Tagged Questions

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

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1
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0answers
2 views

Why the doubly non-central F distribution does not have a mean or variance if the denominator degree of freedom is less than or equal 2 ??

Normally the doubly non-central F distribution is generated by the division of two non-central chi squared Random Variables,, so what is the the problem of using any famous formula to get the mean of ...
1
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1answer
10 views

A hand of six cards is dealt from a standard poker deck. Find formula for p_(XYZ) (x,y,z).

A hand of six cards is dealt from a standard poker deck. Let X denote the number of aces, Y the number of kings, and Z the number of queens. a) write a formula for p_(XYZ) (x,y,z). b) Find ...
2
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1answer
19 views

Frankie and Johnny game. What should Johnny strategy if he wants to minimize his expected loss?

Frankie and Johnny play the following game. Frankie selects a number at random from the interval $[a, b]$. Johnny, not knowing Frankie’s number, is to pick a second number from that same inverval and ...
1
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2answers
25 views

A random variable $X$ uniformly distributed over the interval $[0, 2\pi]$

A random variable $X$ distributed over the interval $[0, 2\pi]$ a) the pdf of $X$ b) the cdf of $X$ c) $P(\frac{\pi}{6} \leq X \leq \frac{\pi}{2})$ d) $P(-\frac{\pi}{6} \leq X \leq \frac{\pi}{2})$ ...
1
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0answers
22 views

Gaussian random vector with 0 mean [duplicate]

Let $X =(X_1,X_2,X_3,X_4)$ be a Gaussian Random Vector with $\mathsf E(X_1)=\mathsf E(X_2)=\mathsf E(X_3)=\mathsf E(X_4)=0$. Show that $$ \mathsf E(X_1 X_2 X_3 X_4) = \mathsf ...
1
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1answer
18 views

Poison distribution variance,probability. and mean.

Let $X$ be the poisson random variable such that $P(X = 2) = 9P(X=4) + 90P(X=6)$ a) find the mean and variance of $X$. b) find P(X $\geq 1$) c) find P(X $\leq 10$) Ok so for the first question I ...
2
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1answer
12 views

How is this Variance found in this old question?

On this question: Probability: Normal Distribution they find these values: $\hat\mu = .05(150) = 7.5\space,\hat\sigma = \sqrt{150(.05)(.95)} = 2.67$ I see how they got $\mu$, but how did they get ...
-2
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0answers
31 views

Gaussian Random vector with zero mean

Let $X =(X_1,X_2,X_3,X_4)$ be a Gaussian Random Vector with $\mathsf E(X_1)=\mathsf E(X_2)=\mathsf E(X_3)=\mathsf E(X_4)=0$. Show that $$ \mathsf E(X_1 X_2 X_3 X_4) = \mathsf ...
2
votes
2answers
35 views

Motivation behind study of martingales

Today I wanted to ask a question which I am sure has been answered in multiple places but for which I do not yet have a very clear understanding. Though martingales is a very well explored area of ...
1
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1answer
19 views

find the mean and variance of this poisson random variable

Let $X$ be the poisson random variable such that $P(X = 2) = 9P(X=4) + 90P(X=6)$ find the mean and variance of $X$. I'm not sure how to approach this problem..am i supposed to multiply each ...
2
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1answer
12 views

Find mass function with 3 dice and 3 different Xs

There are $3$ dice you roll one at a time, $X$ is the number of distinct numbers, as in, $X=1$, you have $(1, 1, 1)$ since there is $1$ distinct # $X=2$, $(1, 2, 1)$ or $(2, 1 ,1)$ etc... $X=3$ all ...
0
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1answer
19 views

Finding the probability of a randomly selected event?

I know I'm over-thinking the following question, I just need to know how to start! In a certain population of women 4% develop symptoms of a classic disease, 20% are smokers, and 3% are smokers and ...
0
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1answer
19 views

Represent probability with multiple distributions. Archer shooting bullseyes problem.

The goal is to come up with two ways to represent this probability: An archer shoots a bulls-eye with probability $0.4$. If the archer shoots ten arrows, what's the probability that at least 3 are ...
0
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1answer
18 views

Limit of a jointly independent sequence of random variables

Can I say the following? If a jointly independent sequence of random variables $X_1,X_2,\dots$ converges to random variable $X$ in the mean square sense, then $X$ is independent of the elements of ...
2
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2answers
21 views

Expected value of X-x for exponential distribution

Assume $X\sim$ exponential$(\lambda)$. In class we noted that $E[X-x|X\geq x]=\frac{1}{\lambda}$. Why is this? I would have thought that $E[X]-E[x]=\frac{1}{\lambda}-x$.
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1answer
19 views

Random Distance on Torus

Let $U=(X_U, Y_U)$ and $V=(X_V, Y_V)$ be two independent random points in $[0,1] \times [0,1]$, where each possible position is equally likely. Now I am interested in the probability that these two ...
0
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2answers
26 views

Find $E[\max (R_1, R_2)]$ when $R_1$ and $R_2$ are independent and uniformly distributed in $[-1,1]$

Find $E[\max (R_1, R_2)]$ when $R_1$ and $R_2$ are independent and uniformly distributed in $[-1,1]$. So first I was thinking something along the lines of $$P(R_1 = n, R_2 \leq R_1)$$ would be ...
1
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1answer
53 views

Let X ∼ Unif (0, 2). What is E[exp(2X/3) − 3]?

