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

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4
votes
2answers
185 views

how to derive the mean and variance of a Gaussian Random variable?

How do we go about deriving the values of mean and variance of a Gaussian Ransom Variable $X$ given its probability density function ?
1
vote
1answer
196 views

$X_n \stackrel{d}{\to} X$, $c_n \to c$ $\implies c_n \cdot X_n \stackrel{d}{\to} c \cdot X$

Let $X_n$, $X$ random variables on a probability space $(\Omega,\mathcal{A},\mathbb{P})$ and $(c_n)_n \subseteq \mathbb{R}$, $c \in \mathbb{R}$ such that $c_n \to c$ and $X_n \stackrel{d}{\to} X$. ...
3
votes
2answers
7k views

Finding probability P(X<Y)

How can I find this probability $P(X<Y)$ ? knowing that X and Y are independent random variables.
2
votes
1answer
76 views

The concept of random variable

I'm reading Bernt Oksendal's "Stochastic Differential Equations" (edition 6) and I got quite confused on the conceptions. Please kindly help. I don't understand what is an event in the definition of ...
1
vote
5answers
2k views

The sum of n independent normal random variables.

How can I prove that the sum of $X_1, X_2, \ldots,X_n$ random variables, all of which have normal distributions $N(\mu_i, \sigma_i)$, is a random variable that is itself normally distributed with mean ...
9
votes
1answer
464 views

Conditional expectation on more than one sigma-algebra

I'm facing the following issue. Let $X$ be an integrable random variable on the probability space $(\Omega,\mathcal{F},\mathbb{P})$ and $\mathcal{G},\mathcal{H} \subseteq \mathcal{F}$ be two ...
5
votes
4answers
682 views

Question on the 'Hat check' problem

The famous 'Hat Check Problem' goes like this, 'n' men enter the restaurant and put their hats at the reception. Each man gets a random hat back when going back after having dinner. The goal is to ...
1
vote
1answer
57 views

finding the limits of integration for joint probability

I have three variables $x_1$, $x_2$ and $x_3$. Their joint dist. is $f(x_1,x_2,x_3)= \exp(-x_1-x_3)$, where limits of $x_3 = 0$ to $\infty$, $x_2 = x_3$ to $\infty$ and $x_1 = x_2-x_3$ to $\infty$. ...
5
votes
1answer
122 views

Density function of $\max(X_1,\dots,X_n)$.

I'm making this statistics exercise and I'm not sure about my solution. Find the density function of $Y=\max(X_1,\dots,X_n)$ if they are all i.i.d. This was my take on this question: $F_Y(a)=P(X_1 ...
3
votes
3answers
945 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) ...
2
votes
4answers
229 views

Why does maximum likelihood estimation for uniform distribution give maximum of data?

I am looking at parameters estimation for the uniform distribution in the context of MLEs. Now, I know the likelihood function of the Uniform distribution $U(0,\theta)$ which is $1/\theta^n$ cannot ...
5
votes
2answers
512 views

Summing (0,1) uniform random variables up to 1 [duplicate]

Possible Duplicate: choose a random number between 0 and 1 and record its value. and keep doing it until the sum of the numbers exceeds 1. how many tries? So I'm reading a book about ...
3
votes
1answer
71 views

[Probability]need help to understand the following expression

So assume $Y$ and $X$ are exponentially distributed with parameters $y_1$, and $x_1$ respecitively. assume c is a constant. I am having huge trouble to understand the integration of the following ...
3
votes
2answers
241 views

Prove the monotonicity of the expectation of a binary random variable function

Consider $R$ independent binary random variables $y^1, \ldots, y^R$ over the space $\{-1, +1\}$ such that $\Pr(y^j = 1) = p^j \geq 0.5$ and $\Pr(y^j = -1) = 1 - p^j$, $\forall j = 1,\ldots,R$. ...
2
votes
1answer
114 views

Where is the fallacy in this coupling argument of two Bernoulli variables?

With respect to the scenario introduced in Prove the monotonicity of the expectation of a binary random variable function, let us now suppose that the function: $$\begin{align*} f(\mathcal{J}) = ...
1
vote
3answers
46 views

Find the Mean for Non-Negative Integer-Valued Random Variable

Let $X$ be a non-negative integer-valued random variable with finite mean. Show that $$E(X)=\sum^\infty_{n=0}P(X>n)$$ This is the hint from my lecturer. "Start with the definition ...
1
vote
2answers
68 views

Must the sequence $X_n$ converge to $0$ in probability?

Let $X_1, X_2,\dots$ be a sequence of random variables with $\lim_{n\to +\infty} E[|X_n|] = 0$. Is it correct or wrong that the sequence $X_n$ must converge to $0$ in probability?
1
vote
2answers
1k views

Determine the PDF of Z = XY when the joint pdf of X and Y is given.

