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Questions tagged [random-variables]

Questions about maps from a probability space to a measure space which are measurable.

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Induced measure by Random variable

Given measurable spaces $(\Omega,\mathcal{F},\textit{P}_{\mathcal{F}})$ and $(\mathbb{R},\mathcal{B})$, a random variable on $\Omega$ is defined as a measurable function $\textit{X}:\Omega \to \mathbb{...
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How to determine cumulative distribution function of squared random variable?

I don't understand how can I get the $F(x)$(cumulative distribution funciton) given only $X = RND^2$ where RND means (continuous) random variable, the paper tells the answer is $F(x) = \sqrt x$ ...
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1answer
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How can I find the expectation and variance of $Z=\max\{X,Y\}$ where $X$ and $Y$ are defined through joint probability distribution?

Random variables $X$ and $Y$ and have the joint distribution below, and $Z=\max\{X,Y\}$ $$ \begin{array}{c|lcr} \text{X\Y} & \text{1} & \text{2} & \text{3} \\ \hline 1 & 0.12 & 0....
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1answer
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“Following” $\operatorname{exp}(\lambda)$ random variables “sum” to $\operatorname{Poi}(\lambda t)$ random variable

Lifetime of a bulb is distributed $\operatorname{exp}(\lambda)$. When one light bulb is burned we replace it immidietely. Let $N_t$ be the number of bulbs we've used by time $t$. Prove that $N_t \sim \...
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1answer
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How. to find the c.d.f from a continuous random variable?

Given the following density function for continuous random variable x f(x)=2/3($x^4$+5$x^3$+2$x^2$+3x-2) for x between [2,3] Determine the c.df and use the c.df to determine p(2.25 < x < 2.5) ...
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18 views

Establish bound for a probability using moment generating function

I have the following question Let $X_{1}$, $X_{2}$, ..., $X_{n}$ be independent and identically distributed random variables with moment generating function $M_{X}(t)$, for -h < t < h, ...
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CDF's properties

I need a help to prove all the properties of CDF to better understand properties and characteristics of bidimensional CDF. By definition of CDF, I know that: $F_X:\mathbb{R}\rightarrow [0,1]: F_X(x)=\...
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1answer
31 views

Probability density function of $X$ with a uniform distribution on a unit sphere

Given that $f_\Phi(\phi)=\dfrac{\sin(\phi)}{2}$ for $\Phi\in[0,\pi]$, and $f_\Theta(\theta)=\dfrac{1}{2\pi}$ for $\Theta\in[0,2\pi]$, where $\Phi$ and $\Theta$ are independent. What is the PDF of $X=\...
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Distribution of the square of the sum of independent rayleigh variables

Suppose $\alpha_i$ is the $i$th independent Rayleigh distribution random variable following a Rayleigh probability density function (PDF) as \begin{equation} f_{\alpha_i}(r) = \frac{r}{\sigma_i^2} \...
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1answer
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$\operatorname{supp}(f_{X,Y})=\{(x,y)\in\mathbb{R}^2||x|+|y|\leq1\}$ then $X,Y$ are not independent

Let $Z=(X,Y)$ be a absolutely coninuous random variable such that $$ \\\operatorname{supp}(f_Z)=\{(x,y)\in\mathbb{R}^2||x|+|y|\leq1\} \ $$ Show that $X,Y$ are not independent. I don't have a good ...
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Determine the Expected Value of Uniformly random elements of sets

Answer is D The way I attempted this was that for X = MAX(a,b), the random variable X is equivalent to the max value of a and b. So, from the 2 sets, the probability of getting k from set {1,2...100} ...
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Perfect correlation between two random variables: some clarifications [on hold]

Consider two random variables $X,Y$. $X$ can take value $x_1$ with probability $p$ and $x_2$ with probability $1-p$. $Y$ can take value $y_1$ with probability $p$ and $y_2$ with probability $1-p$. ...
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1answer
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Is the average of iid random process independent stationary increments?

If $I_1, I_2, I_3, ...$ is a iid process. Then can I say that $$M_n = \frac{1}{n}\sum_{i=1}^n I_i$$ Is a independent stationary increments? I think it shouldn't, but I am not sure if my answer is ...
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Expected Value and Random variables for Uniformly Random Permutation Sets [duplicate]

Question Let $n$ and $k$ be integers such that n is even, $n\ge2$ and $1\le k\le n$. You are having a party where $n$ students attended. a) $k$ of these $n$ students are politically correct ...
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1answer
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Proof verification: Tightness of a family of random variables converging in distribution.

