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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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Exponential distribution in convergence in probability

Let $(X_n)_{n\in N}$ be a sequence of independent random variables such that $X_n \sim Exp\{n\}$ and let $Y_n := (1/n)\sum_{i=1}^{n}X_i$ for $n \in N$. Does the sequence ($Y_n$) converge in ...
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1answer
15 views

Applying the Union Bound in a conditional probability involving a stopping time

Let $b>0$. Consider a sequence of independent and identically distributed random variables $\{X_n\}_{n=1}^\infty$ and the corresponding random walk $S_n = \sum_{k=1}^n X_k$. Define the stopping ...
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0answers
18 views

Showing that two random variables are independent

I have the following problem: Given two independent standard normal random variables, call them $X$ and $Y$, how can I show that $Z = X^2 + Y^2$ and $W=\frac{X}{Y}$ are also independent? I know ...
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numerically solve summation of random variables integral

I want to add two random variables, $X,Y$ together $Z=X+Y$. They may be correlated. The integral is $$f_z(z)=\int^{\infty}_{-\infty}\int^{z-x}_{-\infty}f_{x,y}(x,y)dxdy=\int_{-\infty}^{\infty}f_{x,y}(...
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2answers
31 views

Independence of a combination of iid random variables

I encountered 2 questions regarding independence of 2 random variables. 1) A and B are i.i.d normal and independent, Let $X = 3A + 2B$ and $Y = 2A - 3B$, prove that they are independent 2) Will they ...
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1answer
16 views

expected time for happening the first events of three independent Poisson random variables

suppose we have three independent Poisson random variables $X_1$ and $X_2$ and $X_3$ with the same $\lambda$. We want to have the expected time we need to wait so all of three of them be more than ...
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1answer
19 views

For two continous independent r.v.s what does $\int_Y f_x(y)dy$ mean?

Does the $y$ variable act as an indicator function such that $f_X(y)$ only has values for elements $\omega \in X\cap Y$ ? Is $\int_Y f_x(y)dy$ the probability distribution of $X$ over the sample ...
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1answer
24 views

division of two PDFs

I am trying to solve the following question. I have come up a solution to this but need to know that is my approach correct in solving the question. The question is as follows A Test is 1 hour long, ...
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0answers
12 views

Why do we use the Poisson Probability Mass Function any longer?

My understanding is the Poisson Distribution is helpful to model experiments where there are many independent trials of Bernoulli experiments and each trial has a small chance of success. It is used ...
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1answer
23 views

distribution of fractions of partial sums of exponential random variables

Let $X_1, X_2, ...$ are iid exponential random variables, $S_k=\sum_{i=1}^{k}X_i$. I want to find the distributions of $S_k/S_n$ for $k=1,...,n-1$. i first used transformation $Y_i=S_k/S_n , (k=1,......
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2answers
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Expected Values of Continuous Random Variables

Suppose I have a function $L(c)$ , where $c$ is a continuous random variable. Let $L(c)=c+log(E[\theta])$ where $\theta$ is a continuous random variable and $E$ representing the corresponding ...
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Randomized Submatrix of a Sparse Matrix

I have a sparse square matrix $A$ with size $n \times n$ and number of nonzero entries $nnz$. The goal is making a sub-matrix $B$ with $s$ nonzeros which are randomly chosen from $A$. Duplicates are ...
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I need to write python code for one conditional probability question ?Need idea how i can write

Question is There are 2 identical boxes which have money in it (One has an amount twice than the other ). You may pick one box and keep the money it contains. Having chosen a box at will, but before ...
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25 views

Which distribution has the following variance formula?

Given a set of data $x_i$, for $i=1\ldots N$. I know their variance is calculated by $$\sigma^2 = A \sum_{i=2}^N \ln^2\left(\frac{x_i}{x_{i-1}}\right)$$ where $A$ is a constant. how can I determine ...
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How to determine p in a Poisson Random Variable

This is Problem 2.2 from Tsitsiklis, Bertsekas, Introduction to Probability, 2nd edition. You go to a party with 500 guests. What is the probability that exactly one other guest has the sam ...
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1answer
28 views

Let X be distribution over N (the set of non-negative numbers), with mass P(X=i) = a/2^i, what is the value of a?

