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

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3
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0answers
117 views

Show that $(X_{n},Y) \to^{\mathcal{D}} (X,Y)$ AND if $X=h(Y)$ where $h$ is a Borel function that $X_{n}\to^{P} X$

Let $X_{n}$, $X$, and $Y$ be real-valued r.v.'s all defined on the same space $(\Omega, \mathcal{A},\mathbb P)$. Assume that $\lim_{n \to \infty}\mathbb E\{f(X_{n})g(Y)\}=\mathbb E\{f(X)g(Y)\}$ ...
1
vote
0answers
38 views

Calculating expectation of a random variable knowing expectation of a function of it

Let $X$ be a random variable taking values from $\mathbb{R}^+$. If $\mathbb{E}[a^{X}]$ is given for some positive constant $a$, then what can be said about $\mathbb{E}[X]$? I am particularly looking ...
0
votes
2answers
35 views

Conditional Probability with Independent Discrete Random Variables

Let X, Y be two independent Poisson random variables with lambda of X = 1, lambda of Y = 2. Find P(X = 40 | X + Y = 100). I know P(X|Y) = P(X, Y) / P(Y), and since X and Y are independent P(X, Y) = ...
3
votes
0answers
51 views

Is Hoeffding's bound tight in any way?

The inequality: $$\Pr(\overline X - \mathrm{E}[\overline X] \geq t) \leq \exp \left( - \frac{2n^2t^2}{\sum_{i=1}^n (b_i - a_i)^2} \right) $$ Is this bound (or any other form of hoeffding) tight in ...
0
votes
1answer
31 views

Var(X)<inf implies E|X|<inf?

$\textrm{Var}(X)<\infty$ implies $E|X|<\infty$ ? And what about the converse?
0
votes
1answer
125 views

What is the difference between a random vector and a stochastic process?

I am a little confused about random vectors and stochastic processes. I read their definitions in Wikipedia (random vector,stochastic process) and I cannot understand their differences . I would ...
0
votes
1answer
26 views

The weak law of great numbers from the central limit theorem.

I couldn't derive the weak law of large numbers from the central limit theorem for iid random variables with $0 < \operatorname{Var}(X) < \infty$. The central limit theorem gives $$\frac{\sum ...
1
vote
1answer
91 views

Bernoulli distribution vs the probability mass function

What is the difference between the two? They seem to mean the same thing to me. The probability mass function can be used to find the probability of getting a tail from a coin flip: X{1 ...
3
votes
2answers
69 views

$\operatorname{Bin}{(n,U)}$, where $U$ is uniform on $(0,1)$

A question in my probability class: Let $X$ have the binomial distribution $\operatorname{Bin}{(n,U)}$, where $U$ is uniform on $(0,1)$. Show that $X$ is uniformly distributed on $\{0,1,\dotsc, n\}$. ...
1
vote
0answers
115 views

replacing dependent random variables with independent random variables.

I have $x$ and $y$ independent sub-Gaussian random variables and the quadratic form: $z= x^2+xy$ Let $x'$ be an independent and identically distributed copy of $x$. If I use the replacement $z'= ...
1
vote
1answer
46 views

Marginal Density function

The joint density of Y1, Y2 is given by fY1Y2 (y1, y2) = $\frac{1}{2}$ e-(y1+y2)/2 , $0\leq y$2$ \leq y$1 < $\infty$, $0$, elsewhere. The marginal densities of Y1, Y2 is given in the book as : ...
3
votes
1answer
91 views

Markov Chain Initial Distribution

Suppose $\{X_0,X_1,X_2,\dots\}$ is a discrete-time Markov chain taking values in a finite set $\{1,\dots,N\}$ with initial distribution $p_i(0) = P(X_0 = i)$ for $i\in\{1,\dots,N\}$ and transition ...
0
votes
1answer
94 views

Random Variable Problem w/ variance

Three zero mean, unit variance random variables X, Y, and Z are added to form a new random variable, W = X + Y + Z. Random variables X and Y are uncorrelated, X and Z have a correlation coefficient of ...
1
vote
1answer
15 views

Generating $U[0,1]$ from 3 $U[a,b]$ of unknown $a$ and $b$

I received this interesting question from my friend. Suppose we have 3 random number generators, each generates value from the uniform distribution on the interval $[a, b]$. Can we construct random ...
1
vote
1answer
61 views

If $Y=\sum_{n=1}^\infty X_n$ diverges, is $Y$ a random variable?

Let $X_n$ be random variables. By definition, a random variable is a function from the probability space to $\mathbb{R}$. If $Y=\sum_{n=1}^\infty X_n$ diverges, is it correct to call $Y$ a random ...
1
vote
1answer
47 views

Prove that a random variable is nondegenerate

I want to prove that $X$ is not constant w.p. 1. Is it enough to show that $E(X-E(X))\neq 0$?
1
vote
1answer
43 views

Sum of random variables and expectation … difference?

