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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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Limit using Poisson distribution [duplicate]

Show using the Poisson distribution that $$\lim_{n \to +\infty} e^{-n} \sum_{k=1}^{n}\frac{n^k}{k!} = \frac {1}{2}$$
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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 ...
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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-...
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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: $$f_X(x) =\begin{cases} a e^{-a x},& x\geq 0\\ 0,& \text{otherwise}.\end{cases}$$ ...
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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 ...
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 $X>l$, ...
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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 ...
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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 ...
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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?
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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 ...
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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}{...
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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 ...
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{(...