I am struggling with solution for following problem part of course about probabilities random variables, seek your kind help to show how to solve it, Let $X$ be distribution over $N$ (the set of non-...
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Sample from random normal with sliding mean

I have a uniform random variable $x$ and a normal random variable $z = \mathcal{N}(x, \sigma)$ (i.e. the mean is given by $x$). How can I draw samples $(X, Z)$ such that they correspond to their ...
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21 views

The independence of two random uniform distribution random variables

$y_1 = x_1 + x_0$; $y_2 = x_2 + x_0$. Suppose that $x_1$, $x_2$, and $x_0$ are independent with each other. They all follow the uniform distribution in $[0, 1]$. Then, I want to know if $y_1$ and $...
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17 views

Distinguish between gamma and log-normal distributions based on 95th percentile of a random variable

I know mean and variance of a skewed positive random variable $X$ analytically. in literature both gamma and log-normal distributions can be fitted to such a random variable. I know that to find the ...
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Expectation Operator for autocorrelation function

I am studying random processes at the moment and E, the expectation operator comes up a lot. I have a firm understanding of what it does in probability but in the context of auto correlation it just ...
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26 views

Probability - Liminf & limsup [on hold]

Someone could explain me how find limsup/Liminf of a sequence of rvs? Thanks in advance Let Xn a sequence of i.i.d. random variables with PMF p(x)=1/[x(x+1)] for x=1,2,3,… Find liminf Xn and limsup ...
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Convergence of ess-sup w.r.t. a sequence of empirical measures

Given a Polish space $E$ endowed with Borel $\sigma$-algebra. We consider a sequence of empirical measures $(\eta_N)_{N> 0}$ such that for all the bounded measurable test functions $f\in\mathcal{B}...
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37 views

Continuous random varaible: If X and Y are continuous random variables What are the conditions required for X/Y to be a continuous RV?

If there are two continuous random variables $X$ and $Y$. What are the additional conditions required on $X$ and $Y$ under which ratio of these random variables $\frac{X}{Y}$ is also a continuous ...
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1answer
53 views

If $X$ and $Y$ are independent $N(0,\sigma^2)$, then $X^2+Y^2$ and $X/Y$ are independent?

If $X$ and $Y$ are independent, then $X^2+Y^2$ and $X/Y$ are independent? I was solving the problem for the case that $X$ and $Y$ are independent $N(0,\sigma^2)$. So i found that $X^2+Y^2$ is ...
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3answers
61 views

$X$ , $Y$ are uniform, showing $X$, $Y$ are not dependent but $𝑐𝑜𝑣 [𝑋, 𝑌] = 0$

A pair $(𝑋, 𝑌)$ is a point on a single circle ${(𝑥, 𝑦) | 𝑥^2 + 𝑦^2≤1}$ Place this point in the form It is determined randomly and uniformly distributed. What i want to show is 𝑋 and 𝑌 are not ...
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0answers
22 views

CDF of Mixture of Two Random Variables

$\mathbb{P}(Z=X)=\alpha$ and $\mathbb{P}(Z=Y)=1-\alpha$, i.e. $Z$ is a mixture of two random variables $X$ and $Y$. I am stuck in the application of the law of total probability to show that $F_Z(z)=\...
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34 views

Probability of $P1$ winning championship [duplicate]

Two players $P_1$ and $P_2$ are playing the final of a chess championship,which consists of a series of matches.Probability of $P_1$ winning a match is $\frac{2}{3}$ and that of $P_2$ is $\frac{1}{3}$....
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How many roots exist for $y=sec(x)$

In the interval $( - \pi ,\ \pi ]$.There are 2 roots exist mentioned in the book. Could anyone please explain how? Exact question from book : Let $Y=\sec X$ .Compute $f_Y(y)$ in terms of $f_X(x)$ ....
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Functions of exchangeable random vectors

Consider the random vector $\begin{pmatrix} X_0\\ X_1\\ X_2 \end{pmatrix}$ with joint cdf $F$. Consider the random vector $ \begin{pmatrix} Y_3\\ Y_4\\ Y_5 \end{pmatrix}\equiv \begin{pmatrix} X_1-...
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1answer
38 views

How to show $U$ and $V$ are not independent random variable?

$U$ stands for the number of trials to get the first head, $V$ stands for the number of trials to get two heads. I used hand-waving proof, saying that you could not have the two heads trials without ...
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2answers
22 views

Simulate a discrete random variable

We have a discrete random variable $X$ with the following probability distribution \begin{equation*} p(X=i)=p_i,\quad i=1,2,\ldots 1000, \quad \sum_{i=1}^{1000}p_i=1. \end{equation*} How we can apply ...
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1answer
22 views

Equation for the expected value of a discrete random variable

I'm reading the book Introduction to Stochastic Processes, p.24. In proving the expected value for 'any non-negative random variable $X$', the author provides the following equation for the expected ...
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1answer
57 views

Relationship between Rademacher distribution and Normal distribution

Is there any relationship between Rademacher distribution and Normal distribution? The Rademacher distribution is given as The probability mass function of this distribution (https://en.wikipedia....
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1answer
25 views

PDF of sums of independent random variables confusion

Suppose that $X$ and $Y$ are independent continuous RVs with PDFs $f_X$ and $f_Y$ respectively. I want to find the PDF of $Z = X + Y$. The CDF of $X + Y$ is $$F_{X+Y}(z) = P(X + Y \leq z)$$ $$=\...
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1answer
42 views

