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

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75 views

Finding probability density function $W=|Y-X|$.

Suppose $X,Y$ are independent random variables and have density function $U(0,1)$, how can calculate probability density function of $W=|Y-X|\,, 0<w<1$.
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0answers
613 views

probability Y> max of independent random variables with same distribution, different means

I am extremely enthused at finding this website. Thanks to Dilip Sarwate for some comments. I still don't have a final solution, however. Below is edited to better focus the problem. Here is my ...
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1answer
148 views

Probability of elements in a subset of the original set

Let me try and rephrase the question as an example. I'll use bits since its convenient in this case. You have 3 bits A, B and C, that have probability 1/2 of being ...
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1answer
153 views

Stopping time for sum of iid random variables.

Suppose we have $m$-sided biased die. Let $X_i$ be the outcome of the $i$'th roll with the die. Furthermore let $\mathbb{P}[X_i=k]=p_k$ with $k \in \{1,...,m\}$. We define $T=\min\{n\text{ ...
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3answers
125 views

Probability: Permutations

Consider the experiment of picking a random permutation $\pi$ on $\{1,2,...,n\}$, and define the associated random variable $f(\pi)$ as the number of fixed points of $\pi$, i.e, the number of $i$ ...
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2answers
111 views

Finding $E(Y)$ and $\mathrm{Var}(Y)$ given the conditional density of $Y$

Let $X$ have an exponential distribution with mean $\lambda$ and the conditional density of $Y$ given $X=x$ be $$ f(y|x) = \left\{ \begin{array}{l l} \frac{1}{x} & \quad \text{for ...
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1answer
618 views

Upper/lower bound on covariance two dependent random random variables.

X and Y are two dependent random variables. Marginal pmfs f(X) and f(Y) is given, but joint pmf f(X,Y) is not known. Is it possible to find upper/lower bound on covariance cov(X,Y)?
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1answer
19 views

prob on probability [closed]

Suppose there are 10000 soldiers screened for a rare blood disease with 1 out of 1000(p=0.001) Find the probability of 10 soldiers or more test to be positive for this disease. p[x>=10]=1-p[x<10] ...
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0answers
37 views

Large $k$ asymptotic of $\Pr(X=k)$ for a compound Poisson random variable $X$

Let $N \sim \operatorname{Poisson}(\mu)$, and let $X|N = \sum_{k=1}^N Y_k$, where $Y_k$ are iid non-negative integer-valued random variables. The distribution of $X$ is known as compound Poisson ...
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1answer
156 views

Additivity of the Chi-Square distribution

How do I use the moment generating function of the chi-square distribution to show that if $x_1\sim\chi^2(n_1), x_2\sim\chi^2(n_2),\dots ,x_k\sim \chi^2(n_k)$ then ...
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1answer
85 views

Behaviour of Two Coupled Sequences Towards a Stable Distribution

The following question arises from research that I am doing in swarm intelligence. The relationships given come from geometric considerations which, I believe, should not be relevant for this problem. ...
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2answers
37 views

Number of buses stopping till time $t$

Let $\displaystyle \left(X_n\right)_{n\geq 1}$ be i.i.d random variables on $\displaystyle \left(\Omega,\mathcal A,\mathbb P\right)$, $X_1$ being an exponential random variable with parameter $1$. ...
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1answer
48 views

Formula issues when working out chances of getting certain marks [duplicate]

$$P(X = k) = \binom{N}{k} (0.5)^k (0.5)^{N-k} = \binom{N}{k} (0.5)^N$$ Using formula above, I have got the following results for chances for getting certain percentage on a $50$ question paper, each ...
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1answer
28 views

Confusion related to calculating the probability distribution of a variable

I have this confusion related to calculating the probability distribution of a variable. If I have a variable x1 which has a pdf p(x1).Lets assume that the distribution is gaussian with mean X1. I ...
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3answers
152 views

How to correctly write a binomial distribution for a $50$ questions exam [duplicate]

Using binomial distribution I want to know what is the chance of getting $70\%$ or greater in a $50$ question exam, each question having a true/false option to select from. What is the correct formula ...
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1answer
110 views

Skewness of mixture density

I have the mixture density of two normal distributions: \begin{align} f(l)=\pi \phi(l;\mu_1,\sigma^2_1)+(1-\pi)\phi(l;\mu_2,\sigma^2_2) \end{align} The skewness is given by \begin{align} ...
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2answers
58 views