$C_j = 1/s_j \sum_{i=1}^n w_iJ_i^j$. $w_i$ is some (real) number. $J_i^j$ is a 0-1 random variable and the P($J_i^j = 1)= p_i^j$. $l_j = 1/s_j \sum_{i:[n]} w_i p_i^j$. This is a sum over all $i$. ...
0
votes
2answers
30 views

Finding the Expectation - Continuous case

For the RV with PDF, f(x;$\lambda$) = (e-xx$\lambda$)/$\lambda$!, x > 0 Find the Expectation. I tried integration by parts, but it gets complicated. I got some answer like: ...
0
votes
2answers
36 views

Probability that an independent exponential random variable is the least of three

Let $Y_1, Y_2, Y_3$ be independent exponentially distributed random variables, with parameters $\lambda_1, \lambda_2, \lambda_3$ respectively. Why is it the case that: ...
1
vote
1answer
42 views

What is the variance of this random variable?

A clerk drops $n$ matching pairs of letters and envelopes. He then places the letters into the envelopes in a random order. Let $X$ be the number of correctly matched pairs. Find the variance of $X$.
1
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1answer
61 views
2
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1answer
44 views

Var (X) = 0 if and only if X is degenerate.

I was able to prove one way. i.e. X is degenerate if for some a $\epsilon$ $\mathbb{R}$ P{X = a} = 1. => EX = a. P{X=a} = a EX2 = a2 => Var (X) = 0 But the other way around is not clear. Var ...
1
vote
1answer
29 views

Theorem : If the moment of order t exists for an RV X, moments of order 0 < s < t exist.

An Introduction to probability and statistics - Rohatgi Pg. No. 74 Theorem 2 Theorem : If the moment of order $t$ exists for an RV $X$, moments of order $0 < s < t$ exist. The proof is given ...
2
votes
0answers
58 views

How to construct a uniform joint distribution

I have a question that is critical to my work, but I am not sure if it is any possible. Assume that you have two uniform random variables X and Y. The product distribution of Z=XY is not a uniform. ...
2
votes
3answers
447 views

Difference of two binomial random variables

Could anyone guide me to a document where they derive the distribution of the difference between two binomial random variables. So $X \sim \mathrm{Bin}(n_1, p_1) $ and $Y \sim \mathrm{Bin}(n_2, p_2) ...
0
votes
0answers
27 views

Flaw in calculating the expectation [duplicate]

Because we have a lot of homework so me and my friend try to divide it into 2 stacks. However, my sister is playing with us so instead of dividing those homework into 2 equal stacks, she makes one ...
0
votes
2answers
105 views

Expected value of the function of a uniformly distributed random variable

Let X by a uniformly distributed random variable on the interval [0,1]. Find $E[e^Y]$ I am trying to make use of the formula $$E[g(X)] = \int_{-\infty}^{\infty}g(x)xdx$$ so then $$E[e^X] = \int ...
2
votes
1answer
37 views

$X$ is standard normal if $X=Y1_{\{|Y|\le a\}}-Y1_{\{|Y|>a\}}$ where $Y$ is standard normal

$X$ is standard normal if $X=Y1_{\{|Y|\le a\}}-Y1_{\{|Y|>a\}}$ where $Y$ is standard normal. $$F_X(x)=P(X\le x)=P(\{Y\le x\}\cap\{|Y|\le a\})+P(\{-Y\le x\}\cap\{|Y|> a\})$$ How can I simplify ...
0
votes
2answers
64 views

probability that your vote tips the election results

$\textbf{Question:}$ Assume a presidential elections with two candidates $A$ and $B$ and $2n+1$ voters. Assume that each voter will choose $A$ with probability $p$ and $B$ with probability $1-p$. Let ...
0
votes
1answer
102 views

Probability that sum of 2 random numbers are less than an integer

Assume there is an Ideal Random Number Generator which generates any real number between 0 and given integer. Two numbers are generated from the above generator using integer A and B, let's assume the ...
0
votes
1answer
35 views

sum of independent random variables where $N$ is a random variable

I want to show $E[S_N]=E[N]E[X_j]$ where: $X_1,X_2,\ldots$ is a sequence of independent random variables, and $N$ is a random variable independent of the sequence. $S_n=\sum_{i=1}^n X_i$, ...
0
votes
1answer
45 views

What conditions are needed for a uniform bound on the deviation of a random variable from its expectation?

