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.

8
votes
1answer
15k views

Expected value of normal distribution given that distribution is positive

Given $X \sim N(0, \sigma^2)$ (that is, $X:\mathbb{R} \to \mathbb{R}$ is a normal random variable with mean $0$ and variance $\sigma^2$), I'm trying to calculate the expected value of $X$ given that $...
7
votes
2answers
704 views

A reference for a Gaussian inequality ($\mathbb{E} \max_i X_i$)

I am looking for a reference to cite, for the following "folklore" asymptotic behaviour of the maximum of $n$ independent Gaussian real-valued random variables $X_1,\dots, X_n\sim \mathcal{N}(0,\sigma)...
7
votes
1answer
662 views

Distribution of a random real with i.i.d. Bernoulli(p) binary digits?

Let $X_1, X_2, X_3, \ldots$ be an infinite sequence of i.i.d. Bernoulli($p$) random variables, and define the random real number $X = (0.X_1X_2X_3\ldots)_2$. Question(s): What can be proved about ...
6
votes
2answers
120 views

Distribution at First Time a Sum Reaches a Threshold

Consider the following problem. Roll a die many times, and stop when the total exceeds $M$, for some prescribed threshold $M$. Call this time $\tau$, and call the running score after $n$ rolls $X_n$. ...
6
votes
1answer
3k 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 ...
5
votes
2answers
168 views

Generalizing Poisson's binomial distribution to the multinomial case.

If in a binomial distribution, the Bernoulli trials are independent and have different success probabilities, then it is called Poisson Binomial Distribution. Such a question has been previously ...
5
votes
4answers
1k views

Expected number of frog jumps

There is frog jumping forward on a line. Each jump distance is random with a known cumulative distribution function $F$. What is the expected number of jumps to reach (or go beyond) distance $d$ from ...
4
votes
2answers
5k views

Convolution of continuous and discrete distributions

Assume we have two random variables $X$ and $Y$, such that $X \sim P(x)$ and $Y \sim G(y)$. We ask, what is the distribution of $Z = X+Y$. If both of the distributions of $X$ and $Y$ are discrete, ...
4
votes
1answer
2k views

IID Random Variables that are not constant can't converge almost surely

I am trying to prove the following. If $\{ X_n \}$ are iid random variables and not constant, then $R:=P\{ \omega \mid X_n(\omega)\text{ converges} \}=0$ Using independence I know that by ...
3
votes
2answers
3k views

Maximum a Posteriori (MAP) Estimator of Exponential Random Variable with Uniform Prior

What would be the Maximum a Posteriori (MAP) estimator for $ \lambda $ for IID $ \left\{ {x}_{i} \right\}_{i = 1}^{N} $ where $ {x}_{i} \sim \exp \left( \lambda \right), \; \lambda \sim U \left[ {u}_{...
3
votes
2answers
6k views

What is the density of the product of $k$ i.i.d. normal random variables?

Say you have $k$ i.i.d. normal random variables with some mean $\mu$ and variance $\sigma^2$ and you multiply them all together. What is the density function of the result?
3
votes
1answer
481 views

Probability density function of $X^2$ when $X$ has $N(0,1)$ distribution

I am trying to derive Chi-square distribution. The random variale is $$ U^2=\sum_{i=1}^k X_i^2 $$ where $X$ is a random variable with normal standard distribution. What is the distribution of $X^2$...
3
votes
1answer
197 views

To find the probability whether $X$ is rational

I was thinking whether I can replace this sum by an integral , because I can't think of a distribution whose mgf looks like this. Please Help!
2
votes
1answer
2k views

Reliability function, proving exponential distribution

We are given $R(t)$ = $P(X>t)$ for all $x > 0$ and $$R(0) = 1 - Fx(0) = 1\text{ and }\lim\limits_{t \to \infty} R(t) = 0$$ The random variable $X$ also satisfies the memoryless property: $$P(...
2
votes
1answer
94 views

Compute $P(X_1+\cdots+X_k\lt 1)$ for $(X_i)$ i.i.d. uniform on $(0,1)$

Consider $N=\min\{n: S_n>1\}$, where $S_n=X_1+\cdots+X_n$ and $(X_i)_{i=1}^\infty$ is i.i.d. uniform on $(0,1)$. So, $N$ is the first time that $(S_n)_{n=1}^\infty$ crosses $1$. I'd like to ...
1
vote
2answers
62 views

Proof (confusion) of the product of two random variables of two sequences converge to XY

If $X_n \to X$ $Y_n \to Y$ in probability, then $X_nY_n \to XY$. The textbook proof is as follows: By the triangle inequality, for $\varepsilon > 0$, we have $$ \{|X_nY_n − XY | > \varepsilon\}...
7
votes
2answers
10k views

Jointly Gaussian uncorrelated random variables are independent [closed]

Let $X,Y$ be jointly normally distributed and uncorrelated. Why are they independent?
5
votes
2answers
3k 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[...
5
votes
2answers
322 views