Find the covariance of $\sum_{i=1}^{n} a_i X_i$ and $\ T_2=\sum_{i=1}^{n} |X_i|$

Suppose $X_1,\ldots,X_n$ is a random sample from density function $$f(x,\theta)=\displaystyle\frac{1}{2\theta}e^{-{|x|}/{\theta}}, \qquad -\infty<x<\infty$$ If $T_1=\sum_{i=1}^{n} a_i X_i ,\ ...
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2answers
59 views

Integral question related to i.i.d. random variables

I am stuck on this question, I would appreciate any hint I can get to understand this! We have $(\Omega, M, P)$ a probability space. $X_1$ and $X_2$ are i.i.d. and let $S_2 = X_1 + X_2$. For given ...
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0answers
86 views

Finding Fisher Information matrix for AMH Copula

I wanted to calculate the Fisher Information Matrix for a bivariate AMH copula(with 1 parameter) which has Weibull distributed marginals. What I have been doing till now is that I have been replacing ...
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0answers
108 views

Poisson distribution conditional on Poisson-distributed mean

recently I came across a problem in which the probability of a random variable $n \sim \text{Pois}(k)$ is conditional on $k \sim \text{Pois}(\mu)$. After some trying I figured out that the conditional ...
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1answer
1k views

Probability generating function of the negative binomial distribution.

I am using the definition of the negative binomial distribution from here. This is the same definition that Matlab uses. For convenience, $$P(k) = {r + k -1 \choose k}p^r(1-p)^k ,$$ where $p$ is ...
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1answer
281 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 ...
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1answer
2k views

Distribution of a difference of two Uniform random variables?

Let $X$ and $Y$ both be distributed between $[1,2]$, what is the distribution of $Z=X-Y$?
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1answer
75 views

Independence of two products of random variables

Consider the following problem: $$z_1 = a_1 x_1$$ $$z_2 = a_2 x_2$$ where $a_1, a_2$ are i.i.d. (regardless of their distribution; in the actual case study it is a symmetric Bernoulli distribution ...
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0answers
115 views

Approximate CDF of the sum of a gaussian and a truncated gaussian

I am looking for a quick-to-compute approximation of the CDF of $X+Y$, where $X \sim N(0,\sigma_1^2)$ and $Y$ is a truncated gaussian, more specifically, a gaussian with mean $0$, standard deviation ...
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0answers
44 views

is this binomial distribution correct?

I am trying to work out what is the chances of getting the following marks $(100\%, 70\%, 60\%, 50\%)$ in a paper containing $50$ questions, each question containing yes/no options. Using Binomial ...
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1answer
234 views

Exponential probability density functions of independent variables

I just have a small technical question. I am in the midst of solving a problem where I have gotten 2 different exponential probability density functions that are as follows: pdf #1: 3e^(-3x) pdf #2: ...
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1answer
394 views

Find the conditional probability density function.

If I am given X that follows an exponential distribution with mean m and Y that follows a poisson distribution with mean n, how can I use them to find the conditional probability density function of X ...
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1answer
145 views

Conditional probability problem with urn

An urn contains 3 white, 6 red, and 5 black balls. Six of these balls are randomly selected from the urn. Let $X$= number of white balls selected and $Y$= number of black balls selected. Compute the ...
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64 views

Convergence in distribution and ordered statistics

Let $X_1,X_2,\ldots,X_n$ be i.i.d from some distribution $F$. Denote the ordered statistic of the previous as $X_{(1)}, X_{(2)},\ldots, X_{(n)}$. For $t \in \mathbb R$, define $Z_n = \frac{i}{n}$ if ...
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1answer
107 views

Expected value identities

If I'm given the expected value of two random variables, say A and B is 0 and C = 5*A + B. How would I find the expected value of C? And if the variance of A and B is 1, the variance of C? There is ...
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1answer
203 views

Finding covariance from joint distribution

If I'm given a joint distribution of 2 random variables say A and B, how would I find the covariance of A,B? Example joint distribution: ...
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1answer
238 views

Joint moment generating function

How to prove that if joint moment generating functions of $n$ random variables is equal to the product of each of their moment generating functions then the random variables are independent.
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1answer
95 views

Does $0$ correlation imply independence for marginally normal distributions?

Assume $X \sim \mathcal N(\mu_1, \sigma_1^2)$ and $Y \sim \mathcal N(\mu_2, \sigma_2^2)$. If $\rho_{X,Y} = 0$ then $X \bot Y$. Can someone give a hint why this is true ?
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2answers
106 views

Reference on Polynomial Chaos

I need to understand the basics of "Polynomial Chaos" (http://en.wikipedia.org/wiki/Polynomial_chaos), and I'm having trouble finding a good reference on it. I'm looking for something rigorous enough ...
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1answer
150 views

How is conditional density function with two given conditions ($f_{X\mid Y,Z}(x\mid y,z)$) defined?

