-1
votes
0answers
11 views

Prove that $\limsup \frac{|\log N(n) - X_{N(n)}|}{\log n} = 1$, sequence of records

I am currently reading in the book "The Asymptotic Theory of Extreme Order Statistics" by Janos Galambos. However, there is the Theorem 6.4.3 which is Let the population distribution $F(x) = ...
0
votes
0answers
18 views

concentration of the following random variable: number of items that fit in

This is related to this previous question. Let us assume that we have a capacity $n>0$ which tends to infinity. We are given an i.i.d. sequence of nonnegative random variables ...
1
vote
1answer
11 views

What is the relation between fix points of random uniform permuation, and probability of independent events occuring.

Let $A_1,\dots,A_n$ be independent events that occur with probability $1/n$ each. Let $p_{n,k}=P($exactly $k$ events occur). One can show with stirlings formula that ...
0
votes
2answers
13 views

Does it matter here that random variables are jointly distributed?

My lecture notes ask the following (true/false) question on understanding: Jointly normally distributed random variables are independent iff they are uncorrelated. I don't quite understand what ...
1
vote
1answer
24 views

What is the variance of this random variable: number of items

Let us assume that we have a capacity $n$ which tends to infinity. We have an infinite number of random variables $X_1, X_2, \dotsc$, where each $X_i$ is independent and identically distributed with ...
-2
votes
0answers
23 views

Trouble with Conditional probability and expectation [on hold]

I have a few questions in probability that have been bothering me. The first is this: Why is $$E(T-t | T \ge t) = \int_t^\infty \frac{(s-t)f(s)~ds}{P(T\ge t)}. $$ The second is this: How does one ...
0
votes
0answers
12 views

How do I describe probabilities of realised variables in terms of CDFs and PDFs?

I have a game theory problem here where the realisation of the random variable $Y₂$ (termed $y₂$ is observed in the first stage of the game. $Y₂$ is the second highest order statistic for $n-1$ ...
2
votes
1answer
12 views

Beta/Dirichlet question

A generalization of the beta distribution is the Dirichlet distribution. In its bi-variate version, (X,Y) have pdf $f(x,y) = Cx^{a-1}y^{b-1}(1-x-y)^{c-1}, 0<x<1, 0<y<1, ...
2
votes
1answer
43 views

How to calculate conditional probability with inequality

I know that: \begin{equation}\displaystyle P(A=x|A+B=y) = \frac{P(A=x \cap A+B=y)}{P(A+B=y)}\end{equation} Assuming $A$ and $B$ are independent, the intersection of the two events can be resolved as ...
0
votes
2answers
17 views

Simple finding the PDF given function

I am a little confused on how to go about finding the PDF given a condition for a function. So I have the function $$ Y(x)=ae^{-bx} \,\,\,\,\,\,\, a,b,x \geq0 $$ and I need to find the value for X ...
1
vote
1answer
26 views

Central limit theorem kind of statement for records

I am trying to prove the following statement, but I do not know how to go on: Let $F(x)$ be an arbitrary continuous distribution function. Then there are constants $A_n, B_n > 0$ such that, as ...
3
votes
0answers
34 views

Distribution of $\frac{X}{|Y|}$, where X and Y are standard normal r.v.'s

Let X and Y be independent standard normal random variables. What is the distribution of $\large \frac{X}{|Y|}$? Attempt: Let $\large U = \frac{X}{|Y|}$ and $ V = |Y|$. This transformation is not ...
-1
votes
0answers
15 views

Sum of the smallest or greatest k components of a random vector drawn from a symmetric Dirichlet distribution? [on hold]

Is any distribution known for the sum of the smallest or greatest $k$ components of a random vector drawn from a symmetric Dirichlet distribution?
1
vote
2answers
21 views

Integrating a function wrt different measures [duplicate]

Suppose that $(\Omega, \mathcal E, P)$ is a probability space and $X\colon \Omega \to \mathbb R$ is a RV defined on $\Omega$. Denote as $\mu\colon \mathcal B \to [0,1]$ the probability measure on ...
0
votes
1answer
17 views

How can I find the expected value of a random variable with terms that increase until infinity?

