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

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0
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1answer
22 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?
0
votes
1answer
29 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. ...
0
votes
1answer
19 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
23 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
votes
0answers
33 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. ...
1
vote
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
votes
1answer
22 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 ...
0
votes
0answers
12 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
vote
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 ...
1
vote
0answers
18 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 ...
1
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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
vote
0answers
27 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
vote
1answer
24 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. ...
1
vote
0answers
26 views

What is the variance E[A]^2, statistics? [on hold]

$x(t)= A_i$, for $i \leq t < i + 1$ and $\{i = 0, 1 ,2 ,3,.....\}$. $A_i$ are independent variables, pmf of $P(A_i = 1) = P(A_i = -1) = 1/2$. Find the variance $E[A]^2$. I am so stuck on this ...
0
votes
1answer
36 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 $n \to \infty$.
0
votes
1answer
24 views

A joint pdf question [on hold]

I need help over a question. I appreciate all helps.Thank you.
-2
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
votes
1answer
21 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
votes
1answer
20 views

Conditional Probability of random variables! [closed]

X, Y, Z are i.i.d continuous random variables. How can I compute (1) P(X>Y|X>Z) (2) P(X>Y|Y>Z) ? It seems to be easy but at the same time, confusing! Help me. Thank you:)
0
votes
1answer
20 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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0answers
30 views

Calculating Map estimate [closed]

Hello everyone I am stuck on this problem: Given N independent measurements from an experiment that generates exponentially distributed random variables: $$f(x)={1\over y}e^{-x\over y}$$ ...
0
votes
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
24 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
25 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
votes
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
79 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} ...
0
votes
1answer
24 views

Are integer angles (in radians) uniformly distributed?

Suppose that I have a random variable $X = \sin(T)$ where $T$ was drawn from the uniform distribution on $[0,2\pi)$. Upon generating samples for this random variable, the usual practice you see is to ...
0
votes
1answer
34 views

The characteristic function of the random time $N$

The rv's $X_1,X_2,X_3,\ldots,X_n$ are I.I.D and have the following pmf's: $$p_x(-1)=1/4\quad p_x(0) = 1/2,\quad p_x(1) = 1/4$$ The random time $N$ is defined as: $$N = \min\{n \mid X_n = 0\}$$ ...
2
votes
1answer
23 views

Infinitely many “records” of uniform random variables

I am doing the following exercise: Let $(U_n)_{n \geq1}$ be iid uniform random variables on $[0,1]$. Define the event $E_n = U_n>\max \lbrace U_1, U_2, \dots, U_{n-1} > \rbrace$. I.e. the ...
1
vote
1answer
170 views

“Who's Taller” game with random variables

I have an exercise that I cant get my head around. You play the game of who's taller" in class (of n people). You pick always a random opponent among the people you haven't yet played, compare your ...
0
votes
0answers
19 views

Why are Indicator Random Variables better that Random Variables when analyzing algorithms?

I understand the idea behind a random variable and the indicator random variable. BUT my question is why use indicator random variables if we have random variables? How do these indicator random ...
1
vote
2answers
14 views

Variance of sample mean (problems with proof)

Assuming that I have $\{x_1,\ldots, x_N\}$ - an iid (independent identically distributed) sample size $N$ of observations of random variable $\xi$ with unknown mean $m_1$, variance (second central ...
0
votes
0answers
19 views

Show that $E(Y|X)(\omega):=\sum_{x\in X(\Omega)}E(Y|X=x)\chi_{\left\{X=x\right\}}(\omega)$ is a discrete random variable

Let $X,Y$ be two discrete, integrable random variables, which are defined on a probability space $(\Omega,\mathcal{A},P)$. Recall that the conditional expectation of $Y$ given $X=x$ is defined ...
1
vote
1answer
11 views

Limit on the variance of a positive random number

Let's say a friend tells me he needs my help for chucking wood. He tells me that this takes on average 10 minutes. This motivates my following question. Given an expectation value E on a positive ...
1
vote
0answers
44 views

Joint PDF of Chi-Square & Normal Distribution

Let the independent random variables X1 and X2 be N(0,1) and $\chi^2(r)$, respectively. Let $Y_1$ = $X_1/sqrt(X_2/r)$ and $Y_2$ = $X_2$ a) Find the joint pdf of $Y_1$ and $Y_2$. b) Determine the ...
0
votes
1answer
18 views

