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

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2
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0answers
14 views

Can someone explain what a portfolio is in financial math?

I took mathematical probability last semester and now I am taking financial mathematics, but only probability was a pre requisite for financial math (no finance classes were required). These types of ...
1
vote
2answers
30 views

Product of i.i.d. random variables uniformly distributed on $(-1,1)$ converges almost surely to $0$

Let $(\Omega,\mathcal{F},P)$ be a probability space and $(X_n)$ i.i.d. random variables with uniform distribution on $(-1,1)$. If $Y_n=X_1\ldots X_n$, prove that $(Y_n)$ converges to $0$ a.s.
0
votes
1answer
12 views

Convexity of an exponential function.

A random variable $Y_i$ is given such that, $\mid$Y$_i\mid$$\leq$ $c_i$ where i ranges from 1,.....,t and t is some constant. Now, $Y_i$ is expressed as : $Y_i = ((Y_i - c_i) + (Y_i + c_i))/2$ $= ...
1
vote
1answer
13 views

Generate random variate using inverse transform technique of $ f (x) =a (1+|x-2|)$

I need to generate a random variable with density function: $$ f(x)= \begin{cases} a (1+|x-2|) , & {-1 \le x \le 4} \\ 0, & \text{elsewhere} \end{cases} $$ For that I need to inverse the ...
0
votes
0answers
12 views

Finding mean and variance of P=I²R

Question given: The power,P in watts dissipated in an e.c with resistance,R .Given equation to be P=I²R, where I is current and R is fixed 50 ohms. However I is random variable with mean 15 ...
0
votes
0answers
28 views

Mutual information $I((X,Y,Z);A)$ larger for small pairwise mutual informations $I(X;Y), I(X;Z), I(Y;Z)$?

Is the mutual information $I((X,Y,Z);A)$ larger for small pairwise mutual informations $I(X;Y), I(X;Z), I(Y;Z)$? In particular, in the extreme case that the pairwise mutual informations are ...
0
votes
0answers
24 views

Expected value and standard deviation for infinite sample with probability

Problem: A recruiting firm finds that $20$% of the applicants for a particular sales position are fluent in both English and Spanish. Applicants are selected at random from the pool and interviewed ...
1
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0answers
25 views

Reflection principle for simple random walk

Let $(X_n)$ be a sequence of independent random variables, such that $P(X_i=1) = P(X_i=-1) = 1/2$. Then, the reflection principle states that for all $a > 0$, $$P(\max_{1\leq k\leq n} S_k \geq a) ...
2
votes
0answers
22 views

Conditional expectation of another expectation expression.

What is the intuition and the proof behind the given below expression where $U,V,W$ are random variables: $E[V | W]$ = $E[E[V | U,W] | W]$ I know that $E[V | W]$ can be treated as a random variable ...
0
votes
1answer
27 views

The density of a random variable $X$ is $f(x)$ proportional to $x^{-1/2}$ , what is the mean of $X$?

The density of a random variable $X$ is $f(x)$ proportional to $x^{-1/2}$ for $x \in [0,1]$$ and $f(x) = 0$ for $x \notin [0,1]$. Then, the mean of $X$ is $\frac 12$ $\frac 1{\sqrt2}$ $\frac 13$ ...
1
vote
2answers
12 views

Let $X = -10Y + 10$. Let $r_1$ be the correlation between $X$ and $Z$ and $r_2$ be the correlation between $Y$ and $Z$.

Let $X = -10Y + 10$. Let $r_1$ be the correlation between $X$ and $Z$ and $r_2$ be the correlation between $Y$ and $Z$. Then, which of the following is the best answer? $r_1 = r_2$. $r_1 = 10r_2$ ...
1
vote
1answer
20 views

Finding bivariate probability mass function (by counting?)

Suppose that we role $d$ dice. Let $X, Y$ be random variables, where $X = \#$ rolled by the die with the highest value. $Y = \#$ rolled by the die with the second highest value. By convention, we ...
1
vote
2answers
22 views

Likelihood Function for the Uniform Density. $ (\theta-1,\theta+1)$

Let the random variables $X_1,X_2,...,X_n$ iid $U[\theta-1\,,\theta+1]$. So the likelihood function therefore has the form: $L(\theta|X)=\prod_{i=1}^nf(X_i|\theta)=\frac{1}{2^n}I(X_1, . . . , X_n ...
0
votes
3answers
48 views

Suppose $P(|X| < 1) = 1$ and $P(|Y| = 2) = 1$.

