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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23 views

What is the probability of winning this game?

A tennis player has a 60% chance of winning any given point in a tennis game. Calculate the probability that she will win the game within the first 6 points, stating any assumptions you make. (A game ...
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5 views

How to know the percentage of falling of height of Likelihood when I project the edge 1 C.L sigma joint distribution on the 1D Likelihood

I have currently an issue about the height at which the projection of 1 sigma edges in 2D contour should intersect the associated Likelihood. Here a figure to illustrate my issue : At bottom left is ...
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1answer
9 views

Showing that two random variables are standard normal but are not bivariate normal

Question: If $X\sim \mathcal{N}(0,1)$ and we define $Y$ such that $$ Y = \begin{cases} X& \text{ if }|X|<a \\ -X& \text{ if }|X|\geq a. \end{cases} $$ Show that $Y\sim \mathcal{N}(0,1)$ ...
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Verification of answer for distribution for n-th time of arrival when the consecutive intervals are exponentially distributed.

Assume that the time intervals between the arrival of consecutive customers to a store are independent identically distributed random variables with exponential distribution with parameter $\lambda$. ...
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0answers
20 views

Distribution of a certain functional of iid $N(0,1)$ random variables

Suppose that $X_1,\ldots,X_n$ are iid standard normal random variables. Consider the random variable given by $$ \xi_n=\Bigl|\frac1{\sqrt{n}}\sum_{t=1}^nX_t\Bigr|^2-\frac1n\sum_{t=1}^nX_t^2. $$ What ...
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9 views

Find the median of the pdf of the median of the random sample when $n$ is even.

I am trying to find the pdf the $$M = \frac{X_{\left(\frac{n}{2}\right)} +X_{\left(\frac{n}{2}+1\right)}}{2}$$ here we let $X_1, X_2, ..., X_n$ be a random sample from a Uniform$(\theta; \theta + 1)$ ...
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0answers
27 views

On a question of integration formula of bounded distribution functions

$\mathbf {The \ Problem \ is}:$ If $F$ and $G$ are two bounded distribution functions on the interval $[a,b]$ with no common points of discontinuities, then show that $\int_{(a,b]} G(x)dF(x) + \int_{(...
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2answers
38 views

Discrepancy in Calculating Expectation of a Dependent Random Variable

Suppose $X$ a is random variable with pdf $$\begin{cases} f(x), & x\ge 0 \\ 0, & x\lt 0 \end{cases} $$ The random variable $Y$ is defined as $$Y= \begin{cases} A, & X\lt k \\ A+BX, & X\...
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1answer
19 views

strong law of large number with Covariance

I'm supposed to prove that $ \frac{1}{n} \sum_{j=1}^n (X_j-\overline X_n)(Y_j-\overline Y_n)$ converges almost surely to $Cov(X,Y)$ assuming that $ (X_i,Y_i)$ are iid with the same distribution as $(X,...
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Characteristic function of distribution [closed]

What is the characteristic function of $f(x) = \frac{3}{2 \pi} \cdot \frac{1}{1 + x^6}, x \in \mathbb{R}$?
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19 views

Sampling a probability distribution

An actuary is using the following model for the proportion of settled claims after time $t$: The proportion is $X$ where $$P(X \leq x) = x^te^{t(1-x)}$$ Time is measured in small units. This is not a ...
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23 views

Finding of PDF from a given distribution function.

How to find the pdf of $f_{|h|}(v)$ and $f_h(v)$ from a given square channel pdf given as \begin{equation} f_{h^2}(v) = K v^{-\frac{1}{m + 3} - 1} \end{equation} where $m$ and $K$ are constants and $v ...
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19 views

Proving derivative of CDF of $\mu$ is $0$ a.e. on $\mu$-null set, and an analogue in $\mathbb R^d$

I'm trying to solve the following problem (Klenke's Probability Theory: A Comprehensive Course 2nd ed., Exercise 13.1.6): Let $\mu$ be a Radon measure on $\mathbb R^d$ and let $A \in \mathcal B(\...
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16 views

Joint Distribution of a Time Series with lag = 2 ? Am I wrong or the book?

I try to reproduce the calculation of mutual information I(x;y) from the book "Nonlinear Analysis for Human Movement Variability" p319. Specifically calculating the joint distribution of the ...
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1answer
27 views

If $X_i$ follows $ U( \theta, \theta+1)$ and n is even. How do I find the probability distribution of median?

I have used the following transformation to find the joint pdf: $u= X_\frac{n}{2}$ and $ v = \frac{X_\frac{n}{2}+X_{\frac{n}{2}+1}}{2}$ The joint pdf I have found is like below: $f_{(U,V)}(u,v)= \...
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26 views

How to calculate the error due to expected value?

