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

learn more… | top users | synonyms

0
votes
1answer
29 views

Analytic approach to find probability and total value of a set of independent events

I have a forecasting worksheet which describes a set (worksheet) of independent events, all of which have a likelihood of happening given as a probability (e.g. 0.7). Every event also has a yield ...
1
vote
1answer
22 views

For random variables $X$ and $Y$, $F_{X,Y}(x,y)=F_X(x)F_Y(y)$ if and only if $f_{X,Y}(x,y)=f_X(x)f_Y(y)$

I have seen either statement used as a definition of independent random variables. I was trying to prove their equivalence for discrete random variables. I am able to prove that if the joint density ...
-1
votes
0answers
19 views

normal distribution variable plus not normal distribution variable

I just started to learn normal distributions and learned that a variable is normal if it has a pdf looks like : [1/(sqrt(2pi)*sigma)]*e^(-(x-mu)^2/(2*sigma^2). Now, i have X~N(0,1) and I need to ...
0
votes
1answer
25 views

Special copula function

In a paper I encountered the following notation \begin{equation}P(Z\leq z,u\leq Y\leq v)=C(F_{Z}(z),F_{Y}(v)-F_{Y}(u))\end{equation} However I don't see why this holds in relation to uniform random ...
0
votes
0answers
33 views

$\mathrm{Pr}(X_1 + X_2 + \cdots + X_n \le n) .$ [duplicate]

Let $X_1,X_2,\ldots,X_n$ be a sequence of mutually independent random variables. For each $i$ with $1 \leq i \leq n$, we are given that the variable $X_i$ is equal to either $0$ or $n+1$, ...
1
vote
2answers
54 views

Must every probability distribution over a countable set be discrete?

Intuitively I expect this to follow from countable additivity, but there are ideas I can't rule out such as: Select a real number r from the uniform distribution over [0, 1]. If r is exactly 0.5, ...
0
votes
1answer
30 views

Tail of a sequence of RV

Consider the sequence of random variables $\{X_n\}_{n=1,2,\dotsc}$ where $X_n$ is gamma-distributed with shape $n$ and scale $1/n$ (or equivalently, $2nX_n$ is $\chi^2$-distributed with $2n$ degrees ...
0
votes
0answers
5 views

Stationary point of Unnormalized and Normalized KL-divergence minimization

I have encountered a problem, basically related to the http://arxiv.org/abs/1206.6679 . If I want to minimize the normalized KL-divergence KL(Q||P) with Q a multivariate Gaussian distribution. but in ...
0
votes
1answer
29 views

What is the distribution of the dot product of a Dirichlet vector with a fixed vector?

I am trying to get the distribution of a weighted sum when the weights are uncertain: $S = \sum\limits_{i=1}^N w_iC_i = \mathbf{w}\cdot \mathbf{C}$ where vector $\mathbf{w}$ is random with components ...
2
votes
1answer
93 views

Gumbel distribution

Let $(X_i)_{i \geq 1}$ be a sequence of i.i.d. normal $\mathcal{N}(0,1)$ random variables. Let $M_n = \max_{i=1,\ldots,n} X_i$. Show that $$P[\sqrt{2 \log n} M_n - 2 \log n \leq u ] \rightarrow ...
-1
votes
1answer
345 views

Find the joint probability distributed function of random variables

Suppose there are n i.i.d exponential random variables,say $X_{i},i=1,2,\cdots ,n$ with probability density function $$f(x)=\left\{\begin{matrix} e^{-x} &x\geqslant 0 \\ 0& x<0 ...
0
votes
0answers
43 views

chose uniformly at random from the n different brands, independently of previous orders [duplicate]

Michiel's Craft Beer Company (MCBC) sells $n$ different brands of India Pale Ale (IPA). When you place an order, MCBC sends you one bottle of IPA, chosen uniformly at random from the $n$ different ...
0
votes
3answers
57 views

Product of two random variables that has exponential distribution

I am trying to define the probability distribution of $Z$ such as $Z = X_1\cdot X_2$ where $X_1$ and $X_2$ are two independent and identically exponentially distributed variables. $$P(X_1=x) = ...
1
vote
0answers
19 views

Manipulating this probability distribution function

I have a probability distribution function as follows: $$ P(y|x,w, \phi) = \frac{\phi}{2\pi} \exp ^{-0.5 (y-t(x, w)'\phi (y-t(x,w)) } $$ Here $y$ and $x$ are two observed values. $\phi$ is also some ...
1
vote
2answers
21 views

Division of Objects into Different Sized Boxes

Suppose you have a set of N distinguishable boxes with lengths $l_1$,$l_2$...$l_N$. Suppose you try to divide x distinguishable objects among them, such that the probability of any object landing in ...
0
votes
2answers
24 views