Suppose $P(|X| < 1) = 1$ and $P(|Y| = 2) = 1$. Then which of the following is true? The standard deviation of $X$ is smaller than that of $Y$. The mean of $X$ is smaller than that of $Y$. The ...
0
votes
1answer
25 views

inner product of two random vectors

Two random vectors $\mathbf a$ and $\mathbf b$. Vector $\mathbf a$ has uncorrelated entries satisfying $\mathbb E [\mathbf a \mathbf a^{\rm H}]=\sigma^2{\mathbf I}$. Now I need to calculate ${\mathbb ...
0
votes
0answers
17 views

how to derive joint cdf from marginal cdf

Is there any way to derive the joint CDF of $X$ and $XY$ if distributions of $X$ and $Y$ are available? $X$ and $Y$ have similar distribution with different parameters. $X$ and $Y$ are also ...
0
votes
0answers
9 views

approximating random variable

Can anyone explain the construction of the sequence of simple random variables that can be approximated to any random variable? $ X_n(\omega)=\sum_{k=0}^{n2^{n}} k2^{-n}$ where $\, k2^{-n} \leq ...
0
votes
1answer
33 views

Two random variables X and Y follow the same distribution. Then

Two random variables $X$ and $Y$ follow the same distribution. Then The distribution of $X − Y$ must be symmetric about $0$. The median of $X − Y$ must be zero. The median of $X + Y$ is twice of ...
0
votes
0answers
22 views

Dependence between two mixtures of Gaussians with 1-to-1 correspondence between means

Let $N_{ij}\sim N(0,1),\ i,j=1,2$ be independent standard normally distributed variables, $B\sim\text{Bernoulli}(\frac{1}{2})$ a coin flip with $P(B=1)=P(B=2)=\frac{1}{2}$, and ...
4
votes
5answers
187 views

Finding the expected value in the given problem.

It is given that a monkey types on a 26-letter keyboard with all the keys as lowercase English alphabets. Each letter is chosen independently and uniformly at random. If the monkey types 1,000,000 ...
0
votes
0answers
31 views

Finding MLE for $\mu^{2}$

The problem says the following: Let $X = (X_{1}, ..., X_{n})$ be a random sample, where $X_{i} \sim N(\mu_{0},1)$, where $\mu_{0} \in \mathbb{R}$ is unknown. I do not have problems calculating the ...
0
votes
2answers
41 views

Finding the distribution of $n$ times the minimum of $n$ exponential random variables.

I'm having trouble with this question: Let $X = (X_{1}, \ldots, X_{n})$ be a random sample, where each $X_{i}$ is an exponential random variable with mean $\lambda_{0} \in (0,\infty)$, which is ...
0
votes
1answer
26 views

Calculation of E[X^2] for a random sum?

I have this random sum, $Z=\sum_{k=1}^{N}Y_{k}$ where $(Y_1,Y_2,\dots,Y_k,\dots)$ are independent and exponentially distributed random variables (mean is $\mu$). We've defined $N=M+1$ where $M$ is ...
1
vote
1answer
25 views

Continuous mapping theorem to show $g(x_n)$ to $g(x)$ not converges.

Let $X_n$ be a random variable sequence, such that $P(X_n=1)=1/n$ and $P(X_n=1/n)=1-(1/n)$. Let g be a function, such that $ g(x)= 0$ if $ x\le0$, and 1 if $x>0$ Show that $g(X_n)$ not converges ...
1
vote
0answers
21 views

Test a law-of-iterated-logarithm-like result, with numerical simulation

I have a non-standard random walk $S_n$ for which the increments are not exactly independent (I could describe it, but it would be a totally different long and complex topic). I expect it to have ...
0
votes
1answer
33 views

Sets defined by probabilities

I am doing the following problem from Cover and Thomas, Elements of Information Theory, for self-study: Let $X_1,\dots, X_n$ be an i.i.d. sequence of discrete random variables with entropy $H(X)$. ...
1
vote
1answer
20 views

Maximising the logarithmic expectation in coin bets

We are throwing a coin $N$ times and for some reason the probability that we get heads in the $n$-th toss is $p_n\geq\frac 12$. Now starting with capital $X_0$, before each toss we decide to bet a ...
0
votes
0answers
16 views

What is the distribution of sum of the squares of k dependent standard normal random variables?

It is known that the sum of the squares of k independent standard normal random variables is chi-squared distributed, what happens if we look at the sum of the squares of k dependent normal variables? ...
3
votes
1answer
57 views

Numerical evidence of law of iterated logarithm (random walk)

The law of iterated logarithm states that for a random walk $$S_n = X_1 + X_2 + ... X_n$$ with $X_i$ independent random variables such that $P(X_i = 1) = P(X_i = 1) = 1/2$, we have $$\limsup_{n ...
0
votes
1answer
21 views

Expectation and Variance [on hold]

A day trader buys an option on a stock that will return \$100 profit if the stock goes up today and lose \$200 if it goes down. If the trader thinks there is a 75% chance that the stock will go up, ...
1
vote
2answers
29 views

Probability that 4/4, 2/4 Discrete Uniform RVs aren't equal

If I have four discrete $uniform(1,32)$ RVs. I'm trying to figure out the probability that a) None are the same b) exactly two are the same I thought that the probability that none are the same ...
1
vote
2answers
34 views