The joint probability density function of random variables $ X$ and $ Y$ is given by $ p_{XY}(x,y) = 2(1-x) $ when $ 0<x \le 1, 0 \le y \le 1$ and $ p_{XY}(x,y) = 0 $ otherwise. Determine the ...
0
votes
1answer
52 views

Show that if two random variables sequences are pairwise independent then the limits are independent, too.

Two sequences $X_1, X_2, \ldots, Y_1, Y_2,\ldots : (\Omega, \mathcal{F},\mathbb{P}) \to (\mathbb{R}, \mathcal{B}(\mathbb{R}))$ of real random variables such that $\forall n \ X_n, Y_n $ are ...
0
votes
1answer
24 views

Probability question of independent random varaibles

Let $X\sim \mathcal{N}(6,1)$ and $Y\sim\mathcal{N}(7,1)$ be two independent normal variables. Find $Pr(X>Y)$. the answer is $0.2389$ but I do not know how to do it.
0
votes
2answers
1k views

Joint PDF of two random variables and their sum

What is the joint PDF of two uniformly distributed random variables and their sum?
-1
votes
1answer
49 views

Generate random numbers following the exponential distribution in a given interval $[a, b]$

I know that to genarete ramdom variables following exponential distribution just do: $$X=-\frac{1}{\lambda}ln(U)$$ where $U\sim U(0,1)$ Now, to find a distribution restricted to the interval $(a, ...
6
votes
4answers
301 views

What exactly is a random variable?

I don't really understand the definition of a random variable. I also find the wikipedia entry on random variables kind of confusing. Can someone give me a clear explanation of the random variable?
6
votes
2answers
2k views

Infinite expected value of a random variable

How can a positive random variable $X$ which never takes on the value $+\infty$, have expected value $\mathbb{E}[X] = +\infty$?
7
votes
2answers
764 views

Random sum of random variables

Say you sum i.i.d. variables $X_i$ a total of $Y$ times. If you know the distribution of random variables $Y$ and $X_i$, what is the calculation you have to do to get the distribution of the sum?
6
votes
1answer
325 views

Jensen's Inequality (with probability one)

In the following theorem, I have a problem about the second part. That is showing if $f$ is strictly convex then $X=EX$ with probability $1$. While I can see this must be true, I don't know how to ...
4
votes
1answer
103 views

How to find $\mathbb{E}[X\mid\min(X,Y)]$?

Say I have two independent variables $X$ and $Y$ that are exponentially distributed with respective rates $\lambda_X$ and $\lambda_Y$. How do I compute $\mathbb{E}[X\mid \min\{X,Y\}]$?
4
votes
2answers
105 views

Measurability problem of sample distribution function of a contiuum of independent random variable

Let $I = [0,1]$ be the index set of a contiuum of i.i.d random variables. For each $t \in I$, the sample space of $X_t$ is $\Bbb R$ equipped with Borel $\sigma$-algebra and Borel probability measure. ...
4
votes
1answer
133 views

Understanding the definition of a random variable

I'm working through a math stats book on my own (I've always wanted to learn it), but I'm getting confused about the definition of a random variable. The book says that a random variable is a ...
3
votes
0answers
51 views

Why the definition of Variance is such. [duplicate]

Why we define the variance of a random variable $X$ as $\text{var}[X]=\text{E}[(X-\mu)^2]$ instead of $\text{var}[X]=\text{E}[\left|X-\mu\right|]$. Normally we understand the standard deviation ...
0
votes
2answers
104 views

Show that $\lim\limits_{n\rightarrow\infty} e^{-n}\sum\limits_{k=0}^n \frac{n^k}{k!}=\frac{1}{2}$

Show that $\displaystyle\lim_{n\rightarrow\infty} e^{-n}\sum_{k=0}^n \frac{n^k}{k!}=\frac{1}{2}$ using the fact that if $X_j$ are independent and identically distributed as Poisson(1), and ...
4
votes
4answers
2k views

What does it mean to integrate with respect to the distribution function?

