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

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3
votes
1answer
295 views

If $X_n \stackrel{d}{\to} X$ and $c_n \to c$, then $c_n \cdot X_n \stackrel{d}{\to} c \cdot X$

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 $X_n \stackrel{d}{\to} X$. ...
8
votes
2answers
289 views

Conditional expectation equals random variable almost sure

Let $X$ be in $\mathfrak{L}^1(\Omega,\mathfrak{F},P)$ and $\mathfrak{G}\subset \mathfrak{F}$. Prove that if $X$ and $E(X|\mathfrak{G})$ have same distribution, then they are equal almost surely. I ...
6
votes
3answers
3k 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
1answer
5k views

How to compute the sum of random variables of geometric distribution

Let $X_{i}$, $i=1,2,\dots, n$, be independent random variables of geometric distribution, that is, $P(X_{i}=m)=p(1-p)^{m-1}$. How to compute the PDF of their sum $\sum_{i=1}^{n}X_{i}$? I know ...
4
votes
2answers
3k 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 ?
13
votes
2answers
835 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
2k 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 ...
2
votes
1answer
322 views

Convergence of characteristic functions to $1$ on a neighborhood of $0$ and weak convergence

Prove the following statement: $ X_n \Rightarrow 0 $ (convergence in distribution) if and only if $ (\exists\; \epsilon>0: |t|<\epsilon) \;\; \phi_n(t) \rightarrow 1 $, where $\phi_n(t)$ is ...
1
vote
5answers
3k 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 ...
3
votes
1answer
105 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 ...
2
votes
1answer
1k views

Proof of $E(X)=a$ when $a$ is a point of symmetry

I am trying to develop a proof of the following: Given a random variable $X$ with symmetric probability density function $f(x)$, prove that $E(X)=a$ where $a$ is the point of symmetry. A couple of ...
4
votes
2answers
13k 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
520 views

Sum of two independent normal distributed random variables

If $X_i$, $i =1,2$ are independent and have normal distribution with mean $0$ and variance $\sigma_i ^2$. Show that $X_1 + X_2$ has a normal distribution with mean $0$ and variance $\sigma_1^2 + ...
10
votes
1answer
3k views

sum of squares of dependent gaussian random variables

Ok, so the Chi-Squared distribution with n degrees of freedom is the sum of the squares of n independent gaussian random variables. The trouble is, my gaussian random variables are not independent. ...
4
votes
1answer
281 views

Conditional return time of simple random walk

Consider a simple symmetric random walk on $\mathbb{Z}$, $(S_t)_{t \geq 0}$. Call $\tau_k = \min\{t \in \mathbb{N}\, : \, \, S_t =k \}$, the hitting time of $k \in \mathbb{N}$. Call $\tau^* = \min\{t ...
7
votes
2answers
3k 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 $\mathbb{E}[X] = +\infty$?
4
votes
2answers
1k views

Expectation of inverse of sum of random variables

Let $X_i$'s ($i=1,..,n$) be i.i.d. random variables with mean $\mu$ and variance $\sigma^2$. Is there a method that can be used to compute $\mathbb{E}[1/(X_1+...+X_n)]$?
4
votes
2answers
414 views

Difference between two independent binomial random variables with equal success probability

Let $X$ ~ $Bin(n,p)$ and $Y$ ~ $Bin(m,p)$ be two independent random variables. Find the distribution of $Z=X-Y$. see also Difference of two binomial random variables I figured this out: $$ ...
3
votes
4answers
478 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 ...
3
votes
4answers
7k views

Correlation between three variables question

I was asked this question regarding correlation recently, and although it seems intuitive, I still haven't worked out the answer satisfactorily. I hope you can help me out with this seemingly simple ...
2
votes
2answers
148 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 ...
1
vote
1answer
35 views

Geometric random variables $X_1:G(p_1)$ $X_2:G(p_2)$ $X_3:G(p_3)$ are independent, prove the following :

$$P(X_1 < X_2 < X_3)= \frac{(1-p_1)(1-p_2)p_2p_3^2}{(1-p_2p_3)(1-p_1p_2p_3)}$$ To be frank I do not know where to start with this question, I would like an idea to get me going, or better yet ...
1
vote
1answer
72 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
142 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 ...
5
votes
1answer
3k views

Linear transformation applied to a multivariate Gaussian random variable - what is the mean vector and covariance matrix of the new variable?