I am currently trying to work out the probability of winning a game. This is proportional to another value however the I dont have the true value of this other quantity. Instead it is the sum of ...
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1answer
17 views

Probability of arrangement of letters following specific rules

I have letters ABCDEFG (7 letters). I want to find the probability that if I shuffle the letters and arrange them randomly that B is first and A is last (but I can pick any arbitrary two letters). I ...
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0answers
36 views

Probability on sex distribution

The problem comes from Feller's introduction to probability. The problem states: "Let the probability $p_n$ that a family has exactly $n$ children be $ap^n$ when $n \geq 1$, and $p_0 = 1 - ap(1+p+...
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27 views

Change of variables formula: What happens on the boundary?

The (multivariate) Change of Variables Theorem (e.g. Blitzstein & Hwang 2019, Thm 8.1.7) states (loosely) that for a random vector X with continuous pdf $f_X$ and an invertible function $g:A_0\to ...
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0answers
61 views

Normal distribution and inflection points intuition

My professor made a point that I had not seen before. He said to figure out (graphically) where the first standard deviation($\sigma$) in $N(0,\sigma^2)$ you can look at the point of inflection. This ...
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0answers
12 views

Which statistical dimension should I use as the conversion value for a sample of invoices with sample size of $210$? [closed]

I need to assign a conversion value for actions taken on a website. I have already processed the data for a sample size of $210$ paid invoices. The statistical analysis of the income values can be ...
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10 views

Convergence of confidence regions when densities converge in distribution.

Let $f_n$ be a sequence of probability density functions with respect to Lebesgue measure on $\mathbb{R}^d$. Suppose that $f_n$ tend to $f$ in distribution (i.e. $\forall \phi$ continuous and bounded, ...
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0answers
23 views

What is the probability distribution function of mean hitting-time in a d-dimensional Brownian motion?

The Wikipedia article on the Lévy distribution states that "The time of hitting a single point, at distance $\alpha$ from the starting point, by the Brownian motion has the Lévy distribution with ...
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2answers
70 views

How to prove that $X \sim N(\sqrt\theta,1)$,then $X^2\sim\chi^2_1(\theta)$.

How to prove that $X\sim N(\sqrt\theta,1)$,then $X^2$ has non-central chi-square $\chi_1^2(\theta)$. I can obtain the pdf of $X$, $f(x)=\frac{1}{\sqrt {2\pi}}\exp\left(-\frac{(x-\sqrt \theta)^2}{2}\...
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0answers
10 views

Why does the log-normaliser of an exponential family density need to be steep (infinite derivative at the boundary)?

Consider the canonical form for the exponential family distribution; $$ p(x) = \nu(x)\exp(\eta^T\psi(x) - \Phi(\eta)) $$ where $\Phi(\eta)$ is the log-normaliser. A lot of literature that I read ...
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1answer
19 views

Let $X,Y$ be two independent random variables and $P_X$, $P_Y$ are induced probability measure. Prove $P(X+Y\in B)=\int_{\Bbb{R}} P_Y(B-x)\ dP_X(x)$

Let $X,Y$ be two independent random variables and $P_X$, $P_Y$ are induced probability measure. Prove $P(X+Y\in B)=\int_{\Bbb{R}} P_Y(B-x)\ dP_X(x)$ Here $P_X, P_Y$ are probability distribution of $X,...
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2answers
62 views

How do you find $E(X^3)$ of a Poisson Distribution? [closed]

I know the proof to find the variance of a Poisson Distribution, and I tried to use that to find $E(X^3)$, but I can't get it to work. Any help would be great!!
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2answers
27 views

Cumulative distribution function with 3 variables

Let $X$ be the random variable whose cumulative distribution function is $$ F_X (x) = \begin{cases} 0, & \text{for} \space x\lt 0 \\ \frac{1}{2}, & \text{for} \space 0\le x\le 1 \\ ...
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1answer
29 views

coin toss, possibly binomial distribution [closed]

We are given two coins, the first shows head with probability $p∈(0,1)$ while the second with probability $q∈(0,1)$. We toss these coins simultaneously and, if both show head, we stop. If not, we toss ...
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0answers
21 views

Sign of exponent of the moment generating function

The MGF for a random variable $X$ is defined as $$ M_X(t) = \mathbb{E}[e^{tX}].$$ Sometimes I see $\mathbb{E}[e^{-tX}]$, for example here, where the asker wants to know the value $M_X(-r)$ for some $r&...
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4answers
218 views

Cambridge Admissions Exam Statistics 1999

My work: Since $f$ is the pdf we must have $\int_{0}^{1} Ax\,\mathrm dx=1 \implies A=2$. Let $Y$ be the number of currants in my portion. We have $Y\sim B(4,x)$. For the expectation $$E(Y)=4x,$$ ...
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1answer
48 views

Let $F$ be a probability distribution function. Prove that $\int\limits_{-\infty}^{\infty} F(x+a)-F(x)\ dx= a\ \forall a\in\Bbb{R}$ [duplicate]

A real-valued map $F$ is called a probability distribution if $F$ is monotonically increasing $F$ is right continuous $\lim\limits_{x\to\infty} F(x)=1$ and $\lim\limits_{x\to-\infty} F(x)=0$ Again I ...
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0answers
18 views

Function of pdfs

The variable $p$ is given by: $$p(r) = \frac{A}{r^{2}}g(r)$$ Where $A$ is a non-zero constant, $r$ a random variable with a continuous pdf $f_1(r)$, and $g(r)$ a variable that depends on $r$ with a ...
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0answers
26 views

Do products of even Hermite polynomials evaluated in a multivariate Gaussian have non-negative expectation?