If $f(x)$ is a density function and $F(x)$ is a distribution function of a random variable $X$ then I understand that the expectation of x is often written as: $$E(X) = \int x f(x) dx$$ where the ...
3
votes
1answer
35 views

Find a sequence of r.v's satisfying the following conditions

I think part a) can be solved by using $X_n=\frac{1}{n}\chi_{[0,n^2]}$ Not sure about part b).
3
votes
1answer
137 views

Infinite divisibility of random variable vs. distribution

The distribution of any infinitely divisible random variable is itself infinitely divisible. But this link says the converse is not always true. Can you explain?
3
votes
2answers
441 views

Product of independent random variables

The following is a classic example that pairwise independent does not necessarily imply mutually independent: Let $X_1$ and $X_2$ be independent r.v.'s with distributions ...
2
votes
1answer
50 views

Fixed points in random permutation

Suppose two random permutations of the numbers 1 to n placed side by side. a) Calculate the expectation number of fixed points for $n = 5$. b) Find the value of expectation in the amount of fixed ...
2
votes
1answer
74 views

inclusion of $\sigma$-algebra generated by random variables

Consider the following random variables $$X:\Omega\to\mathbb{R}\quad\text{and}\quad Y:\Omega\to \mathbb{R}$$ and $$Z:=XY$$. One may interpret it as follows, i.e. $$Z(\omega) = X(\omega)Y(\omega).$$ ...
1
vote
1answer
101 views

Solution of equation of binomial random variables

Is it possible to find the probability distribution of the random variable $X$ that solves the following equation? $$ X = Bin(X, p) + Bin(X, 1-p), $$ where $Bin(X,p)$ is a random variable distributed ...
1
vote
1answer
92 views

Large Deviations Question

Let $\left(X_n\right)_{n\geq 1}$ be i.i.d random variables on $\left(\Omega,\mathcal A, \mathbb P\right)$, $X_1$ with mean $\mu$, and $$ L(\lambda) = \begin{cases} \log\mathbb E\left(e^{\lambda ...
0
votes
1answer
213 views

Discrete random variable with infinite expectation

Consider a discrete random variable taking only positive integers as values with $$\mathbb{P}[X=n]=\frac{1}{n(n+1)}.$$ (a) Show that $\mathbb{E}[X]=\infty$. (b) Show that $\mathbb{P}[X ...
0
votes
1answer
106 views

Deriving the transformation function of a random variable from the original and the final distributions

Consider a random variable $X$ and consider that this variable can be either real or integral (so I would like to cover both cases: continuos and discrete random variables). Consider to transform this ...
5
votes
5answers
793 views

“Random” generation of rotation matrices

For a current project, I need to generate several $3\times 3$ rotation matrices for input into an algorithm. I thought I might go about this by randomly generating the number of elements needed to ...
4
votes
1answer
51 views

Convergence of Random Variables in mean

If $$E[|X_n-X|^r]\rightarrow0$$ prove that $$E|X_n^r|\rightarrow E|X^r| $$ for every $r\ge 1$ This is the very notation used. I believe it should be: $$E[|X_n|^r]\rightarrow E[|X|]^r $$ Attempt I ...
4
votes
2answers
277 views

Expected Value of Local Maxima and Local Minima

Recently I came across this question: Given a random permutation of integers 1, 2, 3, …, n with a discrete, uniform distribution, find the expected number of local maxima. (A number is a local maxima ...
4
votes
2answers
89 views

Eigenvalues of a Random Matrix

I am studying the theory of random matrices lately, but there is a basic issue troubling my life. I hope someone here explain me this, thank you. A random matrix is defined as a matrix whose entries ...
4
votes
1answer
256 views

Prove that it is a random variable iff it is constant on each partition

Let $\mathcal{G} = \{A_1, \ldots, A_n\}$ be a partition of a set $\Omega$, $\mathcal{F} = \sigma(\mathcal{G})$. Prove that $X : \Omega\to\mathbb{R}$ is a random variable if and only if it is constant ...
3
votes
5answers
267 views

Sum of random numbers is divisible by $10$

Suppose that $15$ three-digit numbers have been randomly chosen and we are about to add them. What is the probability that the sum would be divisible by $10$? If there were only two or three random ...
3
votes
1answer
84 views

Unbiased estimators for the moments of 2 not independent random variables

Let $X$ and $Y$ be two non independent random variables. Suppose to generate $n$ realizations of both variables, and indicate with $(X_i, Y_i)$ their values. Also, let's pose that $X$ and $Y$ are non ...
3
votes
1answer
95 views

Definition of a real-valued random variable

I have trouble understanding the definition of a random variable: Let $(\Omega, \cal B, P )$ be a probability space. Let $( \mathbb{R}, \cal R)$ be the usual measurable space of reals and its ...
2
votes
2answers
44 views

Independent, random variables with equal distribution satisfy: $\lim_{n \to \infty}\mathbb{P}\left(X_{n+1} > \sum_{i = 1}^{n}X_i\right) = 0$

$X_1, X_2, \ldots$ are independent, non-negative, real random variables with equal probability distribution. Show that $$\lim_{n \to \infty}\mathbb{P}\left(X_{n+1} > \sum_{i = 1}^{n}X_i\right) ...