Chains of random variables

Domain: estimating and project management. Let $A_1$ through $A_n$ be random variables with known properties (I'll elaborate later as required). Assume that each $A_i$ is independent but together ...
2
votes
0answers
31 views

A Lemma in the book “ Mathematical Method for financial markets” (Chapter 5, Section 5.7)

In page 307, Section 5.7, Chapter 5 of the book "mathematical methods for financial markets" by Jeanblanc, Yor and Chesney, Lemma 5.7.1 is given as follows: Lemma 5.7.1.1 Let $W$ be a Brownian ...
2
votes
1answer
48 views

Alice plays a game with $ \frac{1}{3} $ odds. Probability Question

Alice plays repeatedly a game that has three results: win, lose, or tie. Each time she plays, she wins with probability $\frac{1}{3}$ and loses with probability $\frac{1}{3}$ (and therefore she ties ...
0
votes
0answers
21 views

CDF of a MLE($\theta$) $min(X_1…x_n)$

ok so i'm doing a MLE problem and I've gotten to this step and my teacher told that it wasn't differentiable so the MLE was just $min(X_1,....X_n)$ but I have no idea how turn that into a CDF. the ...
0
votes
1answer
62 views

Probability of watching TV or Cartoons

so I am trying to solve this problem: Let X and Y equal the respective numbers of hours a randomly selected child watches movies or cartoons on TV during a certain month. From experience, it is know ...
0
votes
1answer
25 views

CDF of a density function with absolute value.

If X is a random variable with the density function $f(x)=\frac{e^{-|x|}}{2}$, what is the CDF of X? My first inclination is to take $\int_0^\infty \mathrm{e}^{x}/2\,\mathrm{d}x$ and ...
3
votes
2answers
39 views

Understanding the Beta-function

I always forget whether the beta function, B$(\alpha, \beta)$, is defined as $\Gamma(\alpha+\beta)/\Gamma(\alpha)\Gamma(\beta)$ or $\Gamma(\alpha)\Gamma(\beta)/\Gamma(\alpha+\beta)$. Is there an ...
1
vote
2answers
42 views

Prove variance in Uniform distribution (continuous)

I read in wikipedia article, variance is $\frac{1}{12}(b-a)^2$ , can anyone prove or show how can I derive this?
0
votes
0answers
33 views

Dependence between a joint probability distribution

Suppose $X$ and $Y$ are correlated random variables in a finite set ${\mathcal A}$, and let $f, g$ be functions that map elements from ${\mathcal A}$ to ${\mathcal B}$ for some finite set ${\mathcal ...
0
votes
0answers
5 views

Performance of an optimum estimator for Gaussian random variables used against Non-Gaussian random variables

Consider an optimum estimator for some parameter where the underlying distribution is following a Gaussian distribution with mean 'mu' and standard deviation 'sigma' (denoted by N(mu, sigma)). Let ...
1
vote
1answer
17 views

Exchangeability with random effects?

Consider a $N\times N$ random matrix $$ \epsilon:= \begin{bmatrix} \epsilon_{11} & \epsilon_{12} & \dots &\epsilon_{1N} \\ \epsilon_{21} & \epsilon_{22} & \dots & ...
0
votes
0answers
22 views

multiplying Gaussian distributions of different dimensions

The multiplication of multivariate Gaussian distributions defined over some parameter vector of a given dimension can be achieved by the following. Assuming that the Gaussian is parametrized by the ...
0
votes
1answer
60 views

Probability distribution of the product of random numbers

For applied mathematics to evolutionary biology I am often faced to have to describe a probability distribution function (PDF) which results from the product of a function in which a parameter is ...
-1
votes
1answer
14 views

What should be the headcount

In a call centre, we need to arrive at the minimum people to be recruited. The no of working days in a month is 26. Each employee is permitted to take 2 days off. What should be the minimum headcount ...
0
votes
1answer
28 views

Calculating Variance and Standard Deviation with probability distribution

The age [in years] $X$ of sewing machines to be reconditioned is a random variable with the following probability distribution: $f(x)=(1/972)x(18-x)$ for $0<x<18,$ and $f(x)=0,$ elsewhere. The ...
1
vote
0answers
21 views

Finding Variance and Sigma

Let $X$ be a random variable with $E(X)=3$ and $E[X(X-1)]=22$ $(i)$ Find $Var(X)$ I am not sure about this one. This is what I did though: $Var(X)=E[X^2]-E[X]^2$ $E[X(X-1)]=22$ ...
2
votes
2answers
71 views

Professor has 4 umbrellas, Markov chain and Probability

OK this problem is making me tear my hair out. I need someone to walk me through this in baby-steps method like 1 + 1 = 2. I am trying to figure out what I don't understand. I know this is going to be ...
0
votes
1answer
28 views

the random heights of north american women

The heights of North American women are normally distributed with a mean of 64 inches and a standard deviation of 2 inches. A random sample of four women is selected. What is the probability that the ...
0
votes
1answer
42 views

