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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.

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13
votes
1answer
4k views

Limit using Poisson distribution [duplicate]

Show using the Poisson distribution that $$\lim_{n \to +\infty} e^{-n} \sum_{k=1}^{n}\frac{n^k}{k!} = \frac {1}{2}$$
23
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3answers
3k views

Existence of independent and identically distributed random variables.

I often see the sentence "let $X_1, X_2, \ldots$ be a sequence of i.i.d. random variables with a certain distribution". But given a random variable $X$ on a probability space $\Omega$, how do I know ...
31
votes
4answers
15k views

Precise definition of the support of a random variable

$\newcommand{\F}{\mathcal{F}} \newcommand{\powset}[1]{\mathcal{P}(#1)}$ I am reading lecture notes which contradict my understanding of random variables. Suppose we have a probability space $(\Omega, \...
10
votes
2answers
1k views

Show $\mathbb{E}[f(X)g(X)] \geq \mathbb{E}[f(X)]\mathbb{E}[g(X)]$ for $f,g$ bounded, nondecreasing

Let $X$ be a random variable and let $g,f$ be real-valued, nondecreasing, and bounded. Show that $\mathbb{E}[f(X)g(X)]\geq \mathbb{E}[f(X)]\mathbb{E}[g(X)]$ Having a hard time seeing where to start ...
23
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3answers
2k views

Probability of picking an odd number from the set of naturals?

I know there's no uniform distribution for a countably infinite set, but I'm wondering if there's still a way to determine the probability of picking from a subset of a countably infinite set. For ...
7
votes
5answers
12k views

Negative binomial distribution - sum of two random variables

Suppose $X, Y$ are independent random variables with $X\sim NB(r,p)$ and $Y\sim NB(s,p)$. Then $$X + Y \sim NB(r+s,p)$$ How do I go about proving this? I'm not sure where to begin, I'd be glad for ...
8
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2answers
2k views

Do moments define distributions?

Do moments define distributions? Suppose I have two random variables $X$ and $Y$. If I know $E\left[X^k\right] = E\left[Y^k\right]$ for every $k \in \mathbb N$, can I say that $X$ and $Y$ have the ...
7
votes
2answers
6k views

Finding the moment generating function of the product of two standard normal distributions

The following question is on my homework assignment that I cannot figure out: Let U and V be independent random variables, each having a normal distribution with mean zero and variance one. Find ...
7
votes
4answers
27k views

Addition of two Binomial Distribution

What is the distribution of the variable $X$ given $$ X = Y + Z, $$where $Y \sim $ Binomial($n$, $P_Y$) and $Z\sim$ Binomial($n$, $P_Z$)? For the special case, when $P_Y = P_Z = P$, I think that X~...
3
votes
1answer
5k views

pdf of a quotient of uniform random variables

Suppose $x_1, x_2$ are IDD random variables uniformly distributed on the interval $(0,1)$. What is the pdf of the quotient $x_2 / x_1$?
8
votes
2answers
3k views

Expected Value of Square Root of Poisson Random Variable

Find the expected value of $\sqrt{K}$ where $K$ is a random variable according to Poisson distribution with parameter $\lambda$. I don't know how to calculate the following sum: $E[\sqrt{K}]= e^{-\...
1
vote
2answers
227 views

Distribution of $\max_{t \in [0,1]} |W_t|$ for Brownian motion

For a standard Brownian motion $\{W_t, t\geq 0\}$, find $\mathbb{P}(\max_{ t \in [0,1]}|W_t| <x)$. Page 79-80 of Billingsley, P., Convergence of probability measures, New York-London-Sydney-...
5
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2answers
624 views

Prove that $\mathbb P(X>Y) =\frac{b}{a + b}$ if $X, Y$ are exponentially distributed with parameters $a$ and $b$.

Let $X, Y$ be an exponentially distributed random variables with parameters $a, b$. Then $X$ has pdf: $$f_X(x) =\begin{cases} a e^{-a x},& x\geq 0\\ 0,& \text{otherwise}.\end{cases}$$ ...
3
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1answer
605 views

Exponential distribution from Poisson

In Poisson distribution, the probability of inter arrival time to be t or less is: $$ P(X\leq t)= 1 - P(X>t) = 1 - P(0 \mbox{ arrivals in } t) = 1 - e^{-\lambda t} $$ and probability of one ...
2
votes
1answer
262 views

Induced distribution measure and induced distribution function where original r.v. is Pareto

Consider for one car owner the insurance policy with the following clauses: Deductible: If the loss $X>d$, then the insurer pays only for loss above $d>0$. Coverage Limit: If the loss $X>l$, ...
1
vote
1answer
231 views

Prove $X(\omega) = \sup\{y \in \mathbb{R}: F(y) < \omega\}$ is a random variable.

