0
votes
0answers
3 views

find E($\bar{Y^4})$ by using moment generating function for a normal distribution with mean μ and variance 1.

Let $Y_1, Y_2, . . . , Y_n$be a random sample from a normal distribution with mean μ and variance 1. I would like to find E($\bar{Y^4})$ by using moment generating function. The setup I have right ...
0
votes
1answer
12 views

Moment Generating Function of the Chi-Squared Distribution

The questions wants us to show that the MGF for the chi-squared distribution is equal to I know that to show that I need to evaluate this integral. I'm not sure where to begin to evaluate it. ...
1
vote
0answers
8 views

probability and applied statics 4 [on hold]

In cairo 30% of residents listen to the local fm radio . ten residents are chosen at random? a) state the distribution of the random variable b) find the smallest value of s so that P (x >or equal ...
0
votes
0answers
29 views

Find E[MSLOF]. Please help.

Find the expected mean squares error of lack of fit. Trial: $$SSLOF=\sum_{1}^mn_i(\bar y_i-\hat y_i)^2\\=\sum_{1}^mn_i(\bar y_i-\bar y)^2-\sum_{1}^mn_i \hat\beta_i^2(x_i-\bar x)^2$$ and ...
0
votes
1answer
25 views

Show that the entries of a matrix are:

For a regression model $y=\beta x$ (note there is no intercept term), show that entries of the matrix $\bf{H} = \bf{X}[\bf{X'}\bf{X}]^{-1}\bf{X'}$ are $h_{ij} = ...
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 ...
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 ...
1
vote
1answer
41 views

Probability of the sum of independent standard normal random variables

Let $X_1, X_2, X_3, X_4$ be independent standard normal random variables and $$Y = X_1^2 + X_2^2 + X_3^2 + X_4^2$$ Find the probability that $Y \leq 3$. For this problem I know that the ...
2
votes
1answer
13 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 ...
0
votes
0answers
16 views

Expected value of joint p.d.f. with unknown constant

X and Y are random variables that have a joint p.d.f. $$p(x,y)=c*(x^9)*(y^6)$$ when $0<=x, y<=1$ and $c>=0$ is a constant that should be found. What is the expected value of $Y$? I am having ...
1
vote
1answer
67 views

Joint density of two functions of random variable

This is online homework, and I'm not always clear on which chapter questions are from, so I might be completely off base. I have two random variables, $X_1$~UNI(5,10) and $X_2$~UNI(4,10), and then ...
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 ...
0
votes
0answers
14 views

Probability of $t$-test rejecting null when $X$ and $Y$ are Cauchy

I have a homework problem that states: The Wilcoxon test is valid for a broad class of distributions, meaning that the actual type I error is as specified. Note that the $t$-test does not have this ...
-1
votes
2answers
16 views

Time series analysis. Understanding the arma model [on hold]

Determine wether $Y_t= 0.7 + 0.4Y_{t-1} + 0.12Y_{t-2} +Z_t$ is a stationary process.
0
votes
0answers
16 views

finding distribution based on 1st and 2nd moment

I am to determine the unbiased probability densities $p_1 (x)$ and $p_2 (x)$ given the only constraints that the magnitude of the first moment of p1 is fixed (i.e. $<x> = a$ for some real a) and ...
1
vote
1answer
30 views

Finding error variance and confidence interval

Two new types of petrol, called premium and super, are introduced in the market, and their manufacturers claim that they give extra mileage. Following data were obtained on extra mileage which is ...
1
vote
1answer
23 views

Time Series Analysis.Calculate the variance mean and autocorrelation of the time series below.

For the following time series, calculate the mean, varia nce and autocorrelation function: (a) Y_t=5+Z_t+ 0.6Z_t-1
0
votes
0answers
9 views

Asymptotic Properties of Transformation of Estimators

I'm trying to find a good explanation/proof for the following statement: If $ \sqrt{n}({\hat{\theta}} - \theta) \to^{d} N(0, \sigma^2)$, then $ \sqrt{n}({g(\hat{\theta}}) - g(\theta)) \to^{d} N(0, ...
0
votes
0answers
24 views

Which hypothesis test to use

Two identical machines are used to make a special coin. We want to know if they have the same variability. A random sample is taken from each machine : $$ \begin{matrix} MachineA & 135 & ...
-3
votes
0answers
21 views

Confidence Interval [closed]

The data for this problem is in the image below. I attempted this problem in the work shown above. I understand the formula to find the confidence interval for this data, but I am stuck at a certain ...
-1
votes
1answer
33 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
17 views

Poisson distribution confidence intervals and hypothesis

I think I have A and B correct but I have troubles with parts C and D. A) What is the p value if we suppose the following : finding golden apples in a tree follows a Poisson P(2) with $\lambda = 2$ ? ...
1
vote
1answer
28 views

Order statistics finding the expectation and variance of the maximum

Let $X_1,X_2...X_n$ be a collection of independent uniformly distributed random variables on the interval from 0 to $\theta$. The question has 3 parts. Find the CDF of $F_{x_n}$(x) of $X_n = ...
1
vote
2answers
34 views