Let $X$, $Y$ and $Z$ be random variables. Given this conditional density function with two conditions; $Y=y$ and $Z=z$: $$ f_{X\mid Y,Z}(x \mid y, z) = f_{X\mid Y,Z}(x \mid Y=y, Z=z) $$ I have a ...
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1answer
73 views

Probability distribution of a random components vector

The vector $\rho$ has components: $(\rho_x,\rho_y,\rho_z)$ in a three - dimensional cartesian reference frame, where: $$\rho_x=a+\nu_x,\rho_y=b+\nu_y,\rho_z=c+\nu_z$$ with: $a,b,c$ constants and ...
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1answer
391 views

Generating random numbers with skewed distribution

I want to generate random numbers with skewed distribution. But I have only following information about distribution from the paper : skewed distribution where the value is 1 with probability 0.9 ...
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1answer
421 views

Joint density of order statistics $f_{X_{(1)}X_{(n)}}(x,y)$ with combinatorics

Problem I need to find $f_{X_{(1)}X_{(n)}}(x,y)$ for the uniform distribution. $X_{(k)}$ denotes the $k^{\text{th}}$ smallest from an n-sample. I already know that the answer is ...
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1answer
8k views

Lambda value of Poisson distribution

I'm a bit confused about the lambda value of a Poisson distribution. I know it means the average rate of success for a given interval. I'm confused about what this value exactly means through. For ...
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2answers
38 views

Using join probability distribution

Say I'm given a probability distribution of two random variables $A$ and $B$. What does it mean to calculate the join probability distribution of $3^{(A-B)}$? The distribution is in fact discrete.
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277 views

Is this question solvable? $2$ non-linear equations and the proof that the solution is unique (with asymmetric bounty option)

As mentioned in the title I want to show the uniqueness of the solution to $2$ non-linear equations. However, it seems that I can not solve this question with my current mathematical knowledge. More ...
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49 views

How do you explain $f(x_4|x_3)f(x_3|x_2)f(x_2|x_1)f(x_1) = f(x_4,x_3,x_2,x_1)$?

Let $x_1=x(n_1)$, $x_2=x(n_2)$, $x_3=x(n_3)$ and $x_4=x(n_4)$ be random Markov processes $(n_1 < n_2 < n_3 < n_4)$. I don't understand the identity given below on their probability density ...
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1answer
139 views

Do Convergence in Distribution and Convergence of the Variances determine the Variance of the Limit?

Suppose we have a sequence $(X_n)_{n\in\mathbb{N}}$ that satisfies: $X_n \rightarrow_d X$, for $n\rightarrow \infty$, where $\rightarrow_d$ denotes convergence in distribution; $\mathrm{Var}(X_n)$ ...
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2answers
152 views

Probability of Event Occurring in Time-Series

So I'm trying to model email openings over the course of a day. For example, I have a data set for one individual that simply has a bunch of time-stamps of when he has opened an email. I don't care ...
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1answer
73 views

Expectation of a multipart function of a random variable

Let $Q$ be a random variable with known CDF: $P(Q \leq q) = F_{Q}(q)$, say, exponential with known $\lambda$. Which is the expectation of $R$, $E[R]$, and the corresponding CDF $F_{R}(r)$, where $R$ ...
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1answer
28 views

probability space over the coin flips

I recently read the definition of differential privacy, which is as follows: a randomized function $K$ gives $\epsilon$-differential privacy if for all data sets $D$ and $D'$ differing in at most ...
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2answers
230 views

Cumulative distribution function & expectation

Let a be a real number and f: $$f:\mathbb{R}\rightarrow \mathbb{R}, \ f(x) = \begin{cases} a3^x & \text{for } x < 0\\ 1& \text{for } x =0 \\ a3^{-x} & \text{for } x > 0\end{cases}$$ ...
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1answer
75 views

negative parameters in a beta distribution

I have a set of observations of credit loss data, where the mean is 37% and variance 25%. Now, I have to find the distribution and the base assumption is it will follow a beta distribution. the issue ...
2
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2answers
802 views

conditional distribution of random variable given its sum with another random variable

I am trying to figure out the following problem: I have two random variables: $X$ with pdf $f_X(x)$ on $[0,A]$ and $Y$ with pdf $g_Y(y)$ on $[0,B]$. Let's denote $Z=X+Y$. What should be the ...