-2
votes
0answers
12 views

Normal distribution exercise [on hold]

In a factory, compacts are filled with a cosmetic powder. We consider the weight of the powder follows a normal distribution $N\sim(\mu, 1.21)$. The value of $\mu$ depends on the setting of the ...
1
vote
1answer
23 views

Conditional expectation and Rao Blackwell

Consider a family of densitites $f(x,\theta)=\frac{\exp(-\sqrt{x})}{\theta}$. Let $X_1$ be a single observation from this family. I have shown that $\sqrt{X_1}/2$ is an unbiased estimator. Now ...
0
votes
1answer
32 views

Hammersley–Clifford theorem

I'm reading this paper http://image.diku.dk/igel/paper/AItRBM-proof.pdf and I got stuck in page 4 with equation (1) that's based on Hammersley–Clifford theorem. I'm not good in reading set theory ...
0
votes
0answers
29 views

$E[X]< (\sum_{n=0}^\infty P[X>n]< E[X]+1$

If X takes only non-negative integer values then I figured out $$E[X]= (\sum_{n=0}^\infty P[X>n]$$ but I'm having hard time proving $$ E[X]⩽ (\sum_{n=0}^\infty P[X>n] ⩽ E[X]+1$$ for any ...
0
votes
0answers
30 views

Show that $Y_i$ is independent of $Y_j$ for any $i$ not equal to $j$

Let $\{X_1,X_2,\ldots\}$ be independent, identically distributed, absolutely continuous random variables. Let $Y_n=I\{X_n>\max(1< i < n)\}$ for $n=2,3,\ldots$ a) Show that $Y_i$ is ...
0
votes
0answers
11 views

Bayesian mean square error

Given a i.i.d sample $X_{1},..,X_{n}$ of bernoulli random variables test 2 hypotheses $H_{0}:p=2/3$ and $H_{1}:p=1/3$. Bayesian prior is $\pi(2/3)=1/3$ and $\pi(1/3)=2/3$. Find the bayesian criterion ...
1
vote
1answer
23 views

$X$ and $Y$ have a joint distribution density function. Working out a marginal density function for $X$ and $Y$

$f_{X,Y}(x,y) = \frac{3}{2}(x^2+y^2)$ if $0 \lt x \lt 1$ and $0 \lt y \lt 1,$ or $0$ otherwise. I want to find the marginal probability density function of $X$ and $Y$ and then find $Pr(0 \lt x \lt ...
2
votes
1answer
12 views

Expected value, variance and probability from a joint distribution function

Lets say I am given the following table that shows the joint probability function of X and Y: $$\begin{array} \\{}&y=1&y=2&y=3 \\x_=1&0.1&0.2&0.1 ...
2
votes
1answer
11 views

Joint distribution probabilities

I have a question that is similar to the following(made up here): The construction of a tower of cards is done is two stages, procrastination and the actual building. The time in minutes needed to ...
-3
votes
1answer
35 views

Roll Dice- Expected Winnings [on hold]

I have a problem like this: At a charity game you pay \$1 to roll a die. If you roll a 6, you get \$5. Otherwise, you get nothing. How do I set up a probability distribution and what is the ...
1
vote
2answers
23 views

Find Normalizing constant

let $f(x,\theta)=C_\theta \exp(-\sqrt{x}/\theta)$ where $x$ and $\theta$ are both positive. Find the normalising constant $C_\theta$. I get $C_\theta=\sqrt{2}/\theta$ but my book says ...
0
votes
1answer
72 views

Challenging question about probability [on hold]

A manufacturer of plastics claims that its waste is managed in such a way that benzene, a harmful chemical, cannot get into the local ground water. People living near the factory are not so sure. A ...
0
votes
3answers
61 views

Finding probability of a random point [on hold]

Consider a square with sides of length 1 and the bottom left corner at (0;0). Choose a point P randomly within the square. Show that the probability that P is closer to (0;0) than to (0.5, 0.5) is ...
-1
votes
1answer
32 views

$E[\hat{\theta}_{MME}] = E[\frac{1- 2\overline{y}}{\overline{y}-1}] = \int_0^1 \frac{1- 2\overline{y}}{\overline{y}-1}(\theta+1)y^\theta dy$..?

Let $Y_1, Y_2,\dots , Y_n$ denote a random sample from the probability density function $$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& ...
0
votes
0answers
9 views

$\chi^2$ P values comes to be zero

I want to find the P Value of $\chi^2$ (Pearson), in order to see if there is a significant difference between the given two following distributions: ...
0
votes
0answers
16 views

Expected value and Differentiation of Characteristic function

Is there an example of random variable that has characteristic function to be differentiable at zero, but has no expected value?
1
vote
0answers
99 views

Compare two estimators by using the their Expected value and variances

Let $Y_1$, $Y_2$, . . . , $Y_n$ denote a random sample from the probability density function $$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& ...
2
votes
1answer
93 views

Let $Y_1, Y_2,\ldots,Y_n$ denote a random sample from the uniform distrib… Help find finding $ \text{Var}\left[\hat{\theta}_{2}\right]$

Let $Y_1, Y_2,\ldots,Y_n$ denote a random sample from the uniform distribution on the interval $(θ, θ + 1)$. Let $$ \hat{\theta}_2 = Y_{(n)} - \frac{n}{n+1}$$ Find the efficiency of $θ^1$ relative ...
0
votes
1answer
41 views

$$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& elsewhere.\end{cases}$$ Find the MLE for $θ$.

