Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.

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8
votes
1answer
275 views

Volume of the intersection of ellipsoids

How do I compute the volume of the intersection of two $n$-dimensional ellipsoids? Given an $n$-vector $c$ and a symmetric positive-definite $n\times n$ matrix $A$, define the ellipsoid ...
0
votes
0answers
19 views

Repetitive sampling from the uniform and an unknown distribution

I am trying to model an experiment. We have n ''players''. Each one picks independently a sample from a continuous uniform distribution in the [0,$2^{64}$], let's call it $u_i$. He also picks a sample ...
0
votes
1answer
106 views

Conditional binomials

I'm trying to proof this, If X ~ B(n, p) and, conditional on X, Y ~ B(X, q), then Y is a simple binomial variable with distribution Y ~ B(n, pq) . Can someone show me how or link some reference.
0
votes
0answers
16 views

Distribution maximum with small sample related to large sample

Suppose the random variables $X_i$, $i=1,\cdots,n$ and $Y_j$, $j=1,\cdots,m$ all have distribution $F(x)$, with order statistics denoted by $X_{(i)}$ and $Y_{(j)}$. Assuming $n<m$ (e.g. $n=m/100$), ...
0
votes
0answers
37 views

Mathematical Probability and Statistics( all the math need)

I would like some suggestions about mathematical techniques and knowledge are required to understand and master 2nd year undergraduate probability and statistics. I am mature student with some ...
1
vote
0answers
29 views

Showing $\lambda_V(x)\leq \min\{\lambda_1(x),\cdots, \lambda_n(x)\}$.

Suppose $X_1, \cdots, X_n$ are independent, nonnegative continuous functions, each $X_i$ has hazard function $\lambda_i(x)$. If $V=\max\{X_1, \cdots, X_n\}$, I need to show that ...
0
votes
1answer
32 views

Normalizing constants for Extreme value distributions

I have a question regarding the normalizing constants $\mu$ and $\sigma$ that appear in the following problem. Let the random variable $Y_n$ be $Y_n=max(a_1,a_{2},\cdots, a_n)$ and $X_{n}$ be ...
0
votes
1answer
40 views

Normalizing a dataset from the interval [0,1] with fixed properties.

So I have a rather large dataset where values are from the interval $[0,1] \in \mathbb{R}$. But the problem is that a big portion of the values are extremely close to $0$. So firstly I'm looking for ...
1
vote
1answer
19 views

Independence of distribution

Let there be a random matrix defined as $\mathbf{H}_1 = X + \boldsymbol\nu$, where, $X$ is deterministic and $\boldsymbol\nu$ is Gaussian white noise. Now let there be another random matrix defined as ...
0
votes
2answers
23 views

Expected Value with Parameter p

The random variable X has the following probability distribution: P[X=-1]= (1-p)/2 P[X=0]= 1/2 P[X=1]= p/2 The parameter p satisfies the inequality $0 < p < 1$. Find the expected value and ...
1
vote
1answer
55 views

What distribution is this?

Top: Uniform, Bottom: ?? Distribution. Ignore the random spikes - those are just binning errors. Looking for a distribution that is on $[0,1]$ and is equal to $0$ at $1$ and some positive ...
0
votes
1answer
19 views

Negative Binomial distribution as a Gamma mixture distribution

Let $f(x;\theta)$ be the poisson frequency function with mean $\lambda$. and $p(\lambda)$ the Gamma distribution with mean $\mu$, and variance $\mu^2/\alpha$. I have to show that ...
3
votes
1answer
59 views

What is the joint probability distribution of number of balls after $n$ draws?

The following problem came into my mind when I am studying the Polya Urn Model. At the beginning, from a bin containing $c_1$ balls labeled $1$, $c_2$ balls labeled $2$, … , $c_m$ balls labeled $m$, ...
-1
votes
0answers
34 views

A partial derivative wrt a limit of a double integral

I'm stuck with this problem. Is there a way to figure out the sign of the partial derivative $\dfrac{\partial x}{\partial a}$ where $x$ is the solution of the following equation: $$ ...
1
vote
2answers
46 views

If $X$ is distributed normally with mean $0$, is it correct to say $X$ and $-X$ “have the same distribution”?

Q: If $X$ is distributed normally with mean $0$, is it correct to say $X$ and $-X$ have the same distribution? In a way, this seems correct: both $X$ and $-X$ have the same probability density ...
0
votes
1answer
26 views

Using the Weibull Distribution, derive $E(X^k)$

If $X$~WEI$(\theta,\beta)$, derive $E(X^k)$ assuming $k\gt-\beta$. Note that $X$~WEI$(\theta,\beta)=\frac{\beta}{\theta^{\beta}}x^{\beta -1}e^{-({x}/{\theta})^{\beta}}$ I am having a very difficult ...
1
vote
0answers
20 views

Deriving joint distribution from expectation

Given two random variables $X$ and $Y$ and let $K$ be a constant value. Assume the expectation $\mathbb{E}[X(Y-K)^{+}]$ is given for all possible values of $K\geq 0$. Is there a way to derive the ...
2
votes
1answer
30 views

expected value and variance of the difference of number of people in a row.

