# Tagged Questions

23 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 ...
47 views

### Probability Question I can't get around.

This is the question from my assignment, which I can't get around. Suppose that a water distribution system is composed of a number of independent pipes. At temperatures below 0 deg C, the pipes ...
16 views

### Unbiased estimate $\lambda^2$

Given a Poisson distribution I want to figure out whether $d:(x_1,...,x_n) \mapsto x_1^2$ and $d':(x_1,...,x_n) \mapsto x_1x_2$ are unbiased estimations for $\lambda^2$ ? I mean it would sound ...
40 views

### Proof that a median minimizes 1-norm. [duplicate]

I was wondering whether there is an easy way to show the following: We have a data set $x_1,...,x_n$ and $m$ is a median if for at least half of the n data points we have that $x_i \le m$ and for ...
24 views

### On track Prerequisite for Statistics and Probability

I do not really have a solid mathematical background because of the range of courses i had back in high school/university that wasn't really scientific oriented. Presently i am doing an MSc in ...
64 views

### Renewal Processes [on hold]

Suppose the lifetime of a component $T_i$ in hours having density $$f(t)= \frac{1}{t\ln 2}$$ for $100 < t<200$. Components are replaced as soon as one fails and assume that this process has been ...
94 views

### Version 2:Help finding the probability that $Ax^2 + Bx + C$ has real roots?

Suppose that $A, B,$ and $C$ are independent random variables, each being uniformly distributed over $(0,1)$. What is the probability that $AX^2 + BX + C$ has real roots? I am given a hint that if ...
30 views

### Check for Independence

Given $$f_{(U_1,U_2)}(u_1,u_2)=\begin{cases} 1/2& -u_1<u_2<u_1 \text{ and } u_1 - 2 < u_2 < 2 - u_1 \text{ and } 0 < u_1 <2\\ 0& \text{otherwise}\end{cases}$$ I found that ...
26 views

### convergence of sample mean [closed]

how can we prove if random variable Xn tend to 0 almost surely so its sample mean tend to 0 almost surely. but if if random variable Xn tend to 0 in probability may its sample mean doesn't tend to 0 ...
34 views

### Random var. Y with pdf $f_Y(y) = 4y^3$. Show that $-2\ln (Y^4)$ ~ $X_{(2)}^2$.

Let Y be a random variable which has pdf $$f_Y(y) = \begin{cases}4y^3, & 0 < y < 1, \\ 0, &\text{elsewhere}.\end{cases}$$ Show that $-2 \ln (Y^4)$ ~ $X_{(2)}^2$. Could anyone get me ...
39 views
+50

### Help for $U = T^2$ has an $F$ distribution with 1 numerator and $v$ denominator degrees of freedom.

If $T$ has a $t$ distribution with $v$ degrees of freedom, then $U = T^2$ has an $F$ distribution with 1 numerator and $v$ denominator degrees of freedom. First, I set $$T = \frac{Z}{\sqrt(W/v)}$$, ...
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### Finding MLE of $f(x;\theta) =1$ if $\theta-1/2<x< \theta+1/2$

Let $X_1,...,X_n$ have density: $f(x;\theta) = \begin{cases} 1 &\mbox{if } \theta-1/2<x< \theta+1/2 \\ 0 & otherwise \end{cases}$ Let $Y_1=min \lbrace X_1,...,X_n \rbrace$ and ...
38 views

### what the central limit theorem says

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 ...
41 views

### Find MLE of $\alpha$ of $f(x;\alpha)=(1+\alpha x) /2$ (stuck at derivative setup)

$X_1,...,X_n$ is an independent sample with common density: $f(x;\alpha)=(1+\alpha x) /2$ where $-1<x<1$ and $-1<\alpha <1$ I have to find the maximum likelihood estimate of $\alpha$. ...
21 views

### Which probability in this hypothesis test?

We have a hypothesis A (null hypothesis) such that $p\le 0.6$ and B such that $p>0.6$. Now we want to develop a deterministic test $\phi$ for 20 people that has a safety of 95%. Hence we would be ...
28 views