Let $Y_1$, $Y_2$, . . . , $Y_n$ denote a random sample from the probability density function $$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& ...
1
vote
1answer
11 views

Confusion about random variables and convergence in probabilty and distribution

I'm studying statistical analysis and there's something fundamental I'm missing about random variables and how they are used in defining convergence in probability or distribution: In my syllabus ...
1
vote
2answers
34 views

$$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& elsewhere.\end{cases}$$ Find an estimator for $θ$ by the method of moments.

Let $Y_1$, $Y_2$, . . . , $Y_n$ denote a random sample from the probability density function $$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& ...
0
votes
2answers
88 views

Show that $ \hat{\theta}_2 = Y_{(n)} - \frac{n}{n+1}$ is unbiased estimators of $θ$.

Let $Y_1$, $Y_2$, . . . , $Y_n$ denote a random sample from the uniform distribution on the interval $(θ, θ + 1)$. Let $$ \hat{\theta}_2 = Y_{(n)} - \frac{n}{n+1}$$ Show that $\hat{\theta}_2$ is ...
1
vote
1answer
28 views

Show that for $0<k<1$ $P(k < \frac{Y_{(n)}}{\theta} \le 1) = 1 - k^{cn}$.

The distribution function for a power family distribution is given by $$F(y)=\begin{cases} 0, & y<0\\ \left(\frac{y}{\theta}\right)^\alpha, &0\le y \le \theta \\ 1, ...
0
votes
1answer
14 views

Basic business statistics

In 2009, the average charge for tax preparation by Hilda was $\mu=187$. Assuming a normal distribution and a standard deviation of $\sigma=20$, what is the probability that a randomly selected ...
0
votes
0answers
57 views

Derive minimum length confidence bounds for a F distribution variance …

Derive minimum length confidence bounds for a F distribution variance $\sigma^2$ and the ratio of two F distribution population variances $\frac{\sigma_1^2}{\sigma_2^2}$. What I got so far is $$ ...
0
votes
3answers
41 views

Given $X$ and $Y$ are independent N(0,1) random variables and $Z = \sqrt{X^2+Y^2}$ from the marginal pdf of $Z$

Let $X$ and $Y$ be independent $N(0; 1)$ random variables. Let $Z = \sqrt{X^2+Y^2}$. (a) Derive the marginal pdf of $Z$ and then using the marginal pdf to compute ${\rm E}[Z^2]$ (b) Can you propose ...
2
votes
1answer
60 views

LR Test for Exponential Family of Distributions

LR Test for Exponential Family of Distributions: The exponential family of distributions, both discrete and continuous, based on a parameter θ is defined by: f (x |theta) = ...
0
votes
1answer
42 views

A point chosen at random from a disc.

Ive been working on this question and have managed to complete parts $(i)->(iii)$ but am struggling with the last two parts. For $(iv)$ I end up getting this when trying to find the distribution ...
0
votes
1answer
9 views

Hypothesis testing using chi-square distribbution

Four players meet weekly and play eight hands of cards. Over a year, one of the payers finds that he has won x of the eight hands with frequency fx given in the following table: x 0 1 2 3 4 5 6 7 8 ...
1
vote
2answers
53 views

Why does $E[X]$ not equal the integral of $f(x)^2$

If $X$ is a random variable with the pdf $f(x)$ and $Y=g(X)$ how come $E[Y]$ is the integral of $g(x)f(x)$ but $E[X]$ is the intergral of $xf(x)$ ??
2
votes
0answers
33 views

Combining confidence intervals for sums of generic random variables

So my fiancee is a civil servant and asked me for help with the following problem. She has been given a collection of upper and lower bounds on expenditure for a collection of projects like: ...
1
vote
1answer
46 views

Find an unbiased estimator

Let $X$ be an r.v defined by $P(X=0)=p$ and $P(X=1)=1-p$. Find an unbiased estimator for $2p$. My solution: $E(X)=1-p$ so $2-2E(X)$ is unbiased. Is this correct?
1
vote
1answer
14 views

T statistic has t(n-1) distribution

I am trying to prove that $T_n=\frac{\bar{X}_n - \mu}{S/\sqrt{n}}\sim t_{n-1}$. One of the assumptions that seems to come up in proofs I saw of this is that ...
0
votes
0answers
30 views