Show CLT for Poisson random variables, using no generating function

Question is as following: $X\sim Po(\lambda)$ $$\frac{X-\lambda}{\sqrt{\lambda}} \,{\buildrel d \over \rightarrow}\, N(0,1)$$ as $\lambda \rightarrow \infty$. Obs. One is asked not ...
5
votes
1answer
5k views

Let (X,Y) have a Dirichlet Distribution with paramters $(\alpha_1, \alpha_2, \alpha_3)$ Establish that X~Beta$(\alpha_1, \alpha_2 + \alpha_3)$

If the joint pdf of (X,Y) is $f(x,y)=\frac{\Gamma(\alpha_1 + \alpha_2 + \alpha_3)}{\Gamma(\alpha_1) \Gamma(\alpha_2) \Gamma(\alpha_3)} x^{\alpha_1 - 1} y^{\alpha_2 - 1} (1-x-y)^{\alpha_3 -1}$ ...
4
votes
1answer
4k views

Relationship between binomial and negative binomial distributions (how to extend the probability space?)

I wonder a technique to extend the discrete probability space. Here's an example from Concrete Mathematics EXERCISE 8.17: Let $X_{n,p}$ and $Y_{n,p}$ have the binomial and negative binomial ...
4
votes
2answers
2k views

On the proof that every positive continuous random variable with the memoryless property is exponentially distributed

The theorem to prove is: $X$ is a positive continuous random variable with the memoryless property, then $X \sim Expo(\lambda)$ for some $\lambda$. The proof is explained in this video, but I will ...
4
votes
0answers
284 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 ...
4
votes
1answer
4k views

Verification of convolution between gaussian and uniform distributions

Let $n \sim \mathcal{N}(\mu, \sigma^2)$ and let $u \sim \mathcal{U}(a,b)$, with $b>a>0$, and suppose that $n$ and $u$ are independent random variables. Let $g = n + u$. The probability density ...
3
votes
0answers
168 views

Random variables and the topology of weak convergence

To see what's going on, I am trying to translate the idea of topology of weak convergence on a random variable setting, just to get some concrete intuition. This is what I have got so far (where the ...
3
votes
1answer
266 views

Symmetric alpha stable distributions with $X_1+X_2+\cdots+X_n \stackrel{d}{=} n^{1/\alpha}X$ as definition

Suppose $X,X_1,X_2,\ldots$ are independent and identically distributed random variables with a symmetric distribution $F$. We say $F$ is symmetric $\alpha$ stable where $0<\alpha\le 2$, if $$X_1+...
3
votes
2answers
1k 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
950 views

Exponential of Squared Brownian Motion

Long time lurker, first time posting! Have a problem, that looks familiar but I can't put my finger on it. Need to calculate $\mathbb{E} [\exp(aW_T^2)|F_t]$ where $W_t$ is an $F_t$ adapted standard ...
2
votes
5answers
5k views

The 3rd raw moment of a binomial distribution

What is the 3rd raw moment (that is, $ E\{X^3\} $) of a Binomial distribution with parameters $n$ and $p$? I am getting $n(n-1)(n-2)p^3 + 3n(n-1)p^2 + np$. Is it correct?
2
votes
1answer
464 views

Distribution of $(XY)^Z$ for $(X,Y,Z)$ i.i.d. uniform on $(0,1)$

Let $X$, $Y$ and $Z$ be i.i.d. uniform (0,1) random variables. What is the distribution of $(XY)^Z$? I've tried to solve it via mgfs, and what I've found is: $$E\left(e^{(XY)^Z}\right)=E\left(E\left(...
2
votes
1answer
3k views

How to prove that convergence in MGF implies Convergence in Distribution?

I know that if the moment generating function of two distribution converges to the same function then the two distribution converges in CDF. But how can we prove this thing explicitly ?
1
vote
1answer
64 views

Showing that $\frac{X_{(i)}}{X_{(n)}},i=1,2,…,n-1$ and $X_{(n)}$ are independent for a population with df $F(y)=y^{\theta}$

Let $X_1,X_2,...,X_n$ be i.i.d with df $F(y)=y^{\theta}, 0<y<1, \theta>0$. Show that $\frac{X_{(i)}}{X_{(n)}}$, for $i=1,2,...,n-1$ and $X_{(n)}$ are independent. I found the population ...
1
vote
1answer
55 views

Proving formula for math expectation

In a book I met a formula for math. expectation of a random variable $\xi$ with distribution function $F(x)$: $$M{\xi}=-\int_{-\infty}^{0}F(x)dx+\int_{0}^{\infty}(1-F(x))dx$$ I wonder how do I prove ...
1
vote
1answer
937 views

Finding integration bounds for density of sum of two independent random variables

Let $X, Y$ be independent random variables, both uniformly distributed over the interval $(0,1)$. That is, $$f_{X}(a)=f_{Y}(a) = \begin{cases} 1 & \text{if $0 < a < 1$} \\ 0 & \text{...
1
vote
2answers
3k views