### Does Jensen's inequality become stricter with respect to the right boundary point?

Let $f(x)>0$ for all $t\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion$ be a density. I will truncate the density to the finite interval $[a,b]$ and will eventually be taking ...
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### Determine the Asymptotic Distribution of the Method of Moments Estimator of $\theta$, $\tilde{\theta}$

I am having difficulty understanding what it means to find the asymptotic distribution of a statistic. I have the correct answer (as far as I know), but I am unconvinced that I understand the process ...
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### Asymptotic Relative Efficiency: Poisson

I'm trying to find the asymptotic relative efficiency of a Poisson process: $\frac{\lambda^t exp(-\lambda)}{t!}$ = P(X=t). When X = t = 0, the best unbiased estimator of $e^{-\lambda}$ is ...
25 views

### Finding the MLE of pareto dist., and trouble interpreting $\prod$ notation properly.

I am generally having trouble understanding how to use product notation when calculating Maximum Likelihood Estimators. The example bellow is from a random sample $X_1,...,X_n$. Find the MLE of ...
70 views

### Stein's Method and Coupling of random variables

Suppose a particle starts at position 5 on a number line and at each period the particle moves one position to the right with probability p and, if the particle is above position 0, moves one position ...
23 views

### Show that if X has a density f such that f’ exists and is integrable?

Show that if $X$ has a density $f$ such that $f'$ exists and is integrable, then its characteristic function has the property : $\phi(t)=ο(t^{-1} )$ as $t\to \infty$. Hint: If $X$ has a density ...
24 views

### PDF of the product of several Poisson distributions [duplicate]

We draw $n$ values from a Poisson distribution and multiply them. - What is the expected of this product - What is the PDF of this product I think the PDF should just be ...
12 views

### Probabilty Mass Function. Function which depends on past outcomes $X$

I randomly draw numbers according to the probability mass function (PMF) $X$ in which all negative values have probability zero. Each value that is drawn from $X$ can be thought of the lifespan of one ...
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### Sample Variance of Sample Mean is an Unbiased Estimate of Population Variance of Sample Mean,$\mathbb E[\mathbb v(\bar y)]=\mathbb V(\bar y)$.

I have a hypothetical data $2,3,4,5$. I have to draw sample of size $2$ and prove that : sample variance of sample mean is an unbiased estimate of population variance of sample mean, that is ...
23 views

### Stop-Loss reinsurance, Determine the premium?

I have a question regarding the stop-loss reinsurance and the detail of this question is given as follow,
32 views

### Two Top Economist Getting 9/10 [duplicate]

Suppose that we now have twenty economists instead of just one, each of whom makes their predictions based on the toss of a fair coin. what is the probability that the second most successful of the ...
16 views

### Error Propagation - functions of the mean.

Given a number of measurements $\{x_i\}$ with values distributed according to a (known) probability distribution $\rho(x)$ with a theoretical mean $\langle x\rangle = \int dx x\rho(x) = f(y)$ and a ...
34 views

### What is the minimum Premium to be asked for a risk X?

Suppose that an insurer has an exponential utility function $u(x) =-2e^{-2x}.$ What is the minimum premium $P^{-}$ to be asked for a risk X? I got some hint for this, but I could not understand ...
42 views

### If $X$ is a random variable, under which conditions is $g(X)$ also a r.v.?

In many instances, functions of random variables appear, and we usually treat them as random variables also. In the 3d edition, pp. 85-86, of this well-known book (now in its 4th edition), we find the ...
13 views

### On stochastic linear regression

Suppose $X$ is a random vector in $\mathbb{R}^k$ with mean 0 and covariance matrix $\Sigma$. Partition the vector as $X=(X_1,X_2)$ where $X_1 \in \mathbb{R}^r$, $X_2 \in \mathbb{R}^{k-r}$. Partition ...