulative joint probability function $...
2
votes
1answer
76 views

Probability you end up at the origin after taking $2n$ steps?

Starting at the origin on the line we take a step of unit to the left or to the right with probability $\frac12$. We do this repeatedly with independent steps. If we take $2n$ steps, what is the ...