Here is the question A company buys a policy to insure its revenue in the event of major snow storms that shut down business. The policy pays nothing for the first such snowstorm of the year and ...
0
votes
0answers
30 views

Use consecutive odds ratios to find the largest k that maximizes the Poisson (u) probability Pu(k). [closed]

Full Problem: Use consecutive odds ratios to find the largest k that maximizes the Poisson (u) probability Pu(k). For what values of u is there a double maximum? What are the two values of k in that ...
1
vote
1answer
28 views

Central limits without replacement in a finite population.

"Everybody knows" that there are lots of variations on the theme of the central limit theorem. The most frequently seen form seems to be this: Suppose $X_1,X_2,X_3,\ldots$ are i.i.d. random variables ...
2
votes
1answer
43 views

Joint distribution of range $(R=X_n-X_1)$ and mid-range $(V=\frac{1}{2}(X_1+X_n)$order statistics

Let $X_1,X_2, · · · , X_n$ be independent and identically distributed Uniform random variables on the interval (0, a) for a > 0, each having a density function $f(x) = \frac{1}{a}$, $0<x<a$. Let ...
-1
votes
0answers
10 views

Distribution of YXY' [closed]

Let us assume that $Y\in\mathrm{R}^{m\times m}$ is a random matrix with distribution $p_y$ and $X\in\mathrm{R}^{m\times m}$ is another random matrix with distribution $p_x$ and is independent of ...
1
vote
0answers
29 views

$U$-Uniform$(0,2\pi)$, Z-Exp$(1)$, $U$ and $Z$ are independent. Then $\sqrt{2Z}\cos U$ and $\sqrt{2Z}\sin U$ are independent standard normal. [closed]

Given that $U$-Uniform$(0,2\pi)$, Z-Exp($1$), $U$ and $Z$ are independent. Show that $\sqrt{2Z}\cos U$ and $\sqrt{2Z}\sin U$ are independent standard normal variables. Thanks in advance for any ...
2
votes
1answer
23 views

Exchangeable/Independent Bernoulli Distribution

Let P be a uniform random variable on the interval $(0,1)$ with density function f(p) = 1, $0<p<1$. Let $X_i|P$, i = 1,2,...,n be independent and identically distributed random variables having ...
0
votes
1answer
31 views

Probability of sample mean [closed]

A town has $500$ real estate agents. The mean value of the properties sold in a year by these agents is $\$800,000$ and the standard deviation is $\$300,000$. A random sample of $100$ agents is ...
0
votes
0answers
51 views

Probability and Expected profit

I really need help for this qn! You are asked to determine the profitability of a new line of sunglasses, which will retail for \$10. The fixed cost of setting up the line is \$2000. The total number ...
0
votes
2answers
32 views

Flipping several biased coins

Assuming I'm flipping $M$ biased coins with different probability for heads $p_i, i=\{1,...,M\}$. What is the probability of having $k$ times head? Is there a distribution function known for this?
0
votes
2answers
40 views

How can I do a constructive proof of this:

Say Z is a non-negative R.V, and P(Z>0)>0. Then exists a a>0 and an b>0 such P(Z>a)>b. I am not sure how to start with the proof, I've been assigning numbers than can qualify for some CDFs but I don´t ...
0
votes
0answers
46 views

Make the sum of random variables converge, while the sum of the variances diverges [closed]

Suppose $X_n$, $n=1,2,3,...$, are independent and $Var(X_n)$ is uniformly bounded by finite constant $C>0$. Construct $X_n$ such that $\sum_nX_n$ converges a.s., but $\sum_nVar(X_n)=\infty$.
0
votes
1answer
32 views

Pdf of the product of an exponential r.v. and a beta r.v.

Let $X$ and $Y$ are 2 independent random variables, where $X$ has an exponential distribution with parameter $1$ and $Y$ is $\beta(a,b)$ distributed. What is the Pdf of $W=XY$ ? Thanks !
1
vote
0answers
34 views

Expectation of absolute value of Brownian motion

I'm working on this problem that I can't seem to figure out. The problem involves a 1-dimensional Brownian motion, $B_t$, where the subscript denotes the time, and it asks me to show that the ...
0
votes
2answers
29 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
22 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 ...
3
votes
1answer
48 views

$P(X^2+Y^2<1)$ of two independent n(0,1) random variables

Suppose that X and Y are independent n(0,1) random variables. a) Find $P(X^2+Y^2<1)$ Attempt: a) Let $U = X^2 + Y^2$, $V = Y$. Then $X = \sqrt{V^2 -U}$, $Y = V$. $J = \left| ...
1
vote
1answer
42 views

