Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [probability-distributions]

Questions on using, finding, or otherwise relating to probability distributions, probability density functions (pdfs), cumulative distribution functions (cdfs), or other related functions.

0
votes
0answers
8 views

Sum of the $N-K$ largest out of $N$ normal random variables with different variables

Given $N$ independent normal random variables, $X_1,X_2,\ldots,X_N$, the ordered sequence is denoted as $X_{(1)}, X_{(2)}, \ldots,X_{(N)}$ where $X_{(1)}\ge X_{(2)}\ge \cdots\ge X_{(N)}$. Let $X_i\...
0
votes
1answer
11 views

Sum of sequences of random variables converging in distribution

Even if $\{X_n\}, \{Y_n\}$ converges in distribution to $X,Y$, I know that $X_n +Y_n$ need not converge to $X+Y$ . Can it happen that $\{X_n+Y_n\}$ doesn't converge in distribution to anything at all ...
0
votes
2answers
9 views

On the limit of a continuous combination of sequences of random variables converging in distribution

Let $\{X_n\}, \{Y_n\}$ be sequences of real valued random variables converging in distribution to $X$ and $Y$ respectively. Let $f: \mathbb R^2 \to \mathbb R$ be a continuous function such that $\{f(...
0
votes
1answer
25 views

Given a sequence $\{ X_{n} : n \in \mathbb{N} \}$ of standard normally distributed random variables, what is the probability that $X_{1}+X_{2}<1$?

I'm looking for a general formula that can be used to compute the probability that a sum of standard normal random variables is above a certain constant $a$. So for example the probability that $X_{1}+...
3
votes
1answer
29 views

Cdf of the area of a stick broken into 2 with UNI(0,1)

Given a stick $L$ broken into 2 with UNI$(0,L)$. I am trying to find the cdf and pdf of $Y$ which denotes the area of rectangle created from the two lengths. ( one side = $L-p$ other is $p$) Now, I ...
1
vote
1answer
30 views

Proving Cauchy random variables

Trying to prove that if a random variable $T$ has Cauchy distribution with probability density function: $$f(x)= \frac{1}{\pi(1+x^2)}$$ then $X = \frac{1}{T}$ and $Y = \frac{2T}{1-T^2}$ are also ...
0
votes
1answer
19 views

Expected Value of Beta Distribution [on hold]

What is the expected value of 1/(1+X) with X ~ Beta(2,3)?
0
votes
1answer
28 views

Probabitlty (random variables )

There is one error in one of five blocks of a program. To find the error, we test three randomly selected blocks. Let $𝑋$ be the number of errors in these three blocks. Compute $E(X)$ and $Var(X)$. ...
1
vote
1answer
17 views

Finding probability distributions associated with moment generating functions [duplicate]

I think the answer to my question is pretty simple, but I'm not able to figure it out. The question is: Find the distribution which corresponds to the moment generating function $\frac{2e^t}{3-e^t}...
0
votes
1answer
19 views

is there a formula to “invert” the binomial distribution - for simulation purposes

My apologies if this should be in one of the programming sites rather than the mathematics one... I decided it was theoretical enough to post here. Feel free to move if someone with authority ...
0
votes
0answers
15 views

CDF evaluated at random variable

Suppose that $Y\sim F_Y$ is continuous. Then, we have: $P(Y\leq y)=F_Y(y)$. We also have that $F_Y(Y)\sim U_{[0,1]}$. But at the same time, $F_Y(Y)=P(Y\leq Y)=1$. So it seems to me that $F_Y(Y)=P(Y\...
0
votes
0answers
17 views

Non-unique random variable and Chebychev inequality

Consider the random variables $X_1$ and $X_2$, both describing the experiment of tossing an unbaised coin as follows: For $X_1$, $X_1 = -1$ means tails and $X_1 = 1$ means heads. Hence $$E(X_1) = 0....
0
votes
3answers
34 views

Sum of Poisson Distribution, why is my solution incorrect

From SOA #212: The number of days an employee is sick each month is modeled by a Poisson distribution with mean $1$. The numbers of sick days in different months are mutually independent. ...
0
votes
0answers
10 views

Cumulative Distribution Function.

I am trying to solve this problem, let me know if my approach is correct. The problem statement is given below: ...
0
votes
0answers
13 views

Can I adjust a weighted average of uniforms to be uniform?

I have uniformly-distributed random variables $x_i$ satisfying $$0\le x_i\le 1$$ I take a weighted mean of these using weights $w_i$ satisfying $$0\le w_i\le 1$$ $$\sum w_i=1$$ The naive weighted ...
0
votes
0answers
26 views

L1 as Probability space

In a continuous optimization problem, I consider the functions belonging to the "space of probabilities" on $\lbrack 0,1 \rbrack^N $ that admit a probability density. I understand that I exclude in ...
0
votes
1answer
23 views

Markov's Inequality and Probability distribution?

