Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.

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3 views

Conditional Distribution: how to set up Limit of Integration of a joint density

I have a question in conditional probability. I'm asked to find the conditional distribution, however, I'm unsure about the answer given and would appreciate someone helping straighten out the theory ...
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0answers
9 views

How to develop a probability distribution/density function of an issue?

Assume that a health insurance company has $1000$ customers. It is estimated that the probability of a customer making a claim is $p = 0.2$ per year, independently of previous claims and other ...
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1answer
23 views

Random variable related to binomial

The number of successes $X$ in $n$ independent trials with the probability of a success is $p$ for each trial is binomially-distributed. I am interested in a scenario that adds dependence to the ...
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2answers
61 views

What is the distribution of Z=min(X,Y)

Let X and Y be independent geometric random variables. What is the distribution of Z=min(X,Y)?
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1answer
11 views

Show that the σ-algebras generated by the collection of all intervals are equivalent

Show that the σ-algebras generated by the collection of all intervals of the form [a,b]⊂R and by the collection of all the intervals of the form (−∞,b]⊂R are equivalent. i am having trouble with ...
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0answers
4 views

Use copula to find the joint distribution of two random variables

Assuming that there are two random variables $x$, $y$ both having an Arcsine marginal distribution $F$, i.e. $F(x)=\frac{2}{\pi}\arcsin{\sqrt{x}}$ The density of their Gaussian copula can thus be ...
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0answers
11 views

Existence of independent standard normally distributed r.v.s

For any standard normally distributed random variable $X$, show that we can find another standard normally distributed random variable $Y$ that is independent of $X$. This is one of problems in my ...
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1answer
11 views

Combined Distribution of Random variable

How to compute $P[T1 \le T2 \le t]$ for T1, T2 is independent random variable with exponential distribution in terms of cmf, pdf of T1 and T2? Similarly for $P[T1 \le T2 \le T3.. \le t]$ ? I tried ...
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2answers
21 views

Find the probability density function of $Y=X^2$

Consider the random variable X with probability density function $$f(x)=3x^2$$ if $0<x<1$, and $$f(x)=0$$ otherwise. Find the probability density function of $Y=X^2$. This is the first question ...
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0answers
15 views

CDF of maximum of iid rvs [duplicate]

I am having a small doubt regarding maximum of random variables. I have $$Z= \max\{ X_1, X_2,\dots X_p, \dots X_N\}$$ where all $X_i$ are independent, identically distributed. Now, If for sure, I know ...
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0answers
16 views

Bernoulli random variable uniformly distributed have same distribution as a subsequent binomial variable? [on hold]

Suppose two Bernoulli random variables X1 and X2 are individually uniformly distributed. Is Y= X1 + X2 therefore also uniformly distributed?
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1answer
19 views

Given a probability distribution, how many times do I have to repeat an experiment so see a certain outcome

My question concerns random number generation under certain constraints. I assume that the random number generator is good enough to generate uniformly distributed numbers. This means that each number ...
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1answer
35 views

A generalization of the Glivenko-Cantelli theorem

Let $P$ and $P_n$ be probability measures on $\mathcal{B}(\mathbb{R})$ with distribution functions $F$ and $F_n$. Moreover, let $F$ be continuous and $(P_n)_{n\in\mathbb{N}}$ weakly converge to $P$. ...
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0answers
18 views

How to solve $P(Y_1 ≤ 3/4, Y_2 ≥ 1/2)$. Pls help

Let $Y_1$ and $Y_2$ have the joint probability density function given by $$f (y_1, y_2) =k(1 − y_2), 0 ≤ y_1 ≤ y_2 ≤ 1$$ $$ 0,\,\,\,\, elsewhere.$$ a) Find the value of $k$ that makes this a ...
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0answers
6 views

Signal-extraction knowing both the sum and the sum of the absolute values of normally distributed variables

I have two normally distributed variables $X∼N(μ_{x},σ_{x}²)$ and $Y∼N(μ_{y},σ_{y}²)$. I can observe both the sum of their values and the sum of their absolute values, i.e. $Z₁=X+Y$ and $Z₂=|X|+|Y|$. ...
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0answers
15 views

Dynamic programming for optimal maximum and optimal minimum

We have a sequence of $a_i$ and a choosing rule that is take the first number $x_t\ge a_t$. The definition is = $$ min\{ t|t \in \{ 1,2,\cdots,n\}\,\,,\,\, x_t\ge a_t\}$$ The sequence $a_i$ is ...
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1answer
25 views

Sum of Two Poisson distributions

The probability distribution for the number of goals scored per match by Team A is believed to follow $X \sim Poi(0.8)$. Independently, the number of goals scored by Team B is believed to ...
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1answer
26 views

Joint probability distribution

$Y_1$ and $Y_2$ are jointly distributed with density $f(y_1,y_2)=4y_2^2 \qquad 0 \leq y_1 \leq y_2 \leq 1$ Determine the following: $P$( max {$Y_1,Y_2$} $< 1/2$) $P(Y_1+Y_2 < 1/2)$ ...
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0answers
36 views

Let $X_n>0$ be iid and $P(X_n>t)\sim t^{-\alpha}$, show that $Y_n=n^{-1/\alpha}S_n$ and $1/Y_n$ are tight.

