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

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4
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2answers
1k views

Absolute continuity of a distribution function

This appeared on an exam I took. $Z \sim \text{Uniform}[0, 2\pi]$, and $X = \cos Z$ and $Y = \sin Z$. Let $F_{XY}$ denote the joint distribution function of $X$ and $Y$. Calculate $\mathbb{P}\left[...
3
votes
4answers
459 views

Probability for “drawing balls from urn”

I'm afraid I need a little help with the following: In an urn there are $N$ balls, of which $N-2$ are red and the remaining are blue. Person $A$ draws $k$ balls, so that the first $k-1$ are red ...
3
votes
2answers
1k views

Find the expectation

A box contain $A$ white and $B$ black balls and $C$ balls are drawn, then the expected value of the number of white balls drawn is ? The answer is $\large \frac{ca}{a+b}$. How to approach this one?
3
votes
2answers
2k views

Projection of a 3D spherical distribution function in to a 2D cartesian plane

Consider a 3D spherical Gaussian distribution function that depends on radius only, $$f(r) = \frac{1}{N} e^{-(\frac{r-R_\mu}{\sigma})^2}$$ where $R_\mu$ is the radial offset of the distribution and $...
3
votes
2answers
191 views

A variation on the $F$-distribution

If I have $\frac{X/n_1}{Y/n_2}$ where $X$ and $Y$ are independent chi-squared random variables, with degrees of freedom $n_1$ and $n_2$, respectively, then the distribution of this ratio is given by ...
3
votes
1answer
1k views

Result and proof on the conditional expectation of the product of two random variables

My problem is the following: $X$ and $Y$ are two random variables and $\mathcal{F}$ is a $\sigma$-algebra. Given that $X$ and $Y$ are independent, and that $X$ is independent of $\mathcal{F}$, can I ...
3
votes
3answers
1k views

prove that any positive integer-valued random variable with memoryless property has the geometric distribution for some $p$

How to prove that any positive integer-valued random variable with memoryless property has the geometric distribution for some $p$. By memoryless property, $$P(X=i+s | X>i)=P(X=s)$$ How to get ...
3
votes
1answer
48 views

How to calculate expected number of trials of this geometric distribution

I understand why the expected number of trials until there is a success is given by $$ \sum_{i=0}^{\infty} i p q^{i-1} \ = \ E[\text{number of trials until} \ X=1] = \frac{1}{p} $$ where $p$ is ...
3
votes
2answers
794 views

Uniform measure on the rationals between 0 and 1

I am trying to think of a probability measure on the set of rationals between 0 and 1 ($X:=\mathbb{Q}\cap[0,1]$). I want to achieve something like a uniform measure, i.e. every number should have the ...
3
votes
2answers
5k views

PDF of product of variables?

could anyone please indicate a general strategy (if there is any) to get the PDF (or CDF) of the product of two random variables, each having known distributions and limits? After having scanned ...
3
votes
2answers
81 views

Evolution of a discrete distribution of probability

I am designing a virtual card game and I defined an evolution of probabilities, but I don't have the knowledge on this matter to find out how they will evolve. I hope you help me here, with ...
3
votes
1answer
159 views

Random sample from discrete distribution. Find an unbiased estimator.

$X$ is a discrete random variable with parameter $a > 0$ whose pmf is defined as: $$ f_X(x) = \begin{cases}0.2, &x = a\\0.3, &x = 6a\\0.5, &x = 10a\end{cases} $$ Say we have a random ...
3
votes
1answer
207 views

How to prove these two random variables are independent?

If $X$ and $Y$ are independent Gamma random variables with parameters $(\alpha,\lambda)$ and $(\beta,\lambda)$ respectively, how to show that $U=X+Y$ and $V=X/(X+Y)$ are independent?
3
votes
1answer
194 views

Normal approximation of tail probability in binomial distribution

From the Berry Esseen theorem I know, that $$\sup_{x\in\mathbb R}|P(B_n \le x)-\Phi(x)|\in O\left(\frac 1{\sqrt n}\right)$$ whereby $B_n$ has the standardized binomial distribution and $N$ has the ...
2
votes
1answer
710 views

Probability distribution of tossing a coin until obtaining $k$ heads

My question is the following. We toss a coin, for which probability of obtaining heads is $p \in (0,1]$, until we obtain $k$ heads, not necessarily in a row (generally $k$ heads). Let $X$ be a number ...
2
votes
1answer
140 views

If the sum of two i.i.d. random variables is normal, must the variables themselves be normal?

It is well known that if two i.i.d. random variables are normally distributed, their sum is also normally distributed. Is the converse also true? That is, suppose $X$ and $Y$ are two i.i.d. random ...
2
votes
1answer
226 views

If $ X = \sqrt{Y_{1} Y_{2}} $, then find a multiple of $ X $ that is an unbiased estimator for $ \theta $.