Let X ∼ Unif (0, 2). What is E[exp(2X/3) − 3]? $E[e^{\frac{2X}{3}} - 3] = \int_0^2 \! e^{\frac{2X}{3}} - 3 \, \mathrm{d}x$ $= \frac{3}{2}(e^{\frac{4}{3}} - 5) = -1.8095$ I am integrating over the ...
1
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1answer
19 views

How to express combined discrete-continuous RVs in one pdf?

Let's say we have a random variable $X$ that behaves in two different ways where $X\sim$Bernoulli(1/3) AND $X\sim U(0,1)$. $X$ follows the Bernoulli distribution 25% of the time and the uniform ...
0
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1answer
16 views

How do you say that variable is randomly chosen with a random distribution for range [3, 42]?

This question is only about how to formulate something in English for a bachelor's thesis in computer science. I have a variable $x$ which is randomly initialized. It is chosen from a (continuous) ...
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0answers
24 views

Stochastic processes

Update I am a bit confused whether $y_t$ is independent over time under the following assumptions: Consider, first a RV $A$, that follows this process: $A_t = \rho A_{t-1} + e_t$, where $e_t$ is ...
0
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0answers
21 views

How do I simplify $\sum_i^n(y_i-rx_i)^2$, where $r = \frac{\sum y_i}{\sum x_i}$?

I want to simplify: $$\sum_i^n(y_i-rx_i)^2$$ where $y_i$ and $x_i$ are random variables and $r = \frac{\sum y_i}{\sum x_i}$. I've tried expanding the summand and replacing $r$ with $\frac{\sum ...
0
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0answers
11 views

Conditional Expectation of a vector of random variables given another vector of random variables

Let $\{X_i\},\;i=\{1,2,3,4,5\}$ be a set of 5 continuous random variables . Then how do I calculate $\mathbb{E}[(X_1,X_2)\mid (X_3,X_4,X_5)]$ where $(X_1,X_2),(X_3,X_4,X_5)$ are two vectors of given ...
0
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0answers
11 views

maximization of a function with random variable

I would like to know whether this is true in general, and if not when this can be. I am not sure and so I am mostly asking for confirmation. So, is the following correct ? $$\log [\max_{x} ...
4
votes
2answers
69 views

Uniform distribution, as a sum of biased Bernoulli trials.

Suppose that the probability of $x=0$ is $p$, and the probability of $x=1$ is $1-p=q$. Consider the random sequence $X=\{X_i\}_{i=1}^{\infty}$. We map this sequence by $C$ to a point in the interval ...
2
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1answer
38 views

If the variance is $0$ is it constant?

We know that the variance of a constant is $0$. Is the converse also true? Can we say that if the variance of some random variable is $0$ it is a constant?
2
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1answer
31 views

Conditional expectation constant on part of partition

I have a question about conditional expectation, while looking for the answer here on stackexchange I noticed that there are a few different definitions used, so I will first give the definitions I ...
0
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1answer
37 views

Probability function of X and Y when two balls are drawn with no replacement

Two balls are drawn at random from a box containing ten balls numbered 0, 1, ... , 9. Let random variable X be the larger of the numbers on the two balls and random variable Y be their total. ...
1
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1answer
29 views

Solving the integral which shows the second moment of subtracting two Beta-distributed Random Variables

Peace be upon you In my project I needed to find the second moment of the subtraction of two Beta-distributed random variables. I have computed it and reached to the following integral which I should ...
0
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0answers
32 views

Relation between minimum and sum of two random variable

I am interested in finding a relation that involves two independant random variables, that could be used to describe the sum of these, or the minimum of these. For example, regarding the sum, we know ...
5
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0answers
47 views

Applying PCA on covariance matrix in order to generate a new random variable.