Finding the density function of sum of random variables

The question asks given that $X_1,X_2,...,X_N$ are independent normal variables with mean $\mu$ and $\sigma^2$. Find the $\frac{1}{n}\sum\limits_{i=1}^n X_i$ So I know that the normal distribution is ...
1
vote
0answers
103 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
96 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
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
89 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
29 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
0answers
10 views

A Question on the independence of the sample mean and sample variance

The aim of the following question is to show the given random variable follows a student T distribution. Although it seems quite straightforward at the first sight, I am quite confused about the ...
0
votes
3answers
21 views

Scatter plot : Are the two observed data related

I have a regression question that ask to draw the scattered plot graph and then conclude if the two data lists are related. The two data lists are years of people and their cholesterol level. I went ...
0
votes
2answers
38 views

Continuous probability distributions

The following is a question that is likely to be appear on my exam on Friday (tomorrow) but contextualised into a different scenario. I'm having serious difficulty figuring out how to go about ...
0
votes
1answer
49 views

Conditional Probability - Bayes' Theorem

The following question is based on conditional probability. I have been told that it requires an application of Bayes' theorem, which I understand only slightly. If possible, could someone explain the ...
0
votes
0answers
19 views

Distribution of variance estimator (of normal distribution with known mean)

$X_i$ are independently drawn from a normal distribution with mean $\mu$ and variance $\sigma^2$. $$ \hat{\sigma^2} = \frac{1}{n}\sum_{i=1}^{n} (X_i - \mu)^2 $$ I'm trying to get to the distribution ...
1
vote
1answer
28 views

Conditional probability Maths question

The following questions are an excerpt from a set of questions that are to be completed as preparation to my exam on Friday. Myself and friends believe the answer to part 1i to be 1/7 or 0.143 and the ...
0
votes
0answers
58 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
0answers
63 views

Show that the likelihood ratio test can be distributed Chi-Squared

Show that the asymptotic likelihood ratio test statistic, Chi-Square LRT = -2log(Λ), to test H0: μ = μ0 vs. HA: μ ≠ μ0 is truly Chi-Squared (df=1)-distributed for Y1,…,Yn~ N(μ,σ^2) when σ^2 is known ...
0
votes
1answer
37 views

Probability applied to economics

The following two questions are based on the wondrous Statistical topic probability. After attempting both questions I have yet to answer either correctly. If anyone has encountered similar problems ...
0
votes
1answer
24 views

stats - limiting distribution of $X_i$

Suppose $P(X_{n}=i) = \frac{n+i}{3n+3},$ for $i = 0, 1, 2$. Find the limiting distribution of $X_{n}$. Is the cdf $F_{x_{n}}$ like: " if x = 0, $F_{x_{n}} = \frac{n}{3n+3}$; if x = 1, $F_{x_{n}} = ...
1
vote
1answer
26 views

Question on finding the Jacobian

I have a question which is as follows: The random variable U has the pdf $n \dfrac{u^{n-1}}{\theta^{n}}$ , $ 0 \le u \le \theta$ This is not independent of the parameter θ. Let Y=U/θ. Use the ...
0
votes
2answers
20 views

stats - limiting distribution

Let $0 < p < 1$, and let $X_{n}$ have p.d.f. $f_{n}(x) = ( 1 – p ) ( n + 1 ) ( 1 – x )^{n} + p n x^{n – 1}$, for 0 < x < 1, zero elsewhere. Find the limiting distribution of $X_{n}$. ...
1
vote
1answer
26 views

Probability: 30 balls in a bucket, homework

i need some help with some homework, first time i am doing probability and statistics, id like to know if my 2 answers below are correct, and how i can solve the remaining 2. There are 30 balls in a ...
2
votes
1answer
38 views

Do I use the Standard Deviation of my sample or the population to find the standard error.

A professor is interested in determining if attending college influences the level at which an individual cooperates with the police. The professor is unsure if attending college will teach respect ...
1
vote
1answer
32 views

Standard deviation with exponential distribution

Let x denote the distance that an animal moves from its birth site to the first territorial vacancy it encounters. Suppose that x has an exponential distribution with parameter lambda = 0.01386. a. ...
0
votes
2answers
58 views

Question on finding the MLE [closed]

I would appreciate any input or direction on this: I need to find the MLE of the hazard rate $\lambda$ where $$F(y;\lambda) = \begin{cases}1-e^{-\frac{y}{\lambda}} & y \ge 0, \\0 & y < ...
1
vote
2answers
22 views

Binomial probability on ports

This problem appears very simple, but I am almost positive that it should not be so simple. 10 ports. P1,P2,P3...P10 are connected to a computing device which polls them in order to check which ...
0
votes
1answer
16 views

Probability statements in a true or false format

The following questions are to be answered as preparation to my exam next Friday. I feel I understand the terms, such as "complement", "union", and "intersection", but when confronted with questions ...
-1
votes
0answers
30 views

Normal Distribution - statistics applied to economics

It is known that amounts of money spent on clothing in a year by students on a particular campus follow a normal distribution with a mean of £380 and standard deviation of £50. What is the ...