-1
votes
0answers
16 views

Stat problem help me! [on hold]

Hello! I got a problem when I was solving stat problem. I solved by(c), but after that, I found it hard to solve. Can you guys help me or give me a hint? Thank you anyway!
0
votes
2answers
28 views

Ratio of Gamma random variables

If $X_i$, $i=1,2$ are independent gamma$(\alpha_i,1)$ random variables, find the distribution of $\frac{X_1}{X_1+X_2}$ and $\frac{X_2}{X_1+X_2}$. Attempt: Let $Y_1 = \frac{X_1}{X_1+X_2}$ and ...
1
vote
1answer
21 views

Finding distribution of random variable if X is exponential $(1)$

Let X be an exponential (1) random variable, and define Y to be the integer part of X+1, that is $\hspace{15mm}Y=i+1$ if and only if $\hspace{5mm}i \leq X \leq i+1, i = 0,1,2,...$. Find the ...
1
vote
0answers
22 views

Correlation and First Order Stochastic Dominance

Suppose we have a random variable $X \sim [0,1]$ with a continuous distribution $F_X(x)$. Suppose $I \in \left\{0,1\right\}$ is a discrete random variable with $\text{Prob}(I=1 \ | \ X=x)$ strictly ...
0
votes
1answer
11 views

Percentages in Normal Distribution

A statistics problem involves: Lengths of a certain type of carrot have a normal distribution with mean 14.2 cm and standard deviation 3.6 cm. (i) 8% of carrots are shorter than c cm. Find the value ...
2
votes
1answer
33 views

Expected Value on code

I'm trying to figure out the expected number of times this algorithm will print. I'm stuck on how to go about doing so. I used an indicator variable to keep track of the number of print statements ...
2
votes
1answer
33 views

Using the inverse Gaussian integral to find percentiles

I need some help with the following: Let $$R=\mu+\sigma*\epsilon \hspace{1cm} \epsilon \sim N(0,1)$$ I want to argue that $$ \mu + \sigma*\Phi^{-1}(u)$$ are the percentiles of the model when ...
3
votes
1answer
20 views

Generating random variables with complicated probability distribution functions

I have an interesting question I need to solve, and as much as I try, I cannot wrap my head around it. Given a postive random variable X with p.d.f. ...
0
votes
0answers
24 views

Stat problem! Why is this? [duplicate]

This is a statistics problem. although this is not a problem which needs an answer, I want to know the reason Why this is right. Can you guys help me ? Thanks in advance!
0
votes
0answers
33 views

Joint density of $X_1^2+X_2^2\ \text{and} X_2,\ X_i\sim N(0,1)$

Let $X_1 $ and $X_2$ be iid with a common standard normal distribution. I am looking to find the joint pdf of $Y_1 =X_1^2 +X_2^2$ and $Y_2=X_2$. I know i can use a straight transformation argument but ...
-2
votes
1answer
30 views

Question I couldn't identify to solve this distribution [closed]

A couple decides to have 3 children.If none of them is a girl,they will try again,and if they still don't get a girl,they will try again and continues so on.If X is the number of children,the couple ...
0
votes
4answers
30 views

Proving consequence of $\operatorname{var}(X)+\operatorname{var}(Y)=\operatorname{var}(X+Y)$

How to prove that if $\operatorname{var}(X)+\operatorname{var}(Y)=\operatorname{var}(X+Y)$, then $X$ and $Y$ are independent?
0
votes
1answer
13 views

Related to chi-squared functions

I'm finding difficulty in finding what type of function it is in continuous distributions in probability.Mainly how can i identify whether a function is chi-squared or not?
2
votes
0answers
16 views

Differential Equation for brownian bridge?

For the brownian motion, we know that probability density of the particle's position at time $ t $, $ \rho(x,t) $ satisfies the diffusion equation pde: $ \partial_t \rho = d \; \partial_x^2 \rho $. Is ...
1
vote
0answers
18 views

Finding $a_n, b_n$ so that a sequence converges in distribution to a nondegenerate random variable.