Let $\mathbf{x}$ be a random $n\times1$ real vector, $\mathbf{x}\in\Bbb{R}^n$, which is distributed normally with mean $\bar{\mathbf{x}}$ and covariance matrix $\Sigma_x\in\Bbb{R}^{n\times n}$, i.e. ...
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1answer
24 views

uniqueness of joint probability mass function given the marginals and the covariance

Let X and Y be two nonnegative, integer-valued random variables. Is there a way to find the joint probability mass function, i.e. $$ \mathbb{P}(X= k, Y= h) $$ for some $k,h\geq 0$, given the ...
0
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1answer
24 views

Expectation of a powered complex circular gaussian process

Assuming a complex circular zero-mean gaussian random process (or vector) $\textbf{x}$ $\left(\textbf{x}\sim \mathcal{CN}\left(0,\sigma^2\right)\right)$. $\mathbb{E}\{\textbf{x}\}=0$. The question ...
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0answers
21 views

Probability that one random variable is greater than or equal to another

Assume $X$ and $Y$ are i.i.d. with exponential distribution with parameter $\lambda = 1$ (the probability density functions $p_X (x) = e^{-x}$ and $p_Y (y) = e^{-x}$ in $[0, +\infty)$, $0$ otherwise). ...
1
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1answer
20 views

Elementary Probability: Expected Value

I must say, first, that this question IS a homework assignment and I do not wish an answer here, for I already posssess it. I want to know if there is a general procedure of simplification in this ...
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0answers
22 views

Weak convergence, measurability, uniform integrability

I've been facing the following problems: a) Let $ X_n \rightarrow X $ and $f $ be a measurable, bounded function. Prove that $ \mathbb{E}f(X_n) \rightarrow \mathbb{E}f(X) $ (we also assume that the ...
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0answers
14 views

Minimal and maximal elements of a set consisting of sums of random variables

Consider a set of $n$ correlated random variables, $A =\{X_1, \ldots, X_n\}$. Suppose that I have another set, $B$, of all possible combinations of size $k$ of the random variables. Now, if for each ...
1
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0answers
29 views

Probability of a coin falling on the edges of a square

Let a coin be randomly (and uniformly) dropped onto a square on the floor. Assume the edge length of the square to be $ d $ and the radius of the coin to be $ r < d/4$. I know that the probability ...
1
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1answer
25 views

Game of Red balls two drawings are made, which rule would you choose if playing the game, rule A or rule B?

In the game of redball two drawings are made without replacements from a bowl that has four white ping pong balls and two red ping pong balls. The amount won is determined by how many ping-pong balls. ...
0
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1answer
37 views

Geometric distribution

I am trying to solve following question, but I am stuck. Let $X = Y/n$ where $Y$ is Geom($1/n$) random variable. Find the distribution function of $X$ and find its limit as $n \to \infty$.
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1answer
25 views

A joint pdf question [closed]

I need help over a question. I appreciate all helps.Thank you.
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votes
0answers
69 views

Derive the expected value of $X^{0.5}$

I am doing a question considering a continuous random variable $X$ and have calculated $k=1/2$, $E(X)=3/2$ and $V(X)=5/12$ I am unsure of what the expected value of $X^{0.5}$ is. Consider the ...
0
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1answer
25 views

Using the binomial distribution as the distribution for a sum of Bernoulli random variables?

Knowing that the sum of $n$ independent Bernoulli random variables with parameter $p$ ($p \in (0,1)$) has a binomial distribution $Bin(n,p)$, how can I use the Central Limit Theorem (or any other ...
0
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1answer
21 views

Equivalent definition of random variables

I've come across the following two definitions of random variables and am trying to figure out if they are equivalent or not. Let $\Omega$ denote our sample space and $\mathscr{F}$ denote our ...
1
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0answers
11 views

Bound on difference of two i.i.d. variables [duplicate]

Prove that for every two independent, identically distributed real random varaibles $X,Y$, $$Pr(|X-Y|\leq 2)\leq 3\cdot Pr(|X-Y|\leq 1)$$ [Source: The probabilistic method, Alon and Spencer]
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2answers
16 views

Dependence of random variables

I need to solve the following problem: Let X be a normal random variable with mean  and standard deviation  and let I, independent of X, be such that P{I = 2} = P{I = -2} = 0.5. Let Y = I X. In ...
0
votes
1answer
25 views

Probability - Random viarbles

A notepad manufacturer requires that 90% of the production is of sufficient quality. To check this, 12 computers are chosen at random every day and tested thoroughly. The day's production is deemed ...
0
votes
1answer
26 views

Infinite boundary for random variables

I have a question Suppose that X and Y are random variables with joint pdf is given by and zero otherwise. I need to find marginal and conditional pdf's.But I don't know how to intagrate over an ...
0
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1answer
15 views

Joint distribution of two random variables

I have a question about joint distributions but couldn't find the solution. Suppose that $X$ and $Y$ are two random variables and their joint pdf is given by $$f_{XY}(x,y)=cxy(1-x-y), ...
0
votes
3answers
80 views

Prove that $\|x+y\|_{\infty} \leq \|x\|_{\infty} + \|y\|_{\infty}$.

Suppose $\left(X, \Sigma, \mu \right)$ is a measure space and $x,y \colon X \longrightarrow \mathbb{R}$ are random variables. We define $$\|x\|_{\infty} := \inf_{A \subseteq \Sigma, \mu(A)=0} ...