I need to calculate the expected value and the variance of the following variable: $n$ people sit in a row, among them person 'a' and person 'b'. Define $X$ to be the amount of people between 'a' and ...
0
votes
0answers
7 views

What is a residual distribution vector

I have a general question about a probability distribution of the following form. Let us assume $\mu_0$ is a propability distribution on the set $A=\{1,2,\dots,n\}$ (in my case its the starting ...
4
votes
2answers
53 views

Homework problem - Ways to test if a density function is cumulative density function

I have a problem that states: Let $F : \mathbb R \to R$ be defined by $$F(x) =\begin{cases}e^{\frac{-1}{x}} &\text{if } x > 0\\ 0&\text{if } x \leq 0\end{cases}$$ Is $F$ a ...
0
votes
1answer
28 views

Conditional probability and distribution

Let Y ∼ Exp(1/5). Find P(Y ≤ 18|Y > 13). Could anyone give me any hints?
0
votes
1answer
32 views

Joint distribution of independent random variables

Say I have two independent random variables $X$ and $Y$ both having the exponential distribution. I.e. $f_X(x) = \lambda_1 e^{-\lambda_1 x}, \ x \ge 0, 0$ elsewhere $f_Y(y) = \lambda_2 e^{-\lambda_2 ...
5
votes
1answer
80 views

Proof of Interesting Binomial Identity

In my work I've come across the interesting binomial identity $$ \sum_{n\geq k} \frac{\binom{n}{k}}{\binom{m-1}{k}} \frac{\binom{m-1}{n} \binom{i-m-1}{j-n-1}}{\binom{i-2}{j-1}} = ...
-1
votes
0answers
29 views

Homework help - Random Variable min - can't understand what teacher wants me to do with problem

The problem is: Let X(1), . . . ,X n be independent random variables, with X(i) having an exponential with parameter λ(i) distribution, for any i. Then the distribution of the random variable X = ...
1
vote
6answers
169 views

Producing a CDF from a given PDF

So I have this PDF: $$ f(x)= \begin{cases} x + 3 & \text{ for } -3 \leq x < -2\\ 3 - x & \text{ for } 2 \leq x < 3\\ 0 & \text{ otherwise} \end{cases} $$ To make this a CDF, I ...
0
votes
1answer
40 views

Deducing F-distribution PDF

Let $V\sim \chi^2(n)$ and $W\sim \chi^2(m)$ indep. r.v. I want to find the PDF for $X=\frac{V/n}{W/m}$. For that I define $h(v,w)=(v,v/n\cdot m/w)=(v,x)$. So, $h^{-1}(v,x)=(v,\frac{v \cdot ...
0
votes
2answers
53 views

Homework help finding pdf's of y given pdf's of x - stuck

If anyone can give me the steps on how to find pdf$\,'$s of $y$ given $x$. Let X be a continuous random variable with probability density function given by $$ {\rm f}\left(x\right) ...
2
votes
2answers
61 views

Probability Distributions and Probability

Suppose $X \sim N(3, 4)$, and let $Y = X^2$. Find $\Pr(Y ≥ 12)$. What does $Y$ mean?
1
vote
1answer
18 views

Determine the values of c so that the following functions represent joint probability distributions of the random variables X and Y

Determine the values of c so that the following functions represent joint probability distributions of the random variables X and Y: f(x,y) = c x y, for x = 1,2,3; y = 1,2,3; f(x,y) = c|x-y|, for x ...
2
votes
3answers
87 views

distribution of infinite sum of $\sum (2x_n -1)/2^n$

$\{X_n\}\sim\mathrm{Bernoulli}(\frac {1}{2})$ $$Y=\sum_{n=0} ^{\infty} \frac {2X_n -1}{2^n}$$ Find the distribution of $Y$ $X_n$ are independent
0
votes
2answers
33 views

Probability of CDF and PDF [closed]

Suppose continuous random variable $X$ has a cumulative distribution function $FX$ satisfying $FX(x) = 2x^2 − x^4$ for $0 \leq x \leq 1$. (a) Compute $\displaystyle P\left(\frac{1}{4}\leq X \leq ...
0
votes
1answer
48 views

An exercise on quantile from Michael Wichura's notes

Please help me with this (source and context follows after the question). Thank you! Question: Let $F_1,\ldots,F_n,\ldots$ and $F$ be distribution functions with corresponding quantiles ...
0
votes
1answer
29 views

Limiting distribution of $n(T_n-4p^3(1-p))$

I want to find the limiting distribution of a $n(T_n-4p^3(1-p))$, where $T_n=\displaystyle\frac{4(n-t)t(t-1)(t-2)}{n(n-1)(n-2)(n-3)}$ with $t=\sum X_i$ is the UMVUE of $4p^3(1-p)$ that I found, where ...
0
votes
0answers
26 views

Notation related to Markov kernels

We wish to jointly construct two copies $(X_n)_{n \in \mathbb{N}}$ and $(Y_n)_{n \in \mathbb{N}}$ of a Markov chain on general state space, s.t. for $n=1,2,...$ $\mathcal{L}(X_{n+1}|X_n) = ...
4
votes
1answer
299 views

Maximum order statistic for Binomial distribution

Let $X_i$, $1\le i\le t$, be $t$ independent random variables with Binomial distribution $B(n,\frac1t)$. I would like to find the distribution of $X_{Max}=\max_{i=1}^t(X_i)$ Note that this is the ...
0
votes
1answer
15 views

Is squared Brownian Motion a gaussian process?