Probability that sum of independent uniform variables is less than 1

I would like to determine the probability $\mathbb{P}(X_1+\dots+X_n\leq 1)$, where $X=(X_i)_{1\leq i\leq n}$ is a family of independent uniform random variables on $[0,1]$. My first idea is to do this ...
0
votes
1answer
1k views

Joint density of dependent random variables

I'm dealing with this problem Suppose that $X, Y$ are iid $N(0,1)$ random variables and let $R= \sqrt{X^2+Y^2}$. Find the joint pdf of $R$ and $R^2$. I'm not sure how to solve the joint pdf of ...
7
votes
3answers
22k views

Repeating something with (1/n)th chance of success n times

Is there anything that can be said about how many attempts it will take to correctly guess a random number out of 1000 numbers? If the number wouldn't change the probability would just increase every ...
7
votes
0answers
371 views

Regular Version of Conditional Gaussian Distribution

Let $Z_{1}$ and $Z_{2}$ be two independent normally distributed random variables with expectations $\mu_{1},\mu_{2}\in\mathbb{R}$ and variances $\sigma_{1}^2,\sigma_{2}^2\in (0,\infty)$ . I would ...
6
votes
2answers
7k views

Is the family of exponential distributions closed under scaling?

While reading wikipedia article on Exponential distribution, I found the statement on scaling the random variable. Let $Exp(\lambda)$ be the distribution of the exponential random variable with ...
5
votes
2answers
359 views

Asymptotic Moments of the Binomial Distribution, $E(X/(np))^k = 1 + O(k^2/n)$?

Let $X \sim \text{Binomial}(n, p)$ be the sum of $n$ Bernoulli($p$) random variables. What is the value of $E(X/(np))^k$, where $k$ is a large integer, as $n$ grows large? From calculations the ...
5
votes
1answer
2k views

Relations between Order Statistics of Uniform RVs and Exponential RVs

Say we have $U_1 \dots U_n$ i.i.d. random variables uniform on $[0,1]$ and $Y_1 \dots Y_{n+1}$ i.i.d. random variables distributed as $Y_i \sim Exp(1)$. I know that the joint distribution of the order ...
5
votes
2answers
888 views

Calculate probability $P(\min\left\{X,Y\right\} \leq x)$ and $P(\max\left\{X,Y\right\} \leq x)$

$X,Y$ are independent, identical distributed with $$P(X=k) = P(Y=k)=\frac{1}{2^k} \,\,\,\,\,\,\,\,\,\,\,\, (k=1,2,...,n,...)$$ Calculate the probabilities $P(\min\left\{X,Y\right\} \leq x)$ and ...
5
votes
2answers
922 views

Prove quotient of two $N(0,1)$ is $\text{Cauchy}(0,1)$

Problem: Show that if $X$ and $Y$ are independent $N(0,1)$-distributed random variables, then $X/Y ∈ C(0,1)$. Question: I don't know how to proceed below. I want to prove that the PDF of $X/Y$ is ...
4
votes
2answers
2k views

Question on uniform distribution

Two people agree to meet each other on a particular day, between 5 and 6 PM, They arrive independently on a uniform time between 5 and 6 and wait for 15 mintues. What is the probability that they meet ...
4
votes
2answers
1k views

Can sum of two random variables be uniformly distributed

Say $X$ and $Y$ are two random variables where $X\in [-\alpha,\alpha]$, $Y\in [-\alpha,\alpha]$ and $Z=X+Y$. Is it possible to find two independent random variables with certain pdf (not necessarily ...
4
votes
1answer
717 views

break 1-meter stick randomly into 3 segment, form triangle?

I have a 1-meter long stick. Let $x,y$ be i.i.d with uniform ([0,1]), representing the cut in the stick. What is the probability that the 3 segment form a triangle? Attempt: Pr{triangle} = Pr(...
3
votes
3answers
4k 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 ...
3
votes
1answer
2k views

Joint distribution of dependent Bernoulli Random variables

I have $N$ Bernoulli random variables $X_1, ..., X_{N}$ with known parameters $p_1, ..., p_{N}$. I want generate a joint distribution in which these random variables are not independent as I know that ...
3
votes
1answer
84 views

Joint distribution of $X+Y$ and $\frac{X}{X+Y}$

Let $X$ and $Y$ be two random variables i.i.d $U(0,1)$. Find the joint pdf of $T = X+Y$ and $U = \frac{X}{X+Y}$ and the marginal densities of $T$ and $U$ My attempt: We will have the following ...
3
votes
2answers
298 views

Poisson random variables and Binomial Theorem

I'm working on a problem from Casella and Berger's Statistical Inference. X is distributed as Poisson$(\theta)$ and Y is distributed as Poisson$(\lambda)$, with X and Y being independent. We let U = X ...