Probability that Month selected has $30$ days

In a Leap year a month is selected at random and a day is selected at random and found that its fifth Friday. What is the Probability that selected month has $30$ days. My try: Let $A$ be an event of ...
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1answer
33 views

Expand the inequality $E(|X|^r)^\frac{1}{r}\le E(|X|^s)^\frac{1}{s}$ to $-\infty<r<s<\infty$

For a random variable $X$ and numbers $0<r<s<\infty$ $$E(|X|^r)^\frac{1}{r}\le E(|X|^s)^\frac{1}{s}$$ This almost immediately follows from Holder's inequality for random variables $Y,Z$ ...
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1answer
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Let $X_1\sim Laplace(0,\sqrt{1/2})$ and $X_2 \sim Laplace(1/2,\sqrt{1/2})$. Are $X_1$ and $X_2$ independent?

Let $X_1\sim Laplace(0,\sqrt{1/2})$ and $X_2 \sim Laplace(1/2,\sqrt{1/2})$. Are $X_1$ and $X_2$ independent? I understand that in case of independence, the joint pdf is the product of the marginal ...
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1answer
37 views

Is there a more efficient expected value estimator than the sample average?

I'm wondering if there is a known estimator for expected value that is more efficient than the sample average. If that is not the case for an arbitrary random variable, then maybe there are examples ...
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1answer
18 views

Show that $XY^{1/\alpha}$ is stable with index $\alpha \beta$

Suppose $X$ is symmetric and stable with index $\alpha$ and $Y$ is stable and nonnegative with index $\beta$. Show that $XY^{1/\alpha}$ is stable with index $\alpha \beta$. The text also gives the ...
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2answers
31 views

How do you infer the success probability of a Bernoulli random variable from independent samples

Let's say we have a coin (not necessarily fair) and we flip it 100 times and all of the outcomes were tails. We can immediately conclude that the probability of getting tails is not 0 and we ...
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1answer
24 views

From the marginal to the joint cdf

I have some doubts on the relation between the joint cumulative distribution function and its marginals. Consider a random vector $X$ of dimension $L\times 1$ with cumulative distribution function $...
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Sufficient conditions on the marginals for exchangeability of the joint distribution?

Consider a random vector $X$ of dimension $L\times 1$ with cumulative distribution function $F$ absolutely continuous. Let $F_1,..., F_L$ denote the marginal cdf's. Assume that the probability ...
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1answer
65 views

Random variable $X$ of sum of random numbers [closed]

$r$ different numbers are picked Among positive integers $1,2,3,...,s$. If random variable $X$ is the sum of the $r$ random numbers picked. So there is (n choose r) X. what is $Var(X)$. I tried this ...
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1answer
60 views

Expectation of the reciprocal of a random variable

Let $X_1, X_2, \ldots, X_n$ be i.i.d. Bernoulli random variables with $P(X_i = 1) = p$ for every $i \in \{1, \ldots, n\}$. Let $Y = \sum_{i=1}^n X_i$ and let $c$ be a positive number. I am interested ...
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2answers
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What is a continuous random variable? A Collection of definitions

Although this is a question about what's a continuous random variable, it seems that there are at least 2 definitions being used. The Distribution function is continuous. There exists a non-negative ...
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Let Z ∼ Exponential(4). Compute each of the following. (c) P(Z^2 ≥ 9) (d) P(Z^4 − 17 ≥ 9) [closed]

I got part C by changing it to 1- P(-3 < Z <3), but it doesn't work for D.
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2answers
20 views

renormalization of sum of continuous random variables

I want to sum two random variables. So $Z = X+Y$ and $f_{X+Y}(z)=\int_{-\infty}^{\infty}f_{xy}(x,z-x)dx $ So I wanted to test this out a bit. If $$f_{xy}(x,y) = 6∙10^{-4}(x^2+y^2)$$ when $-5\leq x ...
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1answer
34 views

Find the probability that a random variable is “good” and find the expected value of the number of “good” random variables.

Suppose that $X_1,\cdots,X_n$ are i.i.d. continuous random variables. $X_k$ is called good if we have $X_i < X_k$ for all $i<k$. 1) Find the probability that $X_k$ is good. 2) Find the ...
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21 views

Alternative formula for covariance

I have a basic question on the covariance formula. Consider two random variables $X,Z$ with well defined first moment. Hence, $$ (\star) \hspace{1cm}cov(X,Z)\equiv E\Big[(X-E(X) (Z-E(Z))\Big]=E(XZ)-E(...
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1answer
33 views

R types of ants live in a place. Find the expected values.

$R$ types of ants live in a place. A person is catching ants until he picks an ant of type $1$. Since the number of ants in the area is high, you can assume that picking a new ant each time is ...