The title is quite clear, but a few specifics may help guide where I'm coming from. I have a probability space defined by a Poisson distribution over the non-negative integers with some mean $\mu$. I ...
1
vote
1answer
78 views

Series of independent random variables are independent

In the proof of a theorem my lecturer seemed to have used this fact without first proving it: Let $(X_i)_{i \geq 1}$ be real-valued independent random variables on $(\Omega,\mathscr{F},\mathbb{P})$, ...
1
vote
1answer
21 views

Minimizing f(x) when f(x) can only be probed via a random process

Background: I'm writing a piece of software to run on a mid-tier HPC cluster, to perform automatic parameter optimization. Say I have a function f(x). I need to find x that minimizes f(x). For our ...
-2
votes
1answer
22 views

Discrete random variables problem

I don't know how to answer the following question: Argue that if $X$ and $Y$ are nonnegative random variables, then $E[\max(X,Y)] \leq E[X] + E[Y]$
0
votes
2answers
326 views

Product of two random variables

How can one show that the product $X \cdot Y$ of two real-valued random variables $X,Y$ is again a random variable? We can fix some set generating the Borel sigma algebra on the real line, then take ...
1
vote
1answer
50 views

Solution to Billinglsey (1995) problem 20.22

Let $Y_1\leq Y_2\leq ...$ be random variables s.t. $\mathrm{plim} Y_n = Y$. Show that $Y_n \to Y$ with probability 1. Some hints? My strategy would be to prove that $\sum P(\lvert Y_n -Y \rvert > ...
1
vote
2answers
86 views

Largest Number Drawn - RV

From a box containing N identical tickets numbered 1 to N, n tickets are drawn with replacement. Let X be the largest number drawn. Find E[X]. Rohatgi 3.2 (2nd problem) My solution 2: I thought ...
1
vote
1answer
32 views

Average of independent random variables

Let $X_1, X_2, \dots$ be a sequence of independent random variables. For each $n$, $\mathbb{P}(X_n=-n^2)=1/n^2$ and $\mathbb{P}(X_n=n^2/(n^2-1))=1-1/n^2$. I need to show that ...
1
vote
1answer
36 views

Successive Mean Sums of Random Variables

How we can prove the following: $X_1, X_2, ..., X_n$ are $i.i.d$ random variables having all the moments of order less than 4, so \begin{eqnarray*} \sum_{i=1}^n\sum_{j=1}^n\sum_{k=1}^n\sum_{l=1}^n ...
3
votes
0answers
186 views

Relation between standard deviation and mean in random processes

In a Poisson distribution the square of the standard deviation $\sigma$ is equal to mean $\mu$ ($\sigma^2=\mu$) and in a binomial distribution $\sigma ^2=\mu\,(1-p)$ (with $p$ the probability of ...
0
votes
1answer
545 views

How to compute the sum of random variables of geometric distribution

$X_{i}(i=1,2..n)$ is the independent random variables of geometric distribution, that is, $P(X_{i}=m)=p(1-p)^{m-1}$, then how to compute the PDF of $\sum_{i=1}^{n}X_{i}$ I know intuitively it's a ...
2
votes
1answer
464 views

finding probability generating function and the sum of two independent random variables

Let $X$ be a discrete random variable with probability mass function $$P_X(x) = p(1-p)^x,\qquad x=0,1,2,3,\ldots$$ (a) Find the probability generating function for $X$ and hence ...
0
votes
2answers
36 views

Given random variable $F_{x}(n)=P(X=n)=\frac{c}{n(n+1)}$ calculate $c$ and $P(X>m)$

Given random variable $F_{x}(n)=P(X=n)=\frac{c}{n(n+1)}$ calculate $c$ and $P(X>m)$ for $m=1,2,3...$. First of all $\lim\sum\limits_{n \in \mathbb{N}} F_{x}(n)=1 $, so $$\lim\sum\limits_{n \in ...
0
votes
1answer
75 views

Confusing cont. r.v. problem

Let $X_1$, $X_2$, $X_3$, $X_4$, $X_5$ be independent continuous random variables having a common distribution function $F$ and density function $f$, and set $$I = \mathbb{P}\{X_1 < X_2 < X_3 ...
1
vote
2answers
34 views

Heteroskadasticity and Linear Probability

Question Suppose $(Y,X,U)$ be a random vector such that $$ Y = X'\beta + U. $$ Suppose $Y$ takes values in $\{0,1\}$ and that $E[Y\mid X] = X'\beta$. Is it reasonable to assume that $Var[U\mid X]$ ...
0
votes
2answers
64 views

How to find the exact value of an upper bound for an exponential random variable

Suppose the waiting time between people entering a store can be modeled by the exponential random variable $X$ with parameter $\lambda=5$. If you use markov's inequality you can find the $P(X\ge 20)$ ...
1
vote
1answer
88 views

Upper bound on sum of i.i.d. random variables

Here's a problem I've been struggling with: Let $X_1, X_2, X_3, \ldots$ be an i.i.d. sequence of random variables with finite moment generating function $M(t)$. Define the sum $S_n = X_1 + \ldots ...
1
vote
1answer
60 views

Maximising Entropy of Random Variable taking Positive Integer Values

A random variable $X$ takes positive integer values and $E[X]=6$. What distribution of the random variable $X$ maximises the entropy $H(X)$? What if $X$ can only take a finite number of values? ...
0
votes
0answers
46 views

Hoeffding Inequality

Assume that you have $1000$ fair coins labelled as $C_1,C_2,...C_{1000}$. You flip each coin, $C_i$, $10$ times and calculate the fraction of heads $v_1,v_2, .., v_{1000}$ for each coin. Now, each ...