Problem: Suppose that $ (Y_{1},Y_{2},Y_{3},Y_{4}) $ denotes a random sample of size $ 4 $ from a population with an exponential distribution whose probability density function $ f $ is given by $$ f(...
2
votes
1answer
83 views

Do not exist IID random variables $X, Y$ such that $X-Y \sim U[-1,1]$

This is an exercise from Williams, Proability with martingales. Prove that if $Z$ has the $U[-1,1]$ distribution, then $$\phi_Z(t) = \frac{\sin t}{t}$$ Then prove that do not exist IID random ...
2
votes
0answers
35 views

Joint Density and Covariance between Two Random Variables with the same Mean and Variance

This seems like a deceptively simple question, (and it perhaps is and I am missing something) but I could not find anything on this. Q1) Are there any general results / relationships to get the ...
2
votes
3answers
1k views

estimate population percentage within an interval, given a small sample

Given a small sample from a normally-distributed population, how do I calculate the confidence that a specified percentage of the population is within some bounds [A,B]? To make it concrete, if I ...
2
votes
2answers
432 views

Distance between the product of marginal distributions and the joint distribution

Given a joint distribution $P(A,B,C)$, we can compute various marginal distributions. Now suppose: \begin{align} P1(A,B,C) &= P(A) P(B) P(C) \\ P2(A,B,C) &= P(A,B) P(C) \\ P3(A,B,C) &= P(...
2
votes
1answer
269 views

joint distribution of random vector

I want to find the joint distribution of the random vector $(W_t, \int_0^t W_s \; \mathrm ds)$ where $W_t$ is Brownian motion. I know $W_t \sim N(0,t)$, but I don't know how to calculate the ...
2
votes
0answers
679 views

Probability that a normal distribution is greater than two others

Given 3 independent variables with normal distributions, how can I calculate the probability that one of them will be greater than the other two simultaneously? So, how to calculate $P ((A>B) \...
2
votes
2answers
1k views

Singular Distribution

In reading section 2.2, page 14 of this book, I came across the term "singular distribution". Apparently, a multivariate Gaussian distribution is singular if and only if it's covariance matrix is ...
2
votes
2answers
2k views

Convolution of multiple probability density functions

I have a series of tasks where when one task finishes the next task runs, until all of the tasks are done. I need to find the probability that everything will be finished at different points in time. ...
2
votes
0answers
126 views

Minimum / Maximum and other Advanced Properties of the Covariance of Two Random Variables [closed]

Are there any advanced results established regarding the behavior of the Covariance of two random variables other than the bounds on the correlation and independence when it is zero etc. which are ...
2
votes
0answers
65 views

Simple question on conditional probabilities (multidimensional normal distributions)

Let $X$ and $Y$ in $\Bbb{R}^n$ be two random vectors. We assume that $X\mid Y\sim\mathcal{N}(Y,\Sigma_X)$ and $Y\sim\mathcal{N}(\mu_Y,\Sigma_Y)$ The goal is to sample from the distribution of $X$. ...
2
votes
2answers
333 views

Random sums of iid Uniform random variables

Let $\{X_r : r\ge 1\}$ be independently and uniformly distributed on $[0,1]$. Let $0<x<1$ and define $$N=\min\{n\ge 1 : X_1 + X_2 +\ldots+X_n> x\}$$ Show that $$P(N>n) = \frac{x^n}{n!}$$...
2
votes
1answer
120 views

Likelihood Functon.

$n$ random variables or a random sample of size $n$ $\quad X_1,X_2,\ldots,X_n$ assume a particular value $\quad x_1,x_2,\ldots,x_n$ . What does it mean? The set $\quad x_1,x_2,\ldots,x_n$ ...
2
votes
1answer
118 views

the joint distribution of dependent random variables

Let $X \sim N(\mu_1, \sigma_1)$, $Y \sim N(\mu_2, \sigma_2)$, $Z \sim N(\mu_3, \sigma_3)$. I want to derive a joint distribution for $X/(X+Y+Z)$ and $Y/(X+Y+Z)$. Since two random variables i.e. $X/(...
2
votes
1answer
54 views

How do you Taylor expand the log likelihood function of the Poisson distribution?

This question is an extension to this previous question asked by myself: When doing a maximum likelihood fit, we often take a ‘Gaussian approximation’. This problem works through the case of a ...
2
votes
3answers
144 views

How to compute the powers of $2\times2$ Markov matrices

Consider a Markov chain $(X_n)$ with state space $E=\{1,2\}$. If given a transition matrix $$P=\pmatrix{1-a&a\\b&1-b}\;,$$ with $0<a,b<1$. How to find out the $n$-th power to the ...
2
votes
0answers
77 views

Skellam CDF Increasing vs Decreasing in a parameter

I'm working with the following Poisson difference distribution: $$\text{Prob}\{X_1-X_2 \geq 0\} $$ where $X_1 \sim$ Poisson $(\mu_1)$ is independent from $X_2 \sim$ Poisson $(\mu_2)$. I need to ...
2
votes
1answer
69 views

