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

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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 ...
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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. ...
1
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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 ...
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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 ...
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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 ...
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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 ...
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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 $n$ ...
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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 ...
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2answers
59 views

Finding the probability an equation has real roots.

If $Q$~UNIF$(0,3)$, find the probability that the roots of the equation $g(t)=0$ are real, where $g(t)=4t^2+4Qt+Q+2$. There was a similar question asked that I looked at, but I am still a little ...
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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 ...
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0answers
15 views

Kernel density estimation of a divergent probability density function

I'm working with a 2D probability distribution function (pdf) that will be something like $$P\left(r,\theta\right)\approx\frac{3}{\pi^3}\frac{1}{e^{r}-1},$$ when written in polar coordinates (i.e. ...
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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}} = ...
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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), ...
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2answers
47 views

Deriving Moment Generating Function of the Negative Binomial?

My textbook did the derivation for the binomial distribution, but omitted the derivations for the Negative Binomial Distribution. I know it is supposed to be similar to the Geometric, but it is not ...
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1answer
49 views

Interesting Problem - Computing CDF

A rv X is an exponential distribution with parameter 1 and Y is a uniform distribution between 0 and 1. X and Y are independent. Define Z = min {X, Y}. Compute the CDF of Z ? I really have no idea ...
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0answers
8 views

Is it possible to get the PDFs of each of the three vector components knowing the PDF of the modulus if isotropy is guaranteed?

The PDF of a given vectorial quantity modulus is known. I would like to obtain the PDF of each of the three vector components in the case of isotropy, i.e. the three PDFs are supposed to be equal and ...
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0answers
36 views

One double integral elated problem

The bit I am stuck is the limits in the double integral. I tried X from 0 to uy and Y from 0 to infinity, this is obviously incorrect. I just want to know the complete double integral in the order ...
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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 ...
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1answer
39 views

Compute a conditional probability of normal random variable

Suppose $X, T$ are continuous random variables, and $X \sim \mathcal{N}(0, 1)$, $T$ have density function $f_T$. (But $X,T$ do not have joint density) Is there any way to compute the following ...
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1answer
17 views

Covariance of random variables with identical distribution.

Let $X_1,...,X_n$ be random variables with identical distribution, and for all $i=1,...,n$ $\mathrm{Var}(X_i)$ exist. 1. Show that the covariance between each two random variables exist. 2. Show that ...
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3answers
62 views

Let $U$ be $~U [0,1] $and let $Y = U^{\frac{1}{2}}$

Let $U$ be $\sim \mathcal{U}[0,1]$ and let $Y = U^{1/2}$. I'm having trouble finding the $E(Y)$. How do I go about doing this?
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2answers
25 views

probability-distribution that has its mode equal median

Could anyone tell me any asymmetric distribution whose mode=median? Thanks in advance.
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2answers
33 views

Math probability combination explanation

A group of four components is known to contain two defectives. An inspector tests the components one at a time until the two defectives are located. once she locates the two defectives, she stops ...
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1answer
43 views

Estimate arrival time of a ship given the average of the ships in a day in a Poisson Distribution

I'm working in a simulation of a Port where ships come to specific stations of the port. I already know that the average amount of ships is given by a Poisson distribution and the service time (On ...
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0answers
32 views

Convergence of probability density functions

Assume that a sequence of random variables, $(X_t)_{t\geq 0}$, converges in distribution to a random variable $X_0$, as $t\to 0$. Also assume that $X_t$ and $X_0$ have $C^{\infty}$-probability density ...
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1answer
44 views

binomial coefficient: maximum value

For $n\rightarrow \infty$ we consider $$f(p)=\sum_{j=c}^n {n\choose j} p^j (1-p)^{n-j}.$$ We are interested in $\hat{p}:=\arg \max_p f(p)$. Can we say something about $\hat{p}$ dependent on $n$ and ...
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2answers
133 views

Lies, damned lies, and statistics

A story currently in the U.S. news is that an organization has (in)conveniently had several specific hard disk drives fail within the same short period of time. The question is what is the likelihood ...
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2answers
38 views

Does a continuous probability density function (pdf) have zero values on +infinity and -infinity?

Assume a pdf $f(x)$ is continuous along $-\infty$ to $+\infty$. Does this assumption guarantee that $f(+\infty)=f(-\infty)=0$? How to prove? Thanks in advance.
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0answers
16 views

Vysochanskij Petunin vs. Cantelli inequality for random variables

The well known Cantelli inequality states: $$Pr(|X-\mu|\ge\alpha)\le\frac{2\sigma^2}{\sigma^2+\alpha^2}$$ where $X$ is a real valued random variable, $\mu$ the mean value and $\sigma^2$ the variance ...
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1answer
23 views

monotonicity of binomial coefficient

I am interested in $$f(x):={k-1 \choose x-1} p^{x} (1-p)^{k-x}.$$ How do I find out in which Domain this function is monotonically increasing, in which it is monotonically decreasing? For which $x$ ...
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1answer
19 views

Cumulative Distribution Function and

The demand, $X$, for a firm’s product is a random variable with density $f(x) = 2x$ for $0 ≤ x ≤ 1$. The corresponding cumulative distribution function is $F (x) = x^2$ for $0 ≤ x ≤ 1$. The firm’s ...
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1answer
39 views