How do you transform Gamma to Chi-squared distribution

Here is the question not sure how to turn a Gamma into a Chi-Squared: Suppose $X_1....X_n$ is a sample from the distribution Gamma($\alpha=3,\ \lambda=\theta$) with unknown $\theta > 0$. We wish ...
1
vote
1answer
33 views

determining the size of a test bank given acceptable number of repeats

I have a question for a challenge that I'm trying to create - having some trouble quantifying the size of the challenge's test bank. 20 people are taking a challenge of 9 questions the test bank (n) ...
-1
votes
1answer
39 views

Probability & process [closed]

In this example we have problem involving permutation
0
votes
3answers
28 views

Conditional Expectation of rolling two dice.

I have to find $E(X\mid Y)(y)$ where $X$ is the value of the first roll and $Y$ is the sum of the two dice. I know that $$E(X|Y)(y) = \sum_x{xP(X\mid Y)}=\frac{\sum_xxP(X=x, Y=y)}{P(Y=y)},$$ but ...
0
votes
1answer
41 views

derive the mean and variance of $\bar X$ using means of sums rules

I can't find anywhere what the means of sums rules are so i'm confused with this question The random variables $X_1......X_5$ are jointly multivariate normal. Their expectations are $E(x)= \mu_i$ and ...
1
vote
1answer
166 views

Efficient calculation of the multivariate normal density function

The formula for the multivariate normal density function in the standard form contains $\Sigma^{-1}$ and the determinant of $\Sigma$, which are not very computationally friendly. Is it possible to ...
0
votes
1answer
25 views

Calculate $E[XY]$ of dependent variables

I'm having a little trouble whit a probabilistic exercise. The problem says this: There's a vase whit 10 marbles, 4 black and 6 white. Now I extract 2 of them without reposition. Being $X,Y$ random ...
0
votes
1answer
40 views

Gaussian Approximation of an intractable distribution

I am currently encountering this problem: I have an intractable distribution and I want to minimize the KL divergence of this distribution and a multivariate gaussian distribution. So we just need ...
1
vote
1answer
44 views

Determine a distribution with no parameters?

I'm confused by this question and I was hoping for some guidance some one to point me in the right direction Let $X_1.........X_n$ be a random sample from a population with mean $\mu$, that is ...
0
votes
1answer
17 views

Variance of a linear combinations

I was given the problem above. I am confused on how to find the variance of the linear combinations. A for example would have a mean of 22 correct? Can someone ...
0
votes
0answers
21 views

Limit in probability of $(\bar X)^2$

Hello can some one point me in the right direction with this question? Let $ F(x) = \begin{cases} 1- x^{-4} & x>1 \\ 0 & elsewhere \\ \end{cases} $ be the CDF of a random ...
1
vote
0answers
62 views

Determine the Maximum Likelihood Estimator of $\Theta$ and Consistency

Alright so I have been working with this one for a while and i'm not totally sure where to take it Let $X_1,.....X_n$ be a sample from a distribution with CDF $F(x;\theta)= 1-C/((x-\theta)+3)^4 for$ ...
1
vote
3answers
49 views

Random variable with infinite expectation but finite conditional expectation

I've been very stuck on a question from Probability and Random Processes by Grimmett and Stirzaker for ages - so stuck that I flicked to the back to have a look at the answers. But, I can't seem to ...
0
votes
0answers
15 views

Determine the value of constant $C_n$ such that $(p_k,k=1,…,n)$ is a probability measure on $\{1,…,n\}$

For $\theta\in[0,1]$ and $n\ge 2$, define a sequence $(p_k, k=1,...n)$ by $p_k= \begin{cases} C_n(1-\theta)\min(n,n-k) & if\ k=1,...,n-1\\ \theta & if\ k=n \end{cases}$ ...
0
votes
1answer
27 views

Which probability distribution is this?

Suppose we draw a number $x$ uniformly distributed on $(0,1)$, what is then the following distribution. Furthermore, calculate $F(y)$ and $f(y)$. $$y = \dfrac{x}{1-x}$$ This is a question I came ...
1
vote
0answers
28 views

expected value with integration

For the exponential distribution, $f(x)=(1/\theta) e^{-x/\theta}$ for $x>0,$ and $f(x)=0$ for $x \leq0$ $(i)$ Determine the exact value for the probability $P(0<X<3\theta).$ I need help ...
1
vote
1answer
20 views

Finding the density function from joint density function

I'm reading the conditional distributions section of Probability and Random Processes by Grimmett and Stirzaker and I've come across a brief exercise I can't seem to figure out. We're given earlier ...