Limit of a multiple integral [closed]

$$\displaystyle\lim\limits_{n\to \infty} \int_0^1 \int_0^1...\int_0^1 \cos^2\left(\frac{\pi}{2n}(x_1+x_2+...x_n)\right)dx_1 dx_2...dx_n$$ I don't know how to begin.
2
votes
1answer
541 views

Conditional expectation of an uniformly distributed random variable

Suppose $U_1, \ldots, U_n$ are i.i.d. random variables with $U_1$ distributed uniformly on the interval $(-1, 1)$. Compute $\mathbb{E}(U_1 + \ldots + U_n |\max(U_1, \ldots, U_n) = t)$ for $t \in (-1, ...
2
votes
1answer
303 views

Joint probability distribution (over unit circle)

A couple of two continuous random variables $(X,Y)$ is distributed uniformly over the closed unity circle (so $-1\leq x \leq 1$ , $y$ analog). $U$ is defined as the distance from $O$ to the point $(X,...
2
votes
1answer
86 views

Joint pdf of discrete and continuous random variables

Consider two independent random variables $X$ and $Y$, where $X$ is uniformly distributed on the interval $[0,1]$ and $Y$ is uniformly distributed on the set $\{0,1\}$. Thus, the cdfs are given by $...
2
votes
3answers
108 views

Distribution of the sum of $N$ loaded dice rolls

I would like to calculate the probability distribution of the sum of all the faces of $N$ dice rolls. The face probabilities ${p_i}$ are know, but are not $1 \over 6$. I have found answers for the ...
2
votes
2answers
116 views

What is the pmf of rolling a die until obtaining three consecutive $6$s?

I am trying to find the pmf of rolling a die until 3 consecutive 6s turn up. I am able to find the expected value using a tree diagram, but the pmf is not obvious to me. Let A be the event of not ...
1
vote
1answer
1k views

Find the probability mass function of the (discrete) random variable $X = Int(nU) + 1$. [duplicate]

For a non-negative real number $x$, write $Int(x)$ for the largest integer that is less than or equal to $x$. Let $U$ be a uniform random variable on $(0,1)$ and $n \geq 1$ an integer. Find the ...
1
vote
1answer
70 views

Is there a name for this family of probability distributions?

I am wondering whether a family of probability distributions with the following form of a density function has a name: $$f(x)=C*\operatorname{Exp}(-B|x|^A)$$ where $A$, $B$ and $C$ are positive ...
1
vote
1answer
3k views

maximum of two uniform distributions

I have a question. Let's suppose that the two random variables $X1$ and $X2$ follow two Uniform distributions that are independent but have different parameters: $X1 \sim Uniform(l1, u1)$ $X2 \sim ...
1
vote
2answers
65 views

Sums of independent random variables (more than two) [closed]

I read that the convolution of two iid random variables is $$(f * g) (z) = \int f(z-y) g(y) dy$$ What is the general formula for more than two RVs? For example, for three RVs.
1
vote
1answer
58 views

Finding the marginal posterior distribution of future prediction, $y_{n+1}$

Assume the following bivariate regression model: $y_i = \beta x_i + u_i$ where $u_i$ is i.i.d $N(0, \sigma^2 = 9)$ for $i = 1, 2, ..., n$. Assume a noninformative prior of the form: $p(\beta) \...
1
vote
1answer
152 views

Help with understanding the $\chi^2$-distribution

I'm studying statistics and there's one part in my book I can't understand. I tried to make as good translation as I can of the problematic part...here goes: Chi squared $\chi^2$ distribution Let $...
1
vote
1answer
44 views

proving identity for statistical distance

How do I show the following identity? Let $\vec{\rho}_X$,$\vec{\rho}_Y$ denote the probability distributions over a finite set $R$ respectively. Prove that $\Delta(\vec{\rho}_X,\vec{\rho}_Y)=\max_{...
1
vote
1answer
53 views

Bhattacharya Distance (or A Measure of Similarity) — On Matrices with Different Dimensions

We have a series of observations of different properties (such as heart rate or blood sugar level and others as well) across different days from different people from different geographical regions. ...
1
vote
2answers
674 views

Integrating exponential of exponential function

I would like to find the integral of $\int_0^\infty\exp(-u-\exp(-ku))\,du$ for $k>0$. This is related to the gumbel distribution(http://en.wikipedia.org/wiki/Gumbel_distribution), which shows ...
1
vote
2answers
230 views

Why does the log-normal probability density function have that extra “x”?

For a random variable $X \sim N(\mu, \sigma^2)$, the probability density function is $$f(x; \mu, \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} \cdot \exp\left\{ -\frac{(x-\mu)^2}{2\sigma^2} \right\}$$ ...
1
vote
0answers
102 views

Proof that Derivative of Expected Value is Zero (Using Differentiation show Unconditional Expectation is Constant)

If the expected value of a distribution is constant, it means its derivative with respect to the values it can take must be zero. I was wondering if there is a rigorous proof of the same. Steps Tried ...