Let $X$ be a non-degenerate $n$-dimensional centered Gaussian vector, normalized such that every component has variance one. My simulations suggest that if $H_k$ denotes the (probabilist's) Hermite ...
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1answer
11 views

Normal distribution,expectations in F distribution

Let $𝑋_1, ... , 𝑋_{10}$ be independent and identically distributed normal random variables with mean $0$ and variance $2$. Then $𝐸 ( X_1^2/(X_1^2 +X_2^2 + \dots +𝑋_{10}^2))$ is equal to ... My ...
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2answers
70 views

Moment generating function. A miner is trapped in a mine containing 3 doors.

A miner is trapped in a mine containing 3 doors. The first door leads to a tunnel that will take him to safety after 3 hours of travel. The second door leads to a tunnel that will return him to the ...
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0answers
29 views

Pmf of a number of balls where bucket has a certain capacity.

There are $n$ buckets, each with a possibly different finite capacity $c_i \; (i=1, \dots, n)$. There are $k$ balls, each to be distributed randomly to the buckets. A bucket cannot be filled over its ...
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1answer
23 views

Find $P(X_{(1)} < \mu < X_{(4)})$

Let $X_1, X_2, X3, X4$ be iid random variables from a normal distribution with a mean of $\mu$. Find $P(X_{(1)} < \mu < X_{(4)})$ My try: $P(X_{(1)} < \mu < X_{(4)})$ = $P(\mu < X_{(4)})...
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1answer
18 views

Relation between Geometric and Exponential random variables

Let $Y_n$ be a random variable following the exponential distribution with $\lambda$ as a parameter, $Y_n \sim \mathcal E (\lambda)$ For $\theta > 0$, we define $X_n$ as following : $$X_n = [\theta ...
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0answers
54 views
+50

Distribution of payoff in a card game

Consider a pack of $10$ cards marked $1,2,3,3,4,4,5,6,7,8$ ($3$ and $4$ are repeated). There are two players that chose $3$ cards each (a total of $6$ cards without replacement are drawn). The player ...
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0answers
10 views

Sampling from a skewed normal distribution

Given the real positive values $[min, max, avg]$ with $0<min<avg<max$ and a random uniformly distributed value $v\in[0,1]$, I need to be able to sample efficiently from a continuous random ...
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2answers
38 views

Density of random variable max{ξ1, ξ2}

Let $\xi_1, \xi_2$ be independent and equally distributed absolutely continuous random variables with density $p$. How can I find the density of the random variable $\max(\xi_1, \xi_2)$.
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1answer
47 views

sequential flip of a (fair) coin till we either have a total number of 5 heads or 5 tails. [closed]

The problem goes like this: Two gamblers are playing the following game. Each of them deposits 20$ into a common bank. Then, they sequentially flip a (fair) coin. Once the total number of observed ...
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1answer
24 views

CDF of $1-X-Y$ where $X\sim U[0,1]$ and $Y\sim U[0, 1-X]$

I am trying to prove that the random variable: $Z = 1-X-Y$ where $X\sim U[0,1]$ and $Y\sim U[0, 1-X]$ is $Z \sim U[0,1]$. I'm not sure this is even true, but it feels correct. I'm having trouble with ...
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0answers
11 views

High dimensional distribution have analytic cdf

Is there any symmetrical high dimensional distribution that have analytical form for its cumulative distribution function? We are interested in the orthant probability. Right now we are using the ...
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2answers
20 views

Upper bound on difference between Gaussian CDFs

May I know if there are any non-trivial upper bounds $f$ on the following: $$\Phi(a + \Delta) - \Phi(a) \leq f(a, \Delta)$$ for $\Phi$ the CDF of a standard normal and all $a, \Delta > 0$. Thanks!
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3answers
34 views

Probability question about putting X items in K things

The question is: "8 employees want to book meetings with their manager on randomly chosen days of the work week, from Monday till Friday. What's the probability that the manager has at least one ...
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2answers
29 views

Normal distribution equality proving

I proved that for $\xi \sim \mathcal{N}(0,1): \mathbb{E}(\xi)=0$ and $\text{Var}(\xi)=1$. How can I prove the following: Let $\xi \sim \mathcal{N} (0, 1), a, b \in \mathbb{R}$. Prove that $b\xi + a \...
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0answers
51 views

I'm facing a problem regarding a following question: [closed]

We produce a real number X through the following two-stage experiment. 1. First roll a fair die to get an outcome Y E{1,2,...,6}. 2. Then if Y = k, choose a point uniformly at random in (0,k), denote ...
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1answer
30 views

Monte Carlo integration &expected value

I'm confused as to how evaluating the Monte Carlo integration is the same as estimating the expected value. For example, if $x \sim$ unif[0,1], why does $$\int_{0}^{1}f(x)dx = E(f(x)) ?$$

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