We are given that $X_n>0$ be iid with common distribtuon $X$, and $P(X>t)\sim t^{-\alpha}$, I need to show that the scale of $Y_n$ is $n^{1/\alpha}$. Or in other words show that ...
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2answers
21 views

Support of the conditional distribution of a poisson process

I am working on Problem 5.1.8 of this book. It states: Let $\left\{X(t),t \geq 0 \right\}$ be a Poisson process of rate $\lambda$. For $s,t >0$, determine the conditional distribution of ...
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0answers
38 views

$\mathsf kth$ moment of the standard deviation about the origin from a $\mathsf N(\mu,\sigma^2)$ population

Let T be the standard deviation of a random sample of size n from a $\mathsf N(\mu,\sigma^2)$ normal population. Find the $\mathsf kth$ moment of T about the origin, and state the condition for the ...
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2answers
20 views

Probability of the highest order statistic below the population median.

What is the probability that the highest order statistic of a random sample of size n from any continuous distribution is below the median ( population median ) of that distribution.
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1answer
20 views

Probability: How much days we need to play a game win

Suppose the probability of win a lotery game is : $1/1000$ If a person play the lotery every day with the same combination, how much time he need to wait to win the lotery? Im thinking to use a ...
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1answer
17 views

Multivariate sampling of $F(x_1,…,x_n)$?

Let $$(X_1,...,X_n)\sim F(x_1,...,x_n)$$ (not independent). How can I sample from this distribution? In the univariate case, on can use $F^{-1}(u),u\sim U(0,1)$. However, in the multivariate case ...
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2answers
65 views

Prove that if $X$ is stochastically larger than $Y$ then $E(X)\ge E(Y)$

Prove that if $X$ is stochastically larger than $Y$ (i.e. $P(X > t) \ge P(Y > t)$ then $E(X)\ge E(Y)$. I understand how to solve the problem if $X$ and $Y$ are non-negative random ...
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0answers
32 views

A property of the hazard function of the normal distribution

I have a problem that I can't figure out. Define $$\Gamma\left(x\right):=\frac{\phi(x)}{1-\Phi(x)}$$ where $\phi(x)$, $\Phi(x)$ are the density respectively cumulative distribution function of the ...
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0answers
12 views

question about exponential distribution or exponential random variables

Consider a post office that is run by two clerks. Suppose that when Mr. Anderson enters the system he discovers that Mr. Smith is being served by clerk 1 and Mr. Brown by clerk 2. Suppose also that ...
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2answers
59 views

Show that Y=aX+b is an random variable.

Let X be an random variable on a given probability space and let a,b∈R. Show that Y=aX+b is an random variable. if X has a distribution function F, what is the distribution function of Y? if X ...
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15 views

What is the limiting distribution of this Markov Chain?

Take a Markov Chain with state space $\left\{ 0, 1, \dots, 20 \right\}$. Then we have the rule that given $X_n$: Compute $Z = X_n + 1$ or $Z = X_n - 1$ with probability $\frac{1}{2}$ each (if the ...
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0answers
10 views

4th order statistics of Circularly Symmetric Complex Normal random vector?

Assume that ${\bf z} \in C^{n×1}$ is a CSCG random vector denoted with $C (μ,Σ)$ where $μ$ and $Σ$ are mean and contrivance matrix, respectively, and defined as $μ=E({\bf z})$, $Σ=E({\bf z}{\bf ...
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3answers
31 views

A box contains 5 yellow and 3 red balls, from which 4 balls are drawn one at a time, at random, without replacement.

A box contains 5 yellow and 3 red balls, from which 4 balls are drawn one at a time, at random, without replacement. Let $X$ be the number of yellow balls on the first two draws and $Y$ the number of ...
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0answers
13 views

Simulate from a distribution using Metropolis-Hastings and Rejection Sampling?

We have covered the basics behind rejection sampling as well as Metropolis-Hastings from class, but I am not sure how to use the two in conjunction to solve the following problem: Given $\pi(x) = ...
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1answer
33 views

Mean of Poisson distribution

Let $X$ have a Poisson distribution with double mode at $x=1$ and $x=2$. Find $ P(x=0)$.Here is my solution... $\mu= \frac {p(2) 2!}{p(1)}$. then how can find the mean..thanks
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1answer
23 views

Find the cummulative distribution function and the density function of the random variable: $Y={1\over 1+U}$

Let $U$ have a uniform distribution on $[0,1]$. Find the cummulative distribution function and the density function of the random variable: $Y={1\over 1+U}$ My attempt: $F_Y(x)=P[Y\le x]=P[{1\over ...
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0answers
7 views

What is the transformation that maps a Gaussian distribution to a Beta distribution?