Let F be a distribution function. On $(\Omega, \mathfrak{F}, P)=((0,1), \mathfrak{B}(0,1),\lambda)$ where $\lambda$ denotes Lebesgue measure. Define X: $\Omega \to \mathbb{R}$ by $X(\omega) = \sup(y \...
1
vote
3answers
770 views

$f(x, \theta)= \frac{\theta}{x^2}$ with $x\geq\theta$ and $\theta>0$, find the MLE

Let $X$ be a random variable with density $$f(x, \theta)= \frac{\theta}{x^2}$$ with $x\geq\theta$ and $\theta>0$. a) Show if $S=\min\{x_1,\cdots, x_n\}$ is a sufficient statistics and if it is ...
1
vote
1answer
441 views

Prove symmetry of probabilities given random variables are iid and have continuous cdf

Let $Y_1, Y_2, ...$ be independent and identically distributed random variables in $(\Omega, \mathscr{F}, \mathbb{P})$ s.t. their distributions are continuous and $$F_{Y}(y) := F_{Y_1}(y) = F_{Y_2}(y) ...
0
votes
1answer
99 views

Confidence Intervals - Inconsistent Statistical Results

After my last SE question on confidence Intervals here, which clarified the intuition, I tried then to verify statistical results if they are convincingly compliant with theory. I started with CI for ...
65
votes
9answers
16k views

What do $\pi$ and $e$ stand for in the normal distribution formula?

I'm a beginner in mathematics and there is one thing that I've been wondering about recently. The formula for the normal distribution is: $$f(x)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\displaystyle{\frac{(...
25
votes
1answer
18k views

Characteristic function of a standard normal random variable

The characteristic function of a random variable X is given by $$\Phi_X(\omega) = \mathbb{E}e^{j\omega X}=\int_{-\infty}^\infty e^{j\omega x}f_X(x) dx.$$ One can easily capture the similarity between ...
12
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2answers
3k views

Conditional expectation equals random variable almost sure

Let $X$ be in $\mathfrak{L}^1(\Omega,\mathfrak{F},P)$ and $\mathfrak{G}\subset \mathfrak{F}$. Prove that if $X$ and $E(X|\mathfrak{G})$ have same distribution, then they are equal almost surely. I ...
14
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1answer
2k views

How is the distance of two random points in a unit hypercube distributed?

Let $p_1, p_2 \sim U([0, 1]^n)$ with $n \in \mathbb{N}$ be two points in the $n$-dimensional unit hypercube which are uniform randomly independently sampled. How is the distance $d(p_1, p_2) = \sqrt{\...
18
votes
1answer
6k views

Can we prove the law of total probability for continuous distributions?

If we have a probability space $(\Omega,\mathcal{F},P)$ and $\Omega$ is partitioned into pairwise disjoint subsets $A_{i}$, with $i\in\mathbb{N}$, then the law of total probability says that $P(B)=\...
17
votes
1answer
1k views

Math Intuition and Natural Motivation Behind t-Student Distribution

I am trying to understand with basic mathematical background how the $t$-Student distribution is a "natural" pdf to define. A more accessible explanation than this post, or the daunting Biometrika ...
11
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3answers
6k views

Quantile function properties

I am confused by "Inverse distribution function (quantile function)" section of the wikipedia page on CDFs . It says that $$F^{-1}(F(x)) \leq x\text{ and }F(F^{-1}(y)) \geq y$$ However, I ...
5
votes
1answer
21k views

Continuous uniform distribution over a circle with radius R

I started to do this problem with the standard integration techniques, but I cant help but think that there has got to be something I am not seeing. Since it is a uniform distribution, even though x ...
3
votes
2answers
17k views

Probability of sampling with and without replacement

In sampling without replacement the probability of any fixed element in the population to be included in a random sample of size $r$ is $\frac{r}{n}$. In sampling with replacement the corresponding ...
15
votes
3answers
3k views

Random point uniform on a sphere

If $X=(x,y,z)$ is a random point uniform on the unit sphere in $\mathbb{R}^3$, Are the coordinates $x$, $y$, $z$ uniform in interval $(-1,1)$?
11
votes
5answers
32k views

Convolution of two Gaussians is a Gaussian

I know that the product of two Gaussians is a Gaussian, and I know that the convolution of two Gaussians is also a Gaussian. I guess I was just wondering if there's a proof out there to show that the ...
8
votes
2answers
7k views

Distribution of sine of uniform random variable on $[0, 2\pi]$

Let $X$ be a continuous random variable having uniform distribution on $[0, 2\pi]$. What distribution has the random variable $Y=\sin X$ ? I think, it is also uniform. Am I right?
7
votes
1answer
7k views

mean and variance of reciprocal normal distribution

If $X$ is a normal distributed with mean $\mu$ and variance $\sigma^2$. What would be the mean and variance of $Y = \dfrac{1}{X}$
6
votes
1answer
469 views

For a distribution function $F(x)$ and constant $a$, integral of $F(x + a) - F(x)$ is $a$.

For any distribution function and any $a \geq 0$, $\int_{-\infty}^{\infty} (F(x+a)-F(x))dx = a$. In this case, "distribution function" means a right continuous function F with $F(-\infty) = 0$, $F(\...
5
votes
2answers
3k views

Combinations of characteristic functions: $\alpha\phi_1+(1-\alpha)\phi_2$

Suppose we are given two characteristic functions: $\phi_1,\phi_2$ and I want to take a weighted average of them as below: $\alpha\phi_1+(1-\alpha)\phi_2$ for any $\alpha\in [0,1]$ Can it be proven ...
4
votes
2answers
708 views

What is the expected value of $\min\{|X|,|Y|\}/\max\{|X|,|Y|\}$ assuming $X$ and $Y$ are independent?