Now, $X_1, X_2,\dots$ are iid with the same distribution as the chi-squared distribution with one degree of freedom. Find $a_n$ and $b_n$ so that $a_n \left( \max_{1 \leq i \leq n} X_i - b_n \right)$ ...
1
vote
1answer
13 views

Testing statistic $\frac{MSS(X)}{MSS(Y)}$

Suppose a test statistic $\frac{MSS(X)}{MSS(Y)}$, where $MSS$ denotes Mean Sum of Squares, is to be used for testing the significance of the factor $X$. Do we need the assumption $$\mathbb ...
0
votes
1answer
17 views

Expected Value of a Minimum Function using a Beta Distribution

Let $X$ be a IID random variable with support in $[0,1]$ and CDF given by a Beta distribution, i.e. $X \sim Beta(\alpha,1)$. Suppose we have a function of the form: $$ Z_t = \phi(X_t,y_{t-1}) = ...
1
vote
1answer
26 views

Distribution of the sample variance of n iid exponential variables

I have to check some properties of an estimator, but I can't find its distribution. Let $X_1,...,X_n $ be independent identically distributed exponential variables with parameter $ \theta $, i.e. ...
0
votes
1answer
23 views

Poisson distribution equation

This is probably a very simple and silly question to ask, but I just don't understand the steps for b). I don't quite understand where the negative (-) sign came from? Could somebody please shed some ...
0
votes
2answers
55 views

Given a CDF, find P(-.5<X<.5)

Given the following CDF: \begin{equation*} F(x)= \left\{ \begin{array}{lr} 0 & x<-1, \\ \frac{x+2}{4} & -1 \leq x < 1 \\ 1 & x \leq 1 \end{array} \right. \end{equation*} Compute ...
2
votes
0answers
24 views

Hypergeometric RV - what is the sample/population?

An instructor who taught two sections of engineering statistics last term, the first with 20 students and the second with 30, decided to assign a term project. After all projects had been turned in, ...
4
votes
3answers
78 views

Deriving Mean and Variance of Laplace Distribution

It has been a long time since I have used calculus, and I am trying to understand how the mean and variance of the Laplace distribution with pdf $$f(x|\mu,\sigma) = \dfrac{1}{2 ...
1
vote
1answer
30 views

tricky integrating ranges x1-x2

So, we know the sum of n i.i.d. exponential(lambda) is gamma(n,lambda). But I am looking at a problem with X1-X2. So I get the joint dist of z=x1-x2 and w=x2. Then I integrate out w on range 0 to ...
1
vote
1answer
23 views

Cumulative Poisson Distribution Question

For the following question I figured that the expected time between successive arrival is the mean = 1/10 per hour (or 1 per 6 minutes). My question is regarding the second part; does the fact that ...
0
votes
1answer
33 views

Unbiased sample standard deviation of a custom/unknown probability distribution

Hi i must determine the unbiased sample standard deviation of an unknown probability distribution.I dont have the data of the full population so i must work with a sample. Now according to Wikipedia ...
0
votes
2answers
23 views

Derivative of Survival Function

I am trying to get through statistical survival analysis - sadly I only have high school math. I have the following equation: $ S(t) = Pr\{T ≥ t\} = 1−F(t) = \int_t^\infty f(x) dx$ $f(x)$ is the ...
1
vote
1answer
18 views

Is this form of distribution considered Bimodal or Uniform?

Would this form of distribution be considered Bimodal or Uniform? I have been searching through distribution images and the Bimodal distributions generally appear to refer to a pair of Normal ...
0
votes
1answer
16 views

Kernel density estimation including measureed uncertainty

I am trying to plot the distribution of a measured variable from a scientific experiment, in this case a velocity. After making a simple histogram, I have been reading this, ...
1
vote
0answers
25 views

Proof for the distribution of a two-sample t-test with unequal population variances.

I am having trouble finding documentation showing a proof, or at least some outline for it, illustrating how to derive the distribution and degrees of freedom of the test statistic for a two-sample ...
0
votes
1answer
14 views

How to calculate the median and the quantiles of this distribution?

A median of a distribution of one random variable $X$ of the discrete or continuous type is a value of $x$ such that $P(X < x) \leq 1/2$ and $P(X \leq x) \geq 1/2$. If there is only one such $x$, ...
0
votes
1answer
24 views

Find joint distribution function in region

I can't for the life of me figure this one out, I am stuck on part (c) ... I have this as my starting point ? $$ \frac{45}{304}\int_0^x\int_{2-x}^2 u^2v^2\,\mathrm{du} \mathrm{dv} $$ Here is my ...
2
votes
2answers
77 views

Log normal distribution - Where am I wrong?

Let $X$ be a R.V whose pdf is given by $$f(x)=p\frac{1}{\sqrt{2\pi\sigma_1^2}}\exp\left(-\frac{(x-\mu_1)^2}{2\sigma_1^2}\right)+ ...
3
votes
2answers
82 views

How to give rigorous proofs of these two limit statements?

Let $X$ be a random variable with cumulative distribution function $F(x)$. Then how to rigorously prove the following two limit statements? $\lim_{x \to - \infty} F(x) = 0$. $\lim_{x \to + \infty} ...
0
votes
0answers
16 views

An example of $k$-independent distributions.