I know that Markov's inequality provides an upper bound for probability of a random variable being greater than a certain value, but in what instances/distributions would the upper bound be the exact ...
1
vote
1answer
17 views

Conditional concentration inequality for multivariate normal distribution

Recall that if we have IID $X_1, \ldots, X_n \sim \mathcal{N}(\mu, \sigma^2)$ then we can prove various concentration inequalities such as $ \begin{equation} \mathbb{P}(|\overline{X}_n - \mu| > \...
2
votes
1answer
41 views

Obtaining Laplace distribution from bivariate transformation

So this is the problem Let $X_1$ and $X_2$ be $i.i.d.$ random variables with a common density function $\chi_2^2$. Let $Y_1 = \frac{1}{2}(X_1-X_2)$ and $Y_2 = X_2$. Show that $Y_1$ follows a ...
0
votes
1answer
18 views

Conditional marginal distribution of conditional bivariate normal distribution

I have a bivariate normal distribution$$(X, Y)\sim N(\mu_{x}, \mu_{y}, \sigma_{x}^2, \sigma_{y}^2, \rho)$$ My question is : when $X > k$ ($k$ is a constant),how to get the distribution of $Y$? Can ...
1
vote
1answer
21 views

Checking whether $X$ and $Y$ are independent given the joint pdf and calculating conditional expectation

Consider the joint pdf of $(X, Y)$ given by $$f(x, y) = \begin{cases} y^{-1}\text{exp}\left(-y\right) & 0 < x < y \\[1em] 0, & \text{ otherwise.} \end{cases} $$ a) Compute $\mathbb{E}...
-1
votes
1answer
23 views

Are these two probability equal or not? [on hold]

Suppose there is a distribution $D$. $x$ is extracted directly from $D$. $s = (a_1,a_2,\dots,a_n)$ is $n$ samples i.i.d from $D$. then extract a sample $y$ from $s$. Can $y$ be interpreted as ...
-1
votes
0answers
15 views

Probability of independent variables

I want to calculate the probability as $A=P(W<\dfrac{XY^a}{Z^b}).P(W<cY^a)+ P(W>\dfrac{XY^a}{Z^b}).P(W<dY^a),$ where, W,X,Y,and Z are independent random variables. X is an exponential ...
0
votes
2answers
26 views

CDF of quotient of RVs?

Let A and B be i.i.d exponential distributions with $\lambda=1$. Let $C = A/B$. What is $P(C \leq c)$ Let A and B be independent uniform [0,1] distributions. Let $C = A/B$. What is $P(C \leq c)$ For ...
0
votes
0answers
20 views

Random vector and CDF

I have to compute CDF of random vector (X,Y) using the following probability density function c x>=0, y>=0 and x+y<=2 f(x,y) = 0 otherwise ...
0
votes
1answer
16 views

limiting distribution and continuous transformation

Assume that $\sqrt{n}(b-\beta)$ converges in distribution to a normal variable with zero mean and some variance S (here $b$ is a random variable and $\beta$ is a constant, both are scalar valued). I ...
1
vote
1answer
20 views

Help With Statistics and Distributions!!

So the question asks: Airlines find that each passenger who reserves a seat fails to turn up with probability 0.01 independently of other passengers. Consequently, Bryanair always sell 100 tickets for ...
1
vote
1answer
27 views

Berry-Esseen function bound

ByBerry-Esseen theorem on Wikipedia we know that $$|F_n(x)-\Phi(x)|\le \frac{C\rho}{\sigma^3\sqrt{n}}$$ where $F_n$ is the cumulative distribution function given there. However, in many important ...
1
vote
2answers
38 views

How to prove that $E(Y|D=1)=E(DY)/E(D)$

How to prove that $$E(Y|D=1)=E(DY)/E(D)$$ and $$E(Y|X,D=1)=E(DY|X)/E(D|X),$$ where $D$ is a binary variable and takes value of 0 and 1.
0
votes
1answer
34 views

Why is the expectation of cauchy distribution not defined? (What is the intuition behind it?)

Let $X$ be random variable with pdf $f_X(x) = \dfrac{1}{\pi(1+x^2)}$. I understand that mathematically, the improper integral, $\displaystyle\int\limits_{-\infty}^{\infty}\dfrac{x}{\pi(1+x^2)}dx$ does ...
3
votes
0answers
31 views

Conditional distribution from the sum of uniform distributions

I am trying to find the conditional probability distribution function of $Y$ $$F(Y\mid X_1,X_2)$$ given that $Y$ is distributed uniformly on $[0,1]$, $$X_1=Y+Z_1$$ and $$X_2=Y+Z_2$$ where $Z_1$ and $...
0
votes
1answer
59 views

Probability - Expectation value and Covariance

A group of $n = 10$ men exchange their gloves randomly (altogether there are 20 gloves). Let $X_i$ be the random variable which attains the value 1 if the $i-th$ man got at least one of his gloves ...
1
vote
0answers
15 views

The expectation of a geometric random variable where its parameter is uniform

First thanks for any help editing my text. If a random variable $X$ has a geometric distribution with parameter $P$ where $P$ itself is a random variable and uniformly distributed from $0$ to $1-1/n$,...
0
votes
1answer
30 views