I'm trying to solve the following exercise: Suppose that $X_n \to X$ in distribution. Show that $(X_n)_{n \geq 1}$ is a tight family. First, just to recall the definition for those possibly ...
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Statistics - Question on Sampling

Here's the question The scores, $X_1$ and $X_2$, in papers $1$ and $2$ of an examination are normally distributed with means $24.3$ and $31.2$ respectively and standard deviations $3.5$ and $3.1$ ...
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Convergence in probability of sum of bounded random variables implies finite expectation

Let $(X_i)$ be sequence of independent random variables with $|X_i| < 1$ almost-surely and define $M_n :=\sum_{i=1}^{n}X_i$. Suppose it is known that $M_n$ converges in probability to some random ...
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1answer
30 views

Сoincidence of discrete random variables

Let $\xi, \eta$ be a discrete random values and $\mathbb E| ξ |$, $\mathbb E | η |$ < $+\infty$, and any value of these values ​​are accepted with a non-zero probability. How to prove that from $\...
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Prooving $\mathbb{P}(\xi_1+ \xi_2+…+\xi_n=1)=(\sum_{i=1}^{n}\lambda_i)\Delta + \mathcal{R}\Delta^2$

Let $\xi_1, \xi_2,...,\xi_n$ be independent Bernoulli random variables in $(\Omega,\mathcal{P}(\Omega),\mathbb{P})$ and $$\mathbb{P}(\xi_i=0)=1-\lambda_i\Delta$$ and $$\mathbb{P}(\xi_i=1)=\lambda_i\...
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1answer
26 views

How to find CDF of $Y=|X|\wedge 2$ with $X\sim Laplace(\lambda)$

Given $X$ a bilateral exponential with density $f_X(x)=\frac{1}{2}e^{-|x|}, \forall x\in \mathbb{R}$ and $\lambda=1$, i have to find CDF of $Y=|X|\wedge 2$. I know that $Y$ is not a monotonic ...
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1answer
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Existence of a function that generates a Borel set

How to prove that there exists $\xi: \Omega \to \mathbb R$, which is not a random variable such that for all $x \in \mathbb R$ $\xi^{-1}(x) = \{ \omega | \xi(\omega) = x \}$ is a Borel set?
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1answer
32 views

Finding how 3 hunter can share their prey

Three hunters hit the moving target with probabilities $0,6, 0,4$ and $0,2$. When they saw a deer they shot in one time. After it they saw that only one bullet reached the target. So how they can ...
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Entropy of discrete variable

Let entropy of discrete variable $X$ be $H(X) = -\sum\limits_x{P(X=x)log(P(X=x))}$. Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be any function. 1.Show that $H(X) \ge H(f(X))$ 2.Show that $H(X) = H(f(...
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1answer
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Distributions of peaks on $S_n$

I'm ultimately trying to compute the second moment of $X$, or $E[X^2]$. I don't see another way to do this other than to determine the probability distribution $P(X = k)$. It seems to me that for ...
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1answer
33 views

How to find pdf of X+Y given X and Y are dependent.

The joint pdf is f(x,y) = $$\frac{2}{5}(2x+3y)$$ for $0\leq x \leq 1,0\leq y \leq 1$ Normally if the random variables are independent, you can apply the convolution definition Z = X + Y which looks ...
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Coin toss- expected value.

A coin had tossed three times. Let: $X$-number of tails $Y$-number of heads Find the expected value and variance $Z=XY$ My solution: $E(Z)=E(XY)= 2 \cdot 1/4 + 3 \cdot 1/4=6/4$ Because, I know that $...
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1answer
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Expected number of good presents

Given $b$ boys and $g$ girls. Children give presents to each other. They know who gives a present whom from random permutation of $1,2,\dots b+g$. If child gives present to child with same gender ...
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1answer
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joint PDF of 2 dependent variables w/ convolution

Let $X, Y$ be iid standard uniform variables, and let $T = X + Y$. The goal is to find the joint PDF of $X$ and $T$. The work I've done so far is to find the PDF of $T$ by evaluating the convolution, ...
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1answer
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Compute the Poisson distribution

I have the following problem: " In a city two serious accidents happen per week on average. In particular, we assume that the number of serious accidents is Poisson distributed. Calculate the ...
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Discrete positive moment problem

My teacher claims that, given all factorial moments $E((X)_r) = E(\prod_{i=0}^{r-1}(X-i))$ of a positive discrete random variable $X$ it is possible to deduce the law of said variable. The first ...
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1answer
33 views

Equivalence of the condition that the supremum of i.i.d. RVs are finite a.s.

I am proving the following : Suppose $\{X_n : n\in\mathbb{N}\}$ are i.i.d. random variables. Then $P(\sup_{n\in\mathbb{N}}X_n < \infty) = 1$ if and only if $ \sum_{n\in\mathbb{N}}{P(X_n > M)} &...
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Looking for a proof of : variance of sum is the sum of variances.

For independent random variables X and Y, the variance of their sum or difference is the sum of their variances: I can see why above should be true : if $x_1<X<x_1$ and $y_1 <Y < y_2$, ...
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The relationship between sample and population skewness [on hold]

Get the sample skewness coefficient of a random variable X from a random sample of size n. Explain why it measures asymmetry of the distribution of X. Explain the relation between the sample skewness ...
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72 views

Finding a distribution function of random variable sum

Let $\xi_1, \xi_2, \xi_3$ independent random variables in $(\Omega, \mathcal{F},\mathbb{P}).$ Also, they are evenly distributed in $[0,1]$. I need to find a distribution function of sum $\xi_1+ \xi_2+...
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Expected number of coin flips until all cars move to end of array?