I'm trying to better understand the idea of $k$-independence in distributions. The idea is that a distribution $\mu$ over $\{0,1\}^n$ is $k$-independent if any restriction of $\mu$ to $k$ variables ...
0
votes
1answer
18 views

normality of data

Does the qqplot below suggest that the data is normally distributed? The fact that it's nearly perfectly linear is to me an indication of normality. However, the Anderson-Darling test for some reason ...
4
votes
1answer
59 views

Uniform sampling with replacement item frequency

Suppose we are sampling from $N$ distinct items uniformly with replacement $M$ times. What can be said about the distribution of frequencies of items drawn? For example, if I sort all the frequencies ...
1
vote
1answer
38 views

Question about the Bayesian Inference of a parameter

In order to understand the difference between the Frequentist and Bayesian inference, I was reading the presentation at: http://www.stat.ufl.edu/archived/casella/Talks/BayesRefresher.pdf . In order to ...
0
votes
0answers
47 views

Prove Logarithmic function is part of exponential family

The aim is to prove that the logarithmic distribution with parameter $p (0<p<1)$ is part of the exponential family and hence, give its canonical parameter. To prove a distribution is part of ...
1
vote
1answer
22 views

Show that statistic is (not) sufficient

I need to verify ifthe statistic $|X|$ is or npt sufficient for $\mu$, if $ X \sim N(\mu, 1)$ Using the definition, I've obtained the pdf of X given $ T(X)=|X|:$ $$f_{X|T}(x|t) = ...
0
votes
1answer
11 views

Given MTTF and Number of Items, how to calculate failing parts with Time?

I am wondering how to make use of MTTF Here is the situation, I am given an MTTF for an item type x and a certain demand for that item in the next 25 years, say 100 parts that will be in operation ...
0
votes
1answer
25 views

Let X be a random variable with PDF fx. Find the PDF of the random variable |X| in the following

Here's my question: X is uniformly distributed in the interval $[-1,2]$. Find pdf of $|X|$... So I did P($|X| \le x$) = P($-x \le X \le x$)... From here I'm not too sure how to proceed. I know the ...
2
votes
2answers
73 views

$X = (X_1, X_2)$ is it not a multivariate random variable?

$X=(X_1,X_2,\ldots, X_P)$ is a $p$-dimensional random variable on $(\Omega, S, P) $ iff $X_i$'s are univariate random variables on the same probability space $(\Omega, S, P)$ ." We all know ...
2
votes
1answer
71 views

Proving $E_{\theta}[T(X)] = \frac{\psi'(\theta)}{\eta'(\theta)}$

I'm trying to understand how to prove the following theorem: Let $\{P_{\theta}, \theta \in \Theta\}$ be a family of distributions in the one parameter exponential family with density (pmf) ...
0
votes
1answer
26 views

Pareto Distribution transformation

Suppose $X$ is a random variable with Pareto distribution. There pdf and cdf are: $$f_X(x) = \begin{cases} {\alpha x_m^\alpha \over x^{\alpha +1}}, & \text{if $x\ge x_m$ } \\ 0, & \text{if ...
0
votes
0answers
34 views

Determining $\sigma$ given mean and proportion of a Normal distribution?

The marks of a random sample of students with mean $\mu$ and standard deviation $\sigma$ showed that 15.87% scored higher than 70. The distribution of the marks is Normal with mean $50$ standard ...
0
votes
0answers
26 views

probability that a variable, as a function of choice variables, is among the top k out of n when ordered

Suppose $(h_1,h_2,...,h_n)'$ is an $n\times 1$ vector. Let $h_i=g_iX_i$, where $g_i$ is a choice variable which can vary across $i$ and $X_i$ is a random shock with Pareto Type I distribution. ...
0
votes
1answer
12 views

Choice of distribution given data

I have some data to be analyzed. It's histogram looks unimodal, with the support being positive reals between 0 and 100, most of the values huddled up around the mode.I want to be able to ...
1
vote
1answer
40 views

Probability distribution for n-th smallest value in array with random values

I have a $d$-dimensional ($d>3$) array (vector, set, ...) which is filled with random values taken from uniform distribution (interval $[0,1]$). What is the probability distribution for n-th ...
1
vote
1answer
37 views

Probability that a random variable is among the top k out of n when ordered

Suppose $X_1,X_2,\ldots,X_n $ are $n$ i.i.d. random variables with a continuous distribution $F(x)$ and density function $f(x)$. What is the probability distribution that any given $X_i$ is among the ...
1
vote
2answers
32 views

Exponential distribution - Using rate parameter $\lambda$ vs $\frac{1}{\lambda}$

Sometimes I see the exponential distribution defined as follows: $$f(x) = \lambda e^{-\lambda x}$$ when $x > 0, 0$ otherwise I have also seen it defined like so: $$f(x) = \frac{1}{\lambda} ...