Given a random vector $\mathbf x \sim N(\mathbf{\bar x}, \mathbf{C_x})$ with normal distribution. $\mathbf{\bar x}$ is the mean value vector and $\mathbf{C_x}$ is the covariance matrix of ...
3
votes
1answer
91 views

Pairwise independence of Random variables does not imply indendence

Show by a counterexample that for a family $(X_i)_{i\in I}$ of random variables the independence of all pairs $(X_i,X_j)$ with $i,j\in I, i\neq j$ does not imply the independence of the family ...
2
votes
2answers
143 views

$E_n =\lbrace X_n > X_m \ \forall m < n \rbrace $ are independent

I'm stuck with this exercise. Suppose $(X_n)$ are independent random variables defined on $(\Omega, \mathfrak{F}, P)$ with the same p.d.f. Let $E_1 = \Omega$ and for $n \geq 2$ $$E_n =\lbrace X_n ...
2
votes
2answers
144 views

I want to show $E(X)=\int_{0}^{\infty}P(X\ge x)dx$ for non-negative random variable $X$

Show that for a non-negative random variable $X$, $$\mathbb E(X)=\int_{0}^{\infty}\mathbb P(X\ge x)dx.$$ I started with $$\mathbb ...
1
vote
1answer
65 views

What is the joint distribution of two random variables?

Today I was thinking about this and I have the feeling I am missing something obvious, but I can't seem to solve it. Suppose we have a continuos random variable $X$ with density $f_X(x)$. Let $Y = ...
1
vote
1answer
47 views

What is the chances of a duplicate in this equation

I'm not very good at math; However I have a scenario where I'm trying to find the chance of duplicate for randomly generated data. In a nuttshell I have a "bag" with 62 different items, lets say a ...
1
vote
1answer
73 views

Approximate normal distribution

Let $ X \sim N (0, 1)$. For $x$ large enough, the tail of the distribution of $X$ may be approximated as $$P(X > x) \sim e^{-x^2/2}/(x\sqrt{2\pi})$$ Consider a sequence of independent r.v. all ...
1
vote
0answers
158 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 ...
6
votes
2answers
841 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
41 views

How can I find the distribution of a stochastic variable X^2 if X is normal standard distributed? [duplicate]

I am considering a stochastic variable X that is standard normal distributed i.e. $$ F_X(x) = \int_{-\infty}^x\frac{1}{\sqrt{2\pi}}e^{-\frac{t^2}{2}}dt $$ How do I find out the distribution of $X^2$? ...
3
votes
1answer
77 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
351 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
4answers
275 views

Why is the exptected value (mean) of a variable written using square brackets?

My question is told in a few words: Why do you write $E[X]$ in square brackets instead of something like $E(X)$? Probably it is not a "function". How would you call it then? This question also applies ...
2
votes
1answer
128 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
1answer
111 views

Expected value and indicator random variable

During a period of $n$ days, two persons drink beers, one each. There are $n$ different beers $B_1, B_2, B_3,\ldots, B_n$, where $n\geqslant 1$ is an integer. t Person1 drinks the beers in order, ...
1
vote
1answer
41 views

Show $P(X|Z_1,…,Z_n,Y)\not = P(X|Z_1,…,Z_n) \Leftrightarrow P(Y|Z_1,…,Z_n,X)\not = P(Y|Z_1,…Z_n)$

If we have two random variables $X,Y$ and a set of random variables $\{Z_1,...,Z_n\}$, are there any common proofs of the result in the title? Which theorems does this follow after?
1
vote
0answers
16 views

Density function for RV

The density function for a random variable X is given in terms of a constant c. Find the value of c. What is the corresponding distribution function? Sketch both the density and the distribution ...
1
vote
3answers
270 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
1answer
112 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 ...
1
vote
2answers
75 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
3k 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 ...
1
vote
2answers
2k views

Joint PDF of two random variables and their sum

What is the joint PDF of two uniformly distributed random variables and their sum?
0
votes
3answers
71 views

Distribution of time that a flashlight can operate

The lifetimes of batteries are independent exponential random variables , each having parameter $\lambda$. A flashlight needs two batteries to work. If one has a flashlight and a stockpile of n ...
0
votes
1answer
112 views

Unifying the treatment of discrete and continuous random variable

I have been working on the reconciliation of the treatment of discrete and continuous random variable in a measure theoretic sense. But I found myself blocked on fundamental results. We know that If ...
0
votes
1answer
269 views

Birthday Problem (Poisson Distribution)

I've been reading up on Poisson Distributions and have come across the following problem. My doubts are in Bold: What's the probability that in a room of n people, nobody shares the same birthday? ...
0
votes
1answer
45 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.