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

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Random variable probability

On probability space $\Omega$ with elements a,b,c,d,e. Define $\sigma$-algebra $F$ on $\Omega$-collection of subsets of $\Omega$ and $H=X+Y$. Probability measure by P{a}=P{b}=P{c}=P{d}=1/5,RV X,Y: X{...
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75 views

Show Y has a uniform distribution if Y=F(X) where F(x)=P[X $\le$ x] is continuous in x.

If $ F(x) = P[X\le x] $ is continuous in x, show that $ Y=F(X) $ is measurable and that $Y$ has a uniform distribution $ P[Y\le y] = y, \; 0\le y \le 1 $ My first question is about notation. What ...
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40 views

Using Inequalities (Markov + Chebyshev) for lower bounds

I have an exam in a few hours and realized there's problems on practice problems that we didn't directly have to do for class. I know that the markov bound is: $P(X >= k) <= E[X] / k$ and the ...
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2answers
35 views

Convolution formula proof- Random discrete varaiables [closed]

Let X, Y be discrete random variables and take values at $1, 2, · · · , n, · · · $ $f_{X}(t)=\sum_{k=0}^{k=inf} P(X=k)x^{k}$ is the probability generating function. and this result was given below $...
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35 views

Ratio of Order Statistics goes to infinity?

Is there a sequence of distributions $(F_n)$ with $F_n(0)=0$ such that $\frac{E[\max(X_{1,n},X_{2,n})]}{E[X_{1,n}]} \to +\infty $ as $n\to +\infty$, where $X_{1,n}$ and $X_{2,n}$ are independently ...
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1answer
28 views

Conditional exp. - $\mathbb{E}[Y|X]$=$X$

I have very common task - Consider a probability space $\Omega$ with four elements a,b,c,d. Define $\sigma$-algebra $F$ on $\Omega$-collection of subsets of $\Omega$. Probability measure by P{a}=1/6,...
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2answers
150 views

A DE Shaw Interview Question about Two i.i.d random variables inequality [closed]

If $X$ and $Y$ are i.i.d positive random variables, Prove that $\Bbb E(X/Y) \ge 1$: I use Jensen's inequality $\Bbb E[\exp(\log(X/Y))]$ and get the answer. One can also use the A-G inequality to ...
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2answers
33 views

Moment Generating Function of the Product of Three I.I.D Bernoulli random variables

Let X1, X2, X3 be i.i.d. random variables with distribution $P(X_1 =0)=\frac13, P(X_1 =1)= \frac {2}{3}$ Calculate the moment generating function of $Y = X_1X_2X_3.$ My work: $M_x(t)= E(e^{tx}) = \...
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22 views

Can the expected value of a random variable be a function of other random variables?

Let $Y$, $Z$, and $L$ be random variables and the possible value of a uniformly distributed random variable $X$ can be expressed by the following inequality $$Y+Z-L \le X \le Y+Z$$ Can we say that $...
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42 views

Density of the Sum of Two Exponential Random Variable

For independent random variables X ∼ Exp(1) and Y ∼ Exp(2), find the density of the random variable Z = X + Y . My work: For any exponential distribution with parameter $\lambda$ the function is $f(x)...
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54 views

Conditional expectation $\mathbb{E}[Y|X]$

I have very common task - Consider a probability space $\Omega$ with four elements a,b,c,d. Define $\sigma$-algebra $F$ on $\Omega$-collection of subsets of $\Omega$. enter image description here ...
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1answer
18 views

Finding a distribution function for $Y=ax^2$

X is a discrete random variable with a distribution function $F_x$ and PMF $P_X$. How can I find the distribution function of $Y = ax^2$, a $\neq$ 0.
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114 views

Correlated joint normal distribution: calculating a probability

Given $$ f_{XY}(x,y) = \frac{1}{2\pi \sqrt{1-\rho^2}} \exp \left( -\frac{x^2 +y^2 - 2\rho xy}{2(1-\rho^2)} \right) $$ $Y = Z\sqrt{1-\rho^2} + \rho X$ And $$ f_{XZ}(x,z) = \frac{1}{2\pi } \exp \...
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2answers
42 views

Calculating Expectation and Variance

I think I have an idea of how to solve these problems, but I keep getting stuck. Any help would be greatly appreciated! Also, sorry I know I messed up the formatting with the exponents in the problems....
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17 views

Singular values after projection

Suppose we have a random matrix $\Phi$ where entries of $\Phi$ are i.i.d random variables and we would like to construct a new matrix $\Gamma$ in the following way: Compute $P\phi_i$ where $P_{n\...
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32 views

Poisson distribution arrival problem. With conditional uniform distribution

People arrive at rate 20 per unit time. The bus comes uniformly on $[1/2,3/2]$. Let $X$ all people who take the bus (all of them do). Find $Var(X)$ and $E(X)$ For expectation: $E(X)=\int_{1/2}^{3/2} ...
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1answer
31 views

What does mean and variance of a random variable signify?

