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.

1,241 questions
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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 ...
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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 $X_n$. ...
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 $f(x,y)=\frac{\Gamma(\alpha_1 + \alpha_2 + \alpha_3)}{\Gamma(\alpha_1) \Gamma(\alpha_2) \Gamma(\alpha_3)} x^{\alpha_1 - 1} y^{\alpha_2 - 1} (1-x-y)^{\alpha_3 -1}$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ?
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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(...
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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 ...
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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 ...
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 ...
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 ...