So I need to compute $$E\left[\frac{\min\{|X|,|Y|\}}{\max\{|X|,|Y|\}}\right]$$ given $X,Y \sim$ Normal$(0,1)$ and independent. What I am having trouble seeing is whether $\min\{|X|,|Y|\}$ and $\...
7
votes
1answer
3k views

Distribution of Ratio of Exponential and Gamma random variable

A recent question asked about the distribution of the ratio of two random variables, and the answer accepted there was a reference to Wikipedia which (in simplified and restated form) claims that if $...
5
votes
2answers
647 views

CDF of probability distribution with replacement [duplicate]

I want to get every color of gumball in a gumball machine (where there are 16 types of gumballs, each with a 1/16 chance of obtaining a particular color [assume there are an infinite amount of ...
4
votes
2answers
4k views

Uniform distribution on unit disk

Let $(X, Y)$ be a random point chosen according to the uniform distribution in the disk of radius 1 centered at the origin. Compute the densities of $X$ and of $Y$. I know that the joint density of $...
3
votes
2answers
861 views

Distribution of sums

I'm really having a hard time with this topic in probability theory and I was wondering if someone has any tricks, tips or anything useful to help me understand it. In my notes I am told that $X\sim$...
3
votes
1answer
1k views

Question about order statistics

I saw a paper which says that: Let $Z_i$ be i.i.d. exponential random variables with mean $1$, and let $S_n = Z_1 + \dots + Z_n$ for all $n$. For a fixed $n$, let $U_j = S_j/S_{n+1}$, then $(U_1,\...
2
votes
4answers
9k views

The joint density of the max and min of two independent exponentials

Let $X=\min(S,T)$ and $Y=\max(S,T)$ for independent exponential variables $S$ and $T$. Find the joint density of $X$ and $Y$. Are $X$ and $Y$ independent? How would you suggest I approach this?
2
votes
2answers
266 views

What is the joint distribution of $Z=\min(X,Y)$ and $I_{Z=X}$?

Assume that $X$ and $Y$ are independent random variables with $X \sim \exp(\lambda)$ and $Y \sim \exp(\mu)$. It is impossible to obtain direct observations of $X$ and $Y$. Instead, we observe the ...
1
vote
1answer
7k views

Expectation of $\frac{1}{x+1}$ of Poisson distribution

As the title states, I'm trying to find the expecteed value of $\frac{1}{x+1}$ where $X \sim \mathrm{Poisson}(\lambda)$ My attempt: \begin{align} &\sum \frac{1}{x+1} \cdot \frac{e^{-λ}\cdot λ^x}{...
1
vote
2answers
591 views

Find the pdf of $\prod_{i=1}^n X_i$, where $X_is$ are independent uniform [0,1] random variables.

How do I find the pdf of $\prod_{i=1}^n X_i$, where $X_is$ are independent uniform [0,1] random variables. I know X~U[0,1], -ln(x) is exponential(1). I also know the sum of two or more independent ...
3
votes
1answer
5k views

Conditional probability distribution with geometric random variables [duplicate]

Let X and Y are independent random variables following geometric distribution with parameter p. Find the distribution of X given that X + Y = n. I made it this expression... $$P\{X =i|X+Y=n\}=\frac{(...
2
votes
2answers
568 views

Density of the sum of $n$ uniform(0,1) distributed random variables

I am working on the following problem: Let $X_1, X_2, \ldots, X_n, \ldots$ be iid. random variables, each of Uniform$(0,1)$ distribution. Denote by $f_n(x)$ the density of the random variable $S_n :...
1
vote
2answers
582 views

Uncorrelated, Non Independent Random variables

I don't understand the parts highlighted in green. I understand that the supports imply that X and Y are not independent but not how the graph shows this graph. I'm a bit confused by all aspects of ...
1
vote
4answers
7k views

{Thinking}: Why equivalent percentage increase of A and decrease of B is not the same end result?

original post the examples here are, the most important word -- fundamentally -- the same. example1: the most abstract way to present this example. Why equivalent % increase of A in event1 and % ...
1
vote
2answers
196 views

What defines a “description” of a probability distribution?

Say you take a dice and you roll it twice so that you have a pair (X,Y) where X represents the first roll, Y represents the second. When you have a distribution like max(X,Y), what type of ...
2
votes
2answers
422 views

Distribution of Product of Random Variables with one being the normal distribution.

Let $X$ and $Z$ be independent, with $X\sim N(0,1)$, and with $\textbf{P}(Z=1)=\textbf{P}(Z=-1)=\frac{1}{2}$. Let $Y=XZ$ (i.e., $Y$ is the product of $X$ and $Z$). (a) Prove that $Y\sim N(0,1)$. (b)...