I just started learning applied probability. What does mean and variance actually signify? If we want to relate the event of getting head or a tail on a coin toss to $X = 0, 1$ respectively, our mean (...
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74 views

Bivariate normal distribution $X$ and $Y$

I need help figuring out the following. Let $X$ and $Y$ have the bivariate normal distribution $$ f_{XY}(x,y) = \frac{1}{2\pi \sqrt{1-\rho^2}} \exp \left( -\frac{x^2 +y^2 - 2\rho xy}{2(1-\rho^2)} \...
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59 views

How to calculate the probability distribution of this random variable?

The random variable is defined by $$X=\left|g\right|^2\cdot d^{-\alpha}$$ where $\alpha>2$ is a positive real number, $g$ is complex Gaussian random variable with zero mean and unit variance and $d$...
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28 views

Uniformly Distributed Random Variable over an interval

Let $X$ be a random variable uniformly distributed over a nontrival interval $[c,d]$, and let $Y = aX+b$. For what choice of real constants $a$ and $b$ is $Y$ uniformly distributed over [0,1]? How ...
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25 views

Entropy Inequality $H(X| g(Y))\geq H(X|Y)$

Let $X,Y$ be random discrete variables. $H(X) = -\sum\limits_{x}P\{X=x\}\operatorname{log}_2P\{X=x\}$ be the entropy-function. It is known fact that that $H(g(Y))\leq H(Y)$. I want to prove the ...
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40 views

Prove that the expected value of the quadratic distance is n, where n is the number of steps.

A drunkard is standing in the middle of a very large town square. He begins to walk. Each step is a unit distance in one of the four directions East, West, North, and South. All four possible ...
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2answers
104 views

Clarification on random variables?

If $X$ and $Y$ are dependent random variables, then it is possible that $Var(X+Y) > Var(X) + Var(Y)$. I only know that the two are equal for independent random variables; for dependent variables, ...
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72 views

Normalized partial sums of normal random variables are dense in $\mathbb{R}$

I came across an interesting result appearing as an exercise in some lecture notes I'm reading. Suppose $X_{1},X_{2},...$ are IID $N\left(0,1\right)$ RVs all defined on $\left(\Omega,\mathcal{F},\...
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23 views

What does in mean (intuitively) when a covariance matrix is not invertible?

Given a covariance matrix $\Sigma$ for some random vector $X = \left[ X_1 \ \ X_2 \ \ \cdots \ \ X_n \right]$, what does it mean (intuitively) if $\Sigma$ is not invertible? I know that it means that ...
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1answer
31 views

Two i.i.d Random Variables : Is covariance between a function (applied to one R.V) with the the R.V. the same for both Random Variables?

Let $X,Y$, both $$\Omega \to \mathbb{R}^k$$ be i.i.d random variables. Let $f$ be a function from $\mathbb{R}^k \to \mathbb{R}$. Is $\operatorname{cov}(f(X), X) = \operatorname{cov}(f(Y),Y)$? If we ...
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15 views

Moment Generating Function - Find K

Given: $M_x(w) = K/(2-w)$ Find $K$. That is all what we are given. How exactly do I go about solving this?
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31 views

Determining whether or not random variables are correlated

I'm working on the following problem: Consider random variables $X$ and $Y$ such that exactly one of them is equal to $0$. The other then takes the value $1$ or $-1$ with equal probability (ex:...
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1answer
34 views

Random Variables Prove Independent

Consider three Bernoulli random variables $B_1, B_2, B_3$ which take values $\{0, 1\}$ with equal prob- ability. We construct the following random variables $X, Y, Z: X = B_1 ⊕ B_2, Y = B_2 ⊕ B_3, Z = ...
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1answer
13 views

Transforming and Combining Random Variables Clarification

My textbook has again been very ambiguous about this subject, coud someone verify if my reasoning is correct: A swimmer enters a 100m event in which he will win a prize if his total time is under 4 ...
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zipf and lognormal with a particular correlation

I have been struggling on how to generate a correlated zipf and lognormal distribution. I want to generate a set of data ,say,$(X,Y)$,where $X$ is the popularity of file described by zipf,$X=1,2,3......
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24 views

$F_X(x)+F_Y(y)-1\le F_{X,Y}(x,y)\le \sqrt{F_X(x)F_Y(y)}$

Let $F_{X,Y}(x,y)$ be the joint probability distribution of the random vector $(X,Y)$; $F_X(x)$ , $F_Y(y)$ the marginal distributions Prove that for all $x,y\in \mathbb R$: $$F_X(x)+F_Y(y)-1\le F_{X,...
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50 views

Calculating the expected value and variance of $n$ independent observations of $X$

I am attempting to find the expected value and variance of the random variable $X$ analytically (in addition to a decimal answer). $X$ is the random variable ...
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1answer
50 views

Give an upper bound for $E\left[\exp\left({\frac{nt}{\sum_{i=1}^n k_i}}\right)\right]$