Pdf of the product of an exponential rv and a $f_Y=Ka^{-K}y^{K-1}$ distributed rv …

Let $X$ and $Y$ are 2 independent random variables, where $X$ has an exponential distribution with parameter $1$ and $Y$ has the following Pdf: $f_Y=Ka^{-K}y^{K-1}, 0 \le y \le a $. Someone claims ...
1
vote
0answers
26 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 ...
2
votes
2answers
57 views

Poisson random variables and Binomial Theorem

I'm working on a problem from Casella and Berger's Statistical Inference. X is distributed as Poisson$(\theta)$ and Y is distributed as Poisson$(\lambda)$, with X and Y being independent. We let U = X ...
0
votes
2answers
22 views

Conditional probability in multinomial distribution

Consider a multinomial distribution with $r$ different outcomes, where the $i$th outcome having the probability $p_i$, $i$=1,...,$r$, $\sum_{i=1}^r p_i = 1$. Denote $X_i$ be the number of times the ...
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 ...
0
votes
3answers
63 views

Is it even possible to find the variance of this moment generating function?

This is my moment generating function: $M_x(t) = \frac{6e^t}{t^2} + \frac{6}{t^2} + \frac{12e^t}{t} - \frac{12e^t}{t^3} + \frac{12}{t^3}$. I have to find the mean the variance of it. After taking ...
1
vote
1answer
38 views

An inverse problem on tail probability

This is a question out of curiosity. Assume that $f(x)$ is a density function for which there is a constant $C>0$ so that $$ \int_t^\infty f(x) dx \le C f(t) $$ holds for large enough $t>0$. My ...
1
vote
1answer
26 views

Is there a shorter way for me to answer this joint probability question?

Suppose a box contains 10 green, 10 red, and 10 black balls. We draw 10 balls from the box by sampling with replacement. Let X be the number of green balls, and Y be the number of black balls in the ...
3
votes
2answers
50 views

Random variable $X^2$ determined by moments

Let $X$ be a real random variable, with standard normal distribution. Is the distribution of $X^2$ determined by its moments? In general, if $n \in \mathbb N$, is the distribution of $X^n$ ...
1
vote
0answers
25 views

Exchangeability and pairwise exchangeability

Suppose we have $\{Y_{i}\}_{i=1}^{n}$ that is exchangeable so the joint distribution of this sequence is the same as $\{Y_{\sigma(i)}\}_{i=1}^{n}$, where $\sigma:\{1,...,n\}\rightarrow\{1,...,n\}.$ ...
0
votes
0answers
18 views

Simulating beta random variables

Let's say I estimated the parameters of Beta distribution with a single-period dataset, and want to generate a multi-period sample. How can I do that? To be more specific, I estimate my parameters ...
0
votes
1answer
22 views

Derivative of integral over part of Gaussian distribution

I am currently trying to compute the following derivative and integral: $$ P\psi_\theta = \frac{d}{d\theta}\int_{-k}^k tf_T(t)dt, $$ where $t=x-\theta$ and $X\sim N(\theta_0,\sigma^2)$. $f_T$ above ...
0
votes
1answer
22 views

Variance from the pdf

FOr the interpretation of the variance, it is the fluctuation of the data around the mean. So if I know that mean (say mean=0), and then there are lots of data (70%) points are greater than +/-10 away ...
1
vote
1answer
74 views

Explanation of how probability density functions transform under the change of variable

I've just read about probability density function from this article. In that article, there is some wired concept that I can't understand, please see the section named "Dependent variables and change ...
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?
1
vote
2answers
42 views

Continuous and Discrete random variable distribution function

I have a very basic question in probability. It pertains to the difference between a continuous random variable distribution function and a discrete one. This question has confused me many times. ...
1
vote
1answer
40 views

Probability: basic question and concept

I have always been struggling with the problem, in particular, I usually have great difficulty in differenting when should I multiply n! to take care of the ordering, and when should I not do so. For ...
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
2answers
33 views

What does tightness of convergence mean and why do we use this?

In a paper that I am reading it is written: We have $$(X(t_j))_{j=1}^k \xrightarrow{d} (Y(t_j))_{j=1}^k,$$ where $t_0=0, t_1 < \dotsb < t_k$. It further is written: In order to show that ...