Imagine that we have an array of length $2n$, where the first $n$ entries are a $C$ (representing a toy car) and the remaining $n$ entries are empty. Additionally, we have $n$ fair coins labeled $1$ ...
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Probability: the numeric value of a drawn ball

An urn contains N balls. Let $X$ be a random variable to specify the numeric value of a drawn ball twice for the first time (with replacement). I would like to find the following probability: $P(X=k)$ ...
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1answer
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Compute mean and covariance matrix of $\bar{X}$ from a simple random sample

Given $\{X_\alpha , \alpha =1,...N\}$ a simple random sample obtained from any p-dimensional distribution with mean $\mu$ and covariance matrix $\Sigma$, compute the mean and the covariance matrix of $...
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pdf of product of two function of the Exponential variables

I would like to find this probability: $Pr(Z<2^{2R})$ for $R>0$. So, I try several ways and finally decide to find pdf of Z. If $X$ and $Y$ are independent and exponentially distributed with ...
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Inequalities of Random Variables

Working through some problems from Introduction to Probability, Blitzstein Let X and Y be i.i.d. continuous r.v.s. Assume that the various expressions below exist. Write the most appropriate of$\...
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1answer
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Moment generating function of i.i.d

I was reading this pdf https://ocw.mit.edu/courses/mathematics/18-443-statistics-for-applications-fall-2003/lecture-notes/lec15.pdf I have two questions I know about moment generating function of ...
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The fourth moment of a centered random variable is at least equal to the square of its variance

Let X be a random variable with mean µ and variance $\sigma^2$. Show that $E(X-\mu)^4\geq \sigma^4$ and use this to show that the kurtosis of $X$ is at least $-2$. This looks like a form of ...
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Excepted value of infinite variables [on hold]

We have $X_1, X_2,X_3,...$ an infinite sequence of random variables which is randomly distributed on the set $\{1,2,3,...,10\}$,$X_i$ would be called king if for each j $\lt$ i we know that $X_i\gt ...
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1answer
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Relationship between normal sample variance (mean known or unknown) and Chi-squared

I know that $(n-1)s^2/\sigma^2$ is Chi-squared with $n-1$ degrees freedom. I'm currently working on a question where the population mean $\mu$ is known, i.e. $s^2=n^{-1}\sum_{i=1}^n(X_i-\mu)^2$. My ...
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If $f(t,x)$ is continuous and $B_{t}$ has continuous paths, then $f(t,B_{t})$ converges almost surely

Let $f \colon [0,\infty) \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ be a function which is continuous in both variables $t$ and $x$. Let $(B_{t})_{t \in [0,\infty)}$ be a stochastic process ...
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Excepted value of maximum of random variables

let $\mathit X\sim Geo(p) $ and $\mathit Y\sim Geo(3p)$ independent random variables and $\mathit M=Max\{X,Y \} $. Find $\Bbb E[ M ]$? We know that $\Bbb E[ Y ]=\frac{1}{3p} $ and $\Bbb E[ X ]=...
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1answer
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Expectation and variance of travel time with several options for the transportation

A person is traveling between two places, and has 3 options for transportation. The jth option would take an average of µj hours, with a standard deviation of $\sigma_j$ hours. The person randomly ...
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1answer
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Convolutions: Let U~Unif(0,1) and X~Expo(1), independently. Find the PDF of U +X.

Let U~Unif(0,1) and X~Expo(1), independently. Find the PDF of U +X. Solution: $f_T(t) = \int_{-\infty}^{\infty}f_Y(t-x)*f_X(x)dx$ $= \int_{-\infty}^{\infty}1*\lambda e^{-\lambda x}dx$ Integrate ...
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1answer
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Find PMF of X^2 if X~Dunif(0,1,…,n)

Follow up on this: Find PMF of $X^2$ if $X$~Dunif (I do not have enough "reputation points" to comment, so if this is an inappropriate way to ask for a follow up, please let me know) Is this a ...
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1answer
18 views

Let $U \sim \textrm{Unif}(0, \pi/ 2)$. Find the PDF of $\sin(U)$.

This is almost the same as Suppose that X ∼ U ( $− π/2$ , $π/2$ ) . Find the pdf of Y = tan(X)., but making sure I am understanding the process: Let $U \sim \textrm{Unif}(0, \pi/ 2)$. Find the PDF ...
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1answer
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Prooving $\mathbb{I}_{\{A \Delta B \}}=(\mathbb{I}_{\{ A\}}-\mathbb{I}_{\{ B\}})^2$

Let $A$ and $B$ be two events from $\Omega, \mathcal{P}(\Omega),\mathbb{P})$. I need to show that next equal is true $$\mathbb{I}_{\{A \Delta B \}}=(\mathbb{I}_{\{ A\}}-\mathbb{I}_{\{ B\}})^2$$. I ...