Given an upper bound for $E \left[ \exp \left(\frac{nt}{\sum_{i=1}^n k_i}\right)\right]$ where $k_i$'s are random variables which denote the number of independent Bernoulli trials before we ...
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2answers
31 views

Random Variable - Probability and Statistic

Suppose that $f(x)$ is a continuous and symmetric pdf, where symmetry is the property that $f(x) = f(-x)$ for all $x$. Show that $P(-a ≤ X ≤ a) = 2F(a) - 1$ Does anyone have any idea what this is ...
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34 views

Determine the upper and lower bounds of the expected value of a random variable

Let $p_k$ represent the probability of success in the $k$th trial ($k=1,2,\ldots$). It is only known that the following holds $$\alpha \leq p_k \leq \beta$$. There is no other information about $p_k$. ...
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1answer
45 views

Calculate the negative moment of a random variable X

Assume we have a random variable X with some arbitrary distribution F(x). How can we calculate $\mathbf{E}[X^{-n}]$ for $n\geq 0$? For example, let $X \sim Rayleight(\sigma)$. If we use the ...
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48 views

Probability theory: Is there a generalized version of this property?

I know from undergraduate Probability that given a r.v. $X:\Omega \to R$ then $$ Z:\Omega \to R \text{ is } \sigma(X) \text{-measurable } \Leftrightarrow \exists f:R\to R \text{ Borel-measurable ...
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$X_n-Y_n\rightarrow_{a.s.} 0$ and $Y_n\rightarrow_{a.s.} Z$ imply $X_n\rightarrow_{a.s.} Z$?

Consider two sequences of random variables $\{X_n\}_n, \{Y_n\}_n$ and a random variable $Z$, all defined on the same probability space. Let $\rightarrow_{a.s.}$ denote almost sure convergence. Suppose ...
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176 views

Waiting time: exponential distribution

Smith is waiting for his two friends Lee and Yang to visit his house. The time until Lee arrives is Exp($\lambda_1$) and the time until Yang arrives is Exp($\lambda_2$). After arrival, Lee stays an ...
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41 views

What are the expected value and the standard deviation of the net profit made by the pharmacist on this medicine in any given month?

At the beginning of every month, a pharmacist orders an amount of a certain costly medicine that comes in strips of individually packed tablets. The wholesale price per strip is 100, and the retail ...
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2answers
67 views

Variance of max - min of 2 exponential random variables

Suppose we have 2 random variables, $S\sim Exp(\lambda)$ and $T\sim Exp(\mu)$. Let $U=\min(S,T)$ and $V=\max(S,T)$. What is the variance of $W=V-U$? I calculated: $Var(U) = \frac{1}{(\lambda+\mu)^...
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1answer
95 views

Conditional distribution of random variable X given itself

I'm stuck with something that might seem trivial but gives me headache. What is the distribution of $X|X$, i.e. the conditional distribution of $X$ given $X$? I'm pretty confident that: $$\mathbb P(...
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24 views

Replacing dependent with independent RVs for Hoeffding inequality.

Let $\bar Y= \frac 1n \sum_{i=1}^n Y_i$ where the $Y_i\in \{0,1\}$ are not necessarily independent, but where the marginals satisfy $\Pr[Y_i=1]\leq \Pr[X_i=1]$ for independent (but not identically ...
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33 views

Is the conditional probability $P( S=a_i |\, T=a_i, X=x ) \le$ (or $\ge$) $ P( S=a_i |\, T=a_i )$?

Let $Y$ be a positive continuous random variable (r.v.). We define $S$ as a variable that depends on the value of $Y$. If $Y \in [0,b_1]$, then $S=a_0$, If $Y \in [b_1,b_2]$, then $S=a_1$ and when $Y \...
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31 views

What am I doing wrong on this integral for CDF/PDF of product of random variables?

I am trying to use the integral expression given here Wikipedia: Product Distribution to determine the CDF of the product $Z=XY$ of two independent Uniform(0,1) random variables $X$ and $Y$. I ...
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26 views

Compare two samples tests

Sampling $1000$ birds from a population which has $10$ types of birds the expected outcome is $100$ birds of each type (this is for the sake of simplicity; general case is each distributed with ...
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1answer
57 views

Covariance of multinomial distribution

Let $X = (X_1,\ldots, X_k)$ be multinomially distributed based upon $n$ trials with parameters $p_1,\ldots,p_k$ such that the sum of the parameters is equal to $1$. I am trying to find, for $i \neq j$,...
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51 views

Let $X$ be a $B(n, 1/2)$ random variable with $n>=1$, show that the probability that $X$ is even is $1/2$.

I found the following an answer here. But I don't completely understand it. I get that the binomial random variable calculates the probability of exactly $j$ successes and $n-j$ failures in $n$ ...
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19 views

Random variables in statistics clarification

My textbook has been very unclear about this topic and online reading has not aided in my comprehension; there are a few problems which I can't manage to answer: Select the quantity below that is not ...