P(X>Y) Probability Double Integral

$f(x,y) = \frac{12}{7(x^2 + xy)}$ $ 0 \le x \le 1$ and $0 \le y \le 1 $ I want to know the $P(X>Y)$. I believe the correct solution to this is integrating from 0 to 1 for dy and y to 1 for dx ...
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0answers
37 views

Sums of Power Law random variables

Suppose $F$ be a pareto distribution with scale parameter $x_m$ and shape parameter $\alpha$. Assume $X_1, X_2 , ..., X_n$ are iid random variables drawn from $F$. Let $S_n(k) = X_1 ^k + X_2 ^k + ...
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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 ...
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3answers
42 views

Expected value of moment generating functions: [closed]

How do I do these? I don't understand any of them.. especially the last two. I'm studying for a final soon and need help. I recognize that they are distributions but how do I answer the question?
3
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1answer
58 views

Precise definition of the support of a random variable

I am reading lecture notes which contradict my understanding of random variables. Suppose we have a probability space $(\Omega, \mathcal{F}, Pr)$, where $\Omega$ is the set of outcomes ...
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0answers
39 views

Probabilistic fragmentation

Suppose we have the following problem: We start with an interval of length $1$ and break it into two intervals of lengths $r$ and $1-r$, where $r$ is a random variable in $[0,1]$ with probability ...
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1answer
30 views

If $X,Y$ are independent and geometric, then $Z=\min(X,Y)$ is also geometric

Let $X,Y$ be independent geometric random variables with parameters $\lambda$ and $\mu$. If $Z=\min(X,Y)$. Show that $Z$ is geometric and find its parameter. (Answer $\lambda\mu$) $\displaystyle ...
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1answer
17 views

Correlation coefficient of i.i.d variables

Let $X_1, X_2, X_3, ...$ be i.i.d variables, and for every $i$ $X_i$ has variance. Define $S_k=\sum_{i=1}^{k}X_i$. Calculate $\rho(S_m,S_n)$ for $m\leq n$. Well, I know it should be $\sqrt{ m/n }$, ...
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1answer
34 views

What are the different ways of indicating that a random variable has a specific distribution?

Recently I have seen random variable distributions described in two ways: $$ X \sim Nb(r,p) \\ X \stackrel{d}{=} Nb(r,p) $$ Both indicating that $X$ is a negative binomial random variable with $r$ ...
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1answer
24 views

Probability “average” understanding

This is more of a problem understanding probabilities than an actual question. In a game I am playing I can use a certain item to try to unlock different levels. The item will unlock a new level ...
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0answers
24 views

Extension of Slutsky's Theorem

I regard random variables $X_n$ and $Y_n$ with $(X_n+Y_n) \rightarrow (X+Y)$ (in distribution for $n \to \infty$). Furthermore there exist random variables $(a_n) \rightarrow 1$ and $(b_n) \rightarrow ...
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0answers
49 views

Conditioning on function of random variable and random variable itself

Suppose that $Y_{i}\in\{0,1\}$ is a binary variable, and $X_{i}$ is some random vector in $\mathbb{R}^{d}$ . Why can we say the following: \begin{eqnarray*} ...
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1answer
35 views

Confused how to calculate continous random variable with pdf that has a min

The problem given was: Let $X$ be a continuous random variable with probability density function $$f(x) = \dfrac 1 4 \min \left( 1, \dfrac 1 {x^2} \right)$$ Find $P(−2 \le X \le 4)$. The ...
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1answer
29 views

Finding Probabilities of Distribution Functions

I recently turned in an assignment and had an error on it, or so I'm told, I'm not entirely convinced just yet. The problem was as follows: $$F(x) =\begin{cases}1-\frac{16}{x^2}, & x\ge4 \\ 0, ...
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0answers
50 views

What does “sequence is equidistributed in [0, 2]” mean?

I was reading an article in which they are mentioning this sentence: "sequence is equidistributed in [0, 2]" where the sequence in question, is a sequence of real number (the article in question is ...
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0answers
20 views

The distribution of ratio of two shifted gamma

I am wondering if anyone can help me to find the ratio of this distribution. Assume $S$ and $T$ are independent, where $S\sim Gamma(n-1/2, 4(1+\rho)\sigma^2)$ $S\sim Gamma(n-1/2, ...
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1answer
40 views

Square root of Chi-square distribution tends to $N(0,1)$

The question requires to show that $\sqrt{2\chi^2_n}-\sqrt{2n}$ converges in distribution to $N(0,1)$ as $n \rightarrow \infty$, which I dont know how to proceed. The question also has a first part ...
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1answer
33 views

Expected value of series of uniformly converges random variables [duplicate]

Let $X_1,X_2,X_3,...$ a series of i.i.d. variables with $X_i \sim \mathcal{U}(0,1)$. Let $N=\inf\{n\mid \sum_{i=1}^{n}X_i\geq1\}$ Prove that $E(N)=e$. I don't really have a clue how to even start ...
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0answers
31 views

Finding $p$ of the binomial cdf…

Please bear with me, I'm only a biologist ^.^: I have a need of solving this cdf so as I can plug in known values $Pr, n, k$, and get an answer for $p$. $$f(k;n,p) = Pr(X\le k) = \sum_{i = ...