Distribution of X-Y when X and Y are independent geometric

$X$ and $Y$ are independent geometric distributions with parameter $p$. Find the distribution for $X-Y$ I am aware that there are many similar questions. My problem with this specific transformation ...
0
votes
1answer
24 views

Convergence of Normal distribution in probability

Hi Please help in solving below question. Let $X_n ~ N(0,\frac{1}n)$. Does $X_n $ converge in probability to a constant $c$? If yes, what does it converge to? I came up with a proof but it ...
1
vote
0answers
16 views

Computing the probability example

I have been doing the following example, and there is something I do not understand. It goes like this: From an urn with four white balls and six black balls, one ball is drawn repeatedly with ...
0
votes
1answer
10 views

Probability Question And, Or, and Bivariates

I feel like this is a basic question, but I'm having trouble justifying a property. I have the following result (simplified for the post) which I am trying to get an intuitive understanding for: $P(...
0
votes
2answers
11 views

CDF of the of Joint Uniform Distribution for k random variables.

If i have K independent Random Variable: $X_1,X_2,x_3,\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot, X_k$ What would be the CDF of the sum of their Joint Distribution? $f_{X_1+X_2+X_3+...+X_k} (z)$ z&...
1
vote
1answer
25 views

Random variable and maximum metric.

Let $\Omega := [0,1] \times [0,1]$. Consider on $\sigma$-algebra Borel sets with Lebesgue measure .Let $X(w)$ describe the distance (in maximum metric) between the point and the nearest corner of the ...
0
votes
1answer
54 views

Probability function- Random variable

I need to find a Probability function using $$ F(x) = \begin{cases} 0 & \text{if $x<0$}\\ \frac{1-a^k}{1-a^{n+1}} & \text{if $k-1 \leq x \leq k, k=1,\dots,n, n>1$}\\ 1 & \text{if $x ...
0
votes
1answer
18 views

Prove that random variable has standard normal distribution [on hold]

How do I prove that random variable X has a standard normal distribution given the probability density function? A random variable X has a standard normal distribution if X is absolutely continuous ...
1
vote
2answers
16 views

marginal density of $|X|$

I have a continuous random variable $X$ whose distribution is conditioned on another discrete random variable $S$. The conditional density function of $X$ is $f_{X|S}(x|1) = \alpha e^{-\alpha x}, x \...
1
vote
1answer
13 views

A question about continuity of a specific function with probability measure

Let $X$ be a compact metric space, and $\Theta$ be a finite space, endowed with their own $\sigma$-algebra. Let $f \colon X \times \Theta \to \mathbb{R}$ be a Caratheodory function such that (1) for ...
1
vote
1answer
48 views

Expectation of random 2d walk

Start at the origin, take $n$ independent steps of length 1 in the direction of $\theta_i$, which is uniformly distributed on $[0,2\pi]$. If $X,Y$ is the position after $n$ steps and $D = X^2 + Y^2$,...
0
votes
1answer
24 views

Probability Measures, and inequalities

Let X and Y be random variables defined on the probability space $(\Omega, \mathcal{S}, \mathcal{P})$. I have: (i) $\mathcal{P}(|X+Y| >\epsilon) \leq \mathcal{P}(X>\epsilon/2) + \mathcal{P}(...
2
votes
0answers
17 views

Determine the probability of a conditional p.d.f

So the joint p.d.f is f(x,y) = $c(x^2 + y^2)$ for $0\leq x\leq 1$ and $0\leq y \leq 1$ Now the steps I worked out is applying the formula $f(x|y) = \frac{f(x,y)}{f_{2}(y)}$ Which gave me f(x|y) = $\...
0
votes
0answers
11 views

Why this :$I(x)=\int_{-x}^x {0.5(\exp({-t² {\operatorname{erf}(t^2)}})}dt$ is not error function for $|x| >3$?

This integral : $$I(x)=\int_{-x}^x {0.5(\exp({-t² {\operatorname{erf}(t^2)}})}dt$$ close to $x$ for $|x|<3$ and converge to $1$ for $|x|>3$ from $-\infty \to +\infty$ as shown here such that ...
0
votes
0answers
32 views

MAP estimate of Erlang distribution

I have a hard time approaching this problem. I understand how to find the MAP estimate for common distributions but from the given problem below I have totally confused. I have a set of $N$ ...
0
votes
0answers
10 views

Will the function of these random variables be a sub-Gaussian distribution?

Suppose a function $f(X,Y)$ is $\sigma$-sub-gaussian when the random variables $X$and $Y$ are distributed independently of each other. For example, $f(\mu,\sigma) = exp(-\frac{(x-\mu)^2}{2\sigma^...
0
votes
0answers
16 views

Deriving Poisson Distributions from Multinomial Distribution

Given $(p_{1,n})_{n\in\mathbb{N}}$,$(p_{1,n})_{n\in\mathbb{N}} \subseteq (0,1)$ and $λ_1, λ_2 > 0$ with $np_{1,n}\overset{n \rightarrow\infty }{\rightarrow}λ_1 $and $np_{2,n}\overset{n \rightarrow\...