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

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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. ...
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0answers
17 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 ...
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1answer
20 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
16 views

A joint pdf question [on hold]

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

Derive the expected value of $X^{0.5}$ [on hold]

I am doing a question considering a continuous random variable X and have calculated the expected value and variance from the probability density function given. I am unsure of what the expected value ...
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1answer
18 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 ...
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1answer
18 views

Conditional Probability of random variables! [on hold]

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:)
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1answer
19 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 ...
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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
10 views

Prove that expected value is a limit of generic function

How can I show that for random variable $X$ which is descbribed by generating function $g(x) = \sum_{n\geq 0} x^nPr[X=x]$ holds $$EX = \lim_{x \rightarrow{1^-}} g'(x)$$
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0answers
30 views

Calculating Map estimate [on hold]

Hello everyone I am stuck on this problem: Given N independent measurements from an experiment that generates exponentially distributed random variables: $$f(x)={1\over y}e^{-x\over y}$$ ...
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2answers
15 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 ...
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1answer
21 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 ...
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1answer
23 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 ...
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1answer
14 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), ...
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3answers
76 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} ...
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1answer
23 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 ...
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1answer
30 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: $$N = \min\{n \mid X_n = 0\}$$ ...
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1answer
21 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 ...
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1answer
147 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 ...
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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 ...
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2answers
13 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 ...
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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 ...
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1answer
10 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 ...
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0answers
42 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 ...
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1answer
17 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 ...
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1answer
21 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 ...
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0answers
8 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 ...
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0answers
14 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 ...
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1answer
13 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 ...
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1answer
20 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 ...
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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 ...
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1answer
28 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
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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 ...
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2answers
12 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 ...
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1answer
20 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 } ...
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2answers
67 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 ...
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3answers
45 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 ...
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0answers
29 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). ...
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2answers
59 views

Are X and Y independent random variables?

$\bullet$ Let $Z$ be uniformly distributed on $[-1,1]$. $\bullet$ $X$ is a random variable such that $X=1$ when $Z>0$ and $X=-1$ otherwise. $\bullet$ $Y$ is a random variable such that ...
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2answers
40 views

Expectation of maximum of two geometric random variables

Let $X,Y$ be independent geometric random variables, where $X$ has parameter $p$ and $Y$ has parameter $Q$. What is $E[\max(X,Y)]$, and what is $E[X\mid X\leq Y]$? If we follow the definition of ...
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1answer
43 views

random variable and joint probability

A hamburger chain's game card has ten squares, each of which has a covering that can be rubbed off to reveal what is pictured beneath. Seven squares show different foods, two square show the same ...
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1answer
15 views

Using Monotone Convergence Theorem to extend a result involving random variable

We assume that for a non-negative, bounded, continuous random variable we have $$ E[X]=\int_0^\infty P(X>x) dx $$ Now the task is to extend this result to non-negative, continuous random variables ...
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0answers
14 views

Question about the uniform distribution and random variable operation

If I have a random variable with uniform distribution ranging from 0 to 1, as follows: $ X \sim U\left ( 0 ;1 \right ) $ And if I subtract that random variable $ X $ from 1, will the resulting random ...
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0answers
12 views

Estimate number of data points necessary to generate a normal distribution

I've written a program that generates random normally-distributed variables using the Box-Muller transform. My question is if I can find any formula that relates the number of data points that I ...
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1answer
17 views

limsup of sequence of random variables scaled with n^{-1}

I have been stuck on the following problem: Let $X_i$ iid non-negative random variables (not only necessarily integer-valued) and define $ A:=\limsup \frac{X_i}{i}$. Prove that ...
2
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1answer
31 views

Find the pdf of $X+Y$ (discrete case)

Given the marginal pdfs: $$p_X(k)= e^{-\lambda}\frac{\lambda^k}{k!}\text{ and }p_Y(k)= e^{-\mu}\frac{\mu^k}{k!},\quad k=0,1,2,\ldots$$ find the pdf of $X+Y$. I know for $W=X+Y$, $p_W(w)= \sum_x ...
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2answers
28 views

Question about random variables operations

As we can see in the picture above, what we call random variables looks like much more like a function, in the way that there is an input and then this random variable perfoms a process and gives as ...
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2answers
22 views

Question about probability of a random variable with uniform distribution

Why is it that when a random variable has a uniform distribution the following statement it's true? $ \Pr \left ( X_{1} \leqslant c \right ) = c $ The question arised when I was doing this ...
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0answers
11 views

Isotropic correlation function for a vector valued random field

I'm having trouble with some of the implications of the following theorem. Let $\mathbf{T} (\mathbf{x})$ be a mean-square continuous vector valued random field on $\mathbb{R}^3$ satisfying conditions ...