I am working at the following SP, given by $(X_t)_{t\geq0} = \alpha W_t^2+\beta t$ where $W_t$ is Brownian motion and $\alpha,\beta$ real. I managed to calculate mean and covariance function and now I ...
0
votes
1answer
28 views

Conditioning on independent coin tosses - general solution to brute force method?

Consider 10 independent tosses of a biased coin with a probability of heads, $p$. question (4d): find the probability there are 5 heads in first 8 tosses and 3 heads in last 5 tosses. I managed to ...
1
vote
2answers
31 views

When to use alternate parametrization of Gamma distribution?

In Loss Models, 4th ed., by Klugman et al., the following parametrization is given for the Gamma distribution: $$f(x) = \dfrac{(x/\theta)^{\alpha}e^{-x/\theta}}{x\Gamma(\alpha)}\text{.} $$ When ...
0
votes
1answer
32 views

Taking an integration with joint probability integrand.

I encounter a joint probability and I was wondering Am I allowed to rewrite it as an integral by using law of total probability? $$P( \{ f(X,Y) \le g(x,y)\} \cap \{Y>y\} ) = \text{?} = \int_{\xi = ...
1
vote
1answer
24 views

Expected value of normal distributed variable

I need to calculate the expected value of a modified normal distributed variable but i'm struggling. So maybe someone can help me. Suppose we've got a normal distributed variable $X \sim ...
1
vote
1answer
43 views

Coin toss with dynamic probabilities

So, I got a repeated experiment with two outcomes, i.e. a coin toss, but the probabilities might change every toss and are independent. Typically, they might come in sequences of the same ...
0
votes
1answer
16 views

How to determine the distribution of $U:=(X,Y,Z)$?

I've got a question concerning the distribution of a multi dimensional random variable. I know that $X$ and $Y$ and $Z$ are each normal distributed with certain expectations and variances. ...
4
votes
2answers
89 views

Distribution related to brownian bridge

Let $B(t)$ be a Brownian Bridge and $U$ is uniformly distributed on $(0,1)$. I wish to know the distribution function $B(U)$. Is it possible? As we know, $B(t)\sim N(0,t(1-t))$. But, I haven't a clue ...
1
vote
6answers
53 views

Distribution of a binomial variable squared

If I know $X$ is a binomial random variable, how can I find the distribution of $X$ squared (I know that $P(Y=y=x^2) = p(X=x)$ but does this distribution have a standard name)? In particular, how can ...
1
vote
1answer
57 views

Confused with estimator for random variables.

I am working on a practice exercise in preparation for a final this week. I am really stuck on the following problem: Let $X_1, X_2$ be a random sample for a population with the probability density ...
0
votes
2answers
45 views

If $X,Y$ ~$U(0,1)$ what is the distribution of $Z=0.5x^{2}+0.5y^{2}$?

I have some trouble with it.. the question is: $X,Y$ uniformly distributed $U(0,1)$ than $\frac{1}{2}(x^2+y^2) $~$exp(1)$... I am not even sure it is correct.. I know that if $X,Y$~$N(0,1)$ than it is ...
0
votes
1answer
24 views

Differentiation involving determinant

This question has arisen by following the proof in the appendix of Louis Liporace's paper on maximum-likelihood estimation, where the paper concerns classes of probabilistic functions (elliptically ...
1
vote
1answer
51 views

How to determine distribution

I hope you will be patient with the inarticulate question of a non-mathematician. It's hard to get an answer when you don't even know how to ask the question. But here goes... ...Actually, I have two ...
1
vote
1answer
29 views

Homework help with Standard Normal Distribution

I have a homework problem in which I'm not certain where to start: Let $X$ be a random variable with $N (0, 1)$ distribution. Show that $E(X^n) =\left\lbrace{\begin{array}{cc} 0 & \text{if $n$ ...
0
votes
0answers
14 views

Determine the multivariate distribution of $(\bar{Y_1}-\bar{Y_2},\bar{Y_1}-\bar{Y_3},\bar{Y_2}-\bar{Y_3})$

Assume you have a factor variable $A$ with $k=4$ groups and a normal distributed command variable fullfilling the condition $Y_i=\theta_{A_i}+U_i$ with independent $U_i\sim N(0,\sigma^2), ...