Suppose X is a random variable with Gaussian distribution over domain $\mathbb{R} = (-\infty, +\infty)$, with PDF function $f_X$. And Y is a random variable with Beta distribution over domain ...
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1answer
27 views

Probability of Sample Variance Given Variance

I am trying to solve a problem that I have never seen before and cant seem to find a way to solve it so any help or tips would be appreciated! Here's the Problem: Suppose a considerable amount of ...
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0answers
23 views

Let $X$ be a continuous random variable with cdf $F$. Show that $Y = F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{−1}(Y)$

Let $X$ be a continuous random variable with cdf $F$. Show that $Y = F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{−1}(Y)$. My Sol: $P(Y \leq y ) = P(F(X) \leq y) = P(F^{-1}(F(X)) ...
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1answer
36 views

Shortcut to finding $E(XY)$

The question says "Find $E(Y|X)$ and hence evaluate $E(Y)$ and $E(XY)$" The joint pdf is $$f_{X,Y}(x,y)=\begin{cases} 8xy, & \text{ for } 0< y< x < 1, \\0, & \text{ elsewhere } ...
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1answer
13 views

Distributing dimes to 2 groups of people such that each member of one group gets at least one

I have a study question that I have the answer for, but I just can't understand how or why it is the answer. The question is: $n$ dimes are distributed to $y$ young people and $o$ old people. Every ...
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0answers
15 views

Prove $|\log F(v)|\leq |\log F(0)|+|v|+|v|^2$ for $F$ is the standard normal CDF

Suppose that $F$ is the CDF of a standard normal distribution. Hayashi (2000) claims that the following is true $$ |\log F(v)|\leq |\log F(0)|+|v|+|v|^2\quad\text{for all}\quad v. $$ How does one ...
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0answers
15 views

Distribution of the square of magnitude of a Nakagami random variable [on hold]

Given the random variable $$h \sim \operatorname {Nakagami} (m,1)$$ $$ f_{h}(h)= \frac{2m^m}{\Gamma(m)} h^{2m-1} \text{exp}(-m h^2)$$ What is the distribution of the following function $$g:=|h|^2$$
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1answer
55 views

A die is rolled 24 independent times. Let Y be the sum of the 24 resulting values. What is the variance of Y?

A die is rolled 24 independent times. Let Y be the sum of the 24 resulting values. What is the variance of Y? I know how to find the mean of Y, but I'm having some trouble finding the variance of X ...
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1answer
19 views

Is there an interpretation of the Beta Distribution?

There are cases in probability where one distribution has an "interpretation" in terms of another distribution: X ~ Gamma(k,1/m) for positive integer k, can be interpreted as the distribution of ...
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0answers
35 views

Exponential.Damaged lamps replaced and not replaced

The problem: In a system there are 10 lamps installed.Their life cycle of every lamp is exponentially distributed with mean 5000 hours. A)If all the damaged lamps are being replaced immediately with ...
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0answers
15 views

Expected Value of Continuous Random Walk

I'm currently attempting my MMath master project. However, i'm a little stuck on an expected value of the continuous distribution. Its where i wish to change my random walk from a continuous walk to ...
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1answer
15 views

Convolution CDF formula?

In reference to this post, the pdf of dependent random variables $A+B$ is given by: $$f_{A+B}(z) = \int_{-\infty}^{\infty} f_{A,B}(a,z-a) \mathrm da = \int_{-\infty}^{\infty} f_{A,B}(z-b,b) \mathrm ...
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0answers
26 views

Questions regarding a Gamma distributed Random Variable ( first moment and square density)

Consider the following Gamma distributed RV $$\operatorname{Gamma }(m_S,\theta_S)$$ with the following shape and scale parameters $$m_S = \frac{(\theta_1+\theta_2)^2}{\theta_1^2+\theta_2^2}$$ ...
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1answer
31 views

Showing that $X_n$ ~ $N(0, a_n)$ converge to $0$ when $a_n \to 0$ sufficiently fast

If $X_n$ have distribution $N(0, a_n)$ with $\sum_{n=1}^\infty a_n^b < \infty$ for some $b > 0$, then $X_n$ converge almost surely to $0$. I was able to show (for a previous part of the ...
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0answers
22 views

If the difference of two i.i.d. random variables is normal, must the variables themselves be normal?

I previously asked a similar question about the sum of two i.i.d. random variables, thinking the two cases to be equivalent. But I can't see how to apply the proof of that case to this one. It is ...
1
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1answer
27 views

Maximum of RVS independent and identically distributed

I am having a small doubt regarding maximum of random variables. I have $$Z= \max\{ X_1, X_2,\dots X_p, \dots X_N\}$$ where all $X_i$ are independent, identically distributed. Now, If for sure, I know ...