Question about computing the sample mean and variance values from a sample coming from a Weibull Distribution …

Let's suppose that I have a random sample x from a Weibul distribution with shape parameter k=1 and scale parameter λ=2... How am I supposed to compute the mean value of the sample ? Also what can I ...
0
votes
1answer
20 views

Test for Validity of Artificial Samples

I have a model that actually is learned from some observed samples. Then I use the model to generate several artificial data. My question is: Which test should I use to test if the data is of the ...
2
votes
0answers
31 views

Quantitatively comparing event trains of different lengths for Poissonness

I have a parameterized, effectively black box process that generates a series of events (simulated action potentials). Different parameter values often lead to different numbers of events. How can I ...
1
vote
1answer
120 views

questions on bias of estimator

a) Let $X_{1},...,X_{n}$ be i.i.d Uniform$[0,\theta]$. Show that estimator $\beta(X)=max(X_{1},..,X_{n})$ is a biased estimator for $\theta$.Find an unbiased estimator, based on $\theta$. My attempt: ...
1
vote
1answer
35 views

what value for $c$ yields the estimator for $σ^2$ with the smallest mean square error among all estimators of …

If $S'^2 = \dfrac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n}$ and $S^2 = \dfrac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n-1}$ then $S^{'2}$ is a biased estimator of $σ^2$, but $S^2$ is an unbiased estimator of the ...
0
votes
1answer
94 views

Prove that the usual (1-$\alpha$)% confidence interval for $\sigma^2$ is NOT the shortest interval.

Prove that the usual (1-$\alpha$)% confidence interval for $\sigma^2$ is NOT the shortest interval. In particular, show that the minimum length interval satisfies $f_{(n+3)}(a) = f_{(n+3)}(b)$, where ...
1
vote
0answers
45 views

If $S'^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n}$ and $S^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n-1}$, find $V(S'^2)$.

If $S'^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n}$ and $S^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n-1}$ then $S'^2$ is a biased estimator of $σ^2$, but $S^2$ is an unbiased estimator of the same ...
0
votes
2answers
49 views

show that MSE$(\hat{\theta}) = E[(\hat{\theta} − θ)^2] = V(\hat{\theta}) + (B(\hat{\theta}))^2$.

Using the identity $(\hat{\theta} − θ) = [\hat{\theta} − E(\hat{\theta})] + [E(\hat{\theta}) − θ] = [\hat{\theta} − E(\hat{\theta})] + B(\hat{\theta})$, I need to show that MSE$(\hat{\theta}) = ...
1
vote
1answer
15 views

Marginal Density Question

I am faced with the following question, which I think is quite simple, but I can't put together for some reason. Given that $f(x,y)=(6/5)(x+y^2)$ for $0<x,y<1$, ($f(x,y)=0$ everywhere else), I ...
2
votes
1answer
65 views

If $ X = \sqrt{Y_{1} Y_{2}} $, then find a multiple of $ X $ that is an unbiased estimator for $ \theta $.

Suppose that $ (Y_{1},Y_{2},Y_{3},Y_{4}) $ denotes a random sample of size $ 4 $ from a population with an exponential distribution whose density function $ f $ is given by $$ f(y) = \begin{cases} ...
0
votes
1answer
40 views

Expectation formula proof [closed]

Let $X$ have a normal distribution with mean $\mu$ and variance $\sigma^2$. Prove that $E(X-\mu)^2$=$\sigma^2$
0
votes
0answers
13 views

Relationship between gamma and inverse gamma distributions under some algebraic operations

I have a question about the relationship between gamma and inverse gamma distributions. I have an equation that takes $L/(c*X)$ Where $X$ is a Gamma distribution and $c$ and $L$ are constants I'd ...
0
votes
0answers
49 views

Inverting probability generating function via mellin transform substitution.

The pgf is defined as: $E(z^k)= \sum_{k=0}^{\infty} p(k)z^k$ which is a discretised version of the transform: $\widetilde{p}(z) =\int_{-\infty}^{\infty} z^k p(k) \, \mathrm{d}k$ The Mellin ...
0
votes
0answers
12 views

Testing if alcohol consumption and smoking independent or not. PLEASE HELP! [migrated]

The question asks to test if smoking status and level of alcohol consumption are independent or not using usual five step procure at alpha=0.05 I am having trouble finding expected values. As the ...
0
votes
1answer
30 views

How do I find $\theta$ with bootstrap?

I have two vectors of known values $x$ and $y$. And the relationship between them is $y=\sin(\theta \cdot x)+\epsilon$, $\epsilon \sim N(0,1) $ . The question is how do I estimate $\theta$ with ...
1
vote
1answer
43 views

Probability of Renewal Processes

Suppose that there are two brands of replacement components, Brand X and Brand Y, and that for political reasons a company buys a replacements of both types. When a Brand X component fails it is ...