Finding variance and standard deviation of a random variable in an equation

Suppose that X is a random variable with mean 17 and standard deviation 5. Also suppose that Y is a random variable with mean 45 and standard deviation 11. Find the variance and standard deviation of ...
0
votes
1answer
23 views

Finding the mean of a random variable in an equation given standard deviation and mean

Please help! What do I plug into these equations to solve for the mean of Z?? Suppose that X is a random variable with mean 23 and standard deviation 5. Also suppose that Y is a random variable with ...
0
votes
0answers
9 views

Chernoff bound with three possible outputs

A slot machine return requires a player to put in \$1. It returns \$3 with probability $4/25$, returns \$100 with probability $1/200$, and returns nothing otherwise. Using Chernoff bound, what is the ...
2
votes
0answers
19 views

When can I leave the absolute value from Chebyshev's inequality?

I have a positive random variable which distribution is unkown, but its mean is $10$. I have to find an estimation of its variance, given, that $Pr(X\geq9$)=0.9980 I thought of Chebyshev's ...
0
votes
1answer
14 views

Explanation of this situation with two random variables - $X$ conditionally distributed on $N$?

Let $N$ have a Poisson distribution with parameter $\lambda = 1$. Conditional on $N = n$ let $X$ have a uniform distribution over the integers $0, 1, ..., n+1$. What is the marginal distribution of ...
0
votes
1answer
21 views

Help with the limits of integrals in a function of two random variables

I have this problem, I say that $x=z-y$, hence $F_z(z)=\iint\limits_D f(x,y)\ dx\ dy$, now I think that the limits for the $x$ integral would be from $0$ to $z-y$ and for $y$ would be from $0$ to ...
3
votes
1answer
29 views

General Weak Law of Large numbers

I came across a question regarding the WLLN. Suppose for $X \geq 0$ , $\mathbb{E}[X] = \infty $ , $S_n = \sum_{i \leq n} X_i$, $X_i$ are iid copies of $X$ , and $\frac{\mathbb{E}[X \mathbf{1} _{X ...
1
vote
1answer
29 views

One-sided variant of Chebyshev inequality

For random variable $X$ with standard deviation $\sigma$, and any $t>0$, show that $$\Pr(X-E[X]\geq t\sigma)\leq\dfrac{1}{1+t^2}.$$ Chebyshev's inequality yields $$\Pr(|X-E[X]|\geq ...
3
votes
3answers
42 views

Ratio of expectations for integer-valued random variable

For a nonnegative integer-valued random variable $Y$ with positive expectation, show that $$\dfrac{E[Y]^2}{E[Y^2]}\leq\Pr[Y\neq 0].$$ I suppose that the probability that $Y=i$ is $x_i$, for ...
0
votes
2answers
14 views

Random variable with finite expectation and unbounded variance

What is an example of a random variable with finite expectation and unbounded variance? I'm thinking about putting $1/n$ probability on each of $n$ equally-spaced points. Then as $n$ approaches ...
1
vote
1answer
21 views

A problem on almost sure convergence of an average

I have the following exercise: Let $X_1, X_2 \ldots$ be such that $$ X_n = \left\{ \begin{array}{ll} n^2-1 & \mbox{with probability } n^{-2} \\ -1 & \mbox{with probability } ...
10
votes
2answers
123 views
+50

How to generate points uniformly distributed on the surface of an ellipsoid?

I am trying to find a way to generate random points uniformly distributed on the surface of an ellipsoid. If it was a sphere there is a neat way of doing it: Generate three $N(0,1)$ variables ...
1
vote
3answers
46 views

Computing a complicated variance

In a lottery $n$ numbers are selected from the $N$ numbers $1,2,\cdots,N.$ Find the variance of the sum $S_n$ of the selected numbers. My idea: We want to find $P(S_n=k)$. Now, it would be the ...
0
votes
0answers
41 views

Lack of Memory Property with Random Time

Let Z be an exponential random variable and R an independent nonnegative random variable. Show that Z has the lack of memory property also at the random time R, i.e. P(Z − R > u|Z > R) = P(Z > u). ...