Geometric distribution achieves maximum entropy for given mean

Let $X$ be a random variable with geometric distribution, ie $P(X=k)=p(1-p)^k$. If I calculated it correctly, $X$ has mean $E(X)=\frac{1-p}p$ and entropy $H(X)=-\log p - \frac{1-p}p\log{(1-p)}$ ...
0
votes
0answers
19 views

Variance in the sum of batch-correlated residuals in a regression

I am looking at a regression model of the following form: $Y=intercept+\beta_{Yf.n}X_f+\beta_{Yn.f}X_n +error$ where $X_f$ and $X_n$ are predictors. A value for $Y$ will be sampled from the ...
4
votes
2answers
31 views

Probability of every ball occurring in multiple independent random samples

An urn contains 5 distinct numbered balls. You choose 2 without replacement. You then reset the urn and choose another 2 without replacement. Do this one more time. Now you have three random samples ...
-1
votes
0answers
16 views

$\sigma$-algebras of r.v. $X(\omega)=\sin \omega$ and $Y(\omega)=\cos \omega$ [closed]

Consider the random variables $X,~Y:(-1,1) \rightarrow \mathbb{R}$ such that $X(\omega)=\sin \omega, ~Y(\omega)=\cos \omega$ (the space $(-1,1)$ is equipped with the $\sigma$ algebra of the Borel ...
2
votes
0answers
42 views

$X_{n}$ independent and there is $a_{n}\to0$ s.t $\lim\limits _{m\to\infty}a_{m}\sum_{n=1}^{m}X_{n}$ is finite w.p 1. Then the limit is constant.

I'm trying to prove the following claim: Suppose $X_{1},X_{2},...$ are independent and there exists a sequence $\left\{ a_{n}\right\} _{n\geq1}\subseteq\mathbb{R}$ s.t $\lim\limits ...
0
votes
1answer
42 views

Prove the conditional expectation for iid binary random variables/.

Suppose that $(X_1, X_2, \dots)$ are independent identically distributed binary variables that take on the values $0$ and $1$ with probability $P[X_i = 1] = p$, $0 < p < 1$. We take a new r.v., ...
0
votes
0answers
29 views

heuristic for expected number of visits random walk

What is the heuristic argument that explains why, on $\mathbb{Z}^d$, $d \geq 3$, the expected number of visits of a random walk starting from the origin at $x$ is of order $$ O(|x|^{2-d})? $$
0
votes
1answer
23 views

Cumulative distribution functions and random variable problem

So the question is: If X is a random variable with a cdf $FX (t)$, and Y is the random variable given by Y = aX + b. Express the cdf $FY (t)$ of Y in terms of $FX (t)$. (Consider separately the ...
0
votes
0answers
13 views

Sub-Gaussian Random Variable with Small Variance

Write $X \in sG(\sigma^2)$ if $X$ is sub-Gaussian of parameter $\sigma^2$, that is $\mathbb{E}(e^{\lambda X}) \le e^{\lambda^2 \sigma^2 / 2}$. I'm interested in showing that, given $\epsilon > 0$, ...
1
vote
2answers
54 views

how to calculate expectation $X$ continuous and $Y$ discrete

if has density $f(x,y) = \dfrac{12}{13}x^y,\quad 0<x<1, \quad y=1,2,3$, how to calculate this expectation $E(Y\mid X=x)$? I am confused because $X$ be continuous random and $Y$ be discrete ...
1
vote
0answers
49 views

Infinite sum of indicators almost sure convergence

Let $S_{n}:= \sum_{i=1}^{n} X_{i,n}$ where for each $n, X_{1,n}, X_{2,n},..., X_{n,n}$ are sequences of independents r.v.'s. $$X_{i,n}=\begin{cases}1, & \text{with probability }p_n\\0,& ...
1
vote
1answer
41 views

Computing the expectation of a Tricky R.V. (Brought form Neuroscience).

I need to compute $\Bbb{E}(\tau^{X} \ \Bbb{1}_{\{\tau^{X}<+\infty\}})$ where: $1) $ $\tau^y$ is a r.v. representing the time spent by a particle until it "jumps", ( $ y \in R_{\geq 0} $ is the ...
2
votes
1answer
28 views

Convergence in probability implies Fatou's lemma?

Let $(\Omega, \mathcal{F},P)$ be a probability space and $(X_n)$ be a positive-valued sequence of random variables on $\Omega$. We assume that $(X_n)$ converges in probability to the random variable ...
1
vote
1answer
25 views

Almost everywhere equality of r.v.'s , based on information on mean values.

Let $X, Y$ be two random variables on a probability space $(\Omega, \mathcal{F},P)$, where $\mathcal{F}=\sigma(\mathcal{E})$. We assume that: $\mathcal{E}$ is closed on intersections, i.e. $A\cap ...
1
vote
1answer
31 views

Does this sum of normally distributed random variables necessarily result in a continuous R.V?

Originally I had asked whether two continuous random variables can sum to a discrete random variable. More specifically, I am wonder whether, if we Let $X_n \sim \text{iid } N(0,\sigma_x^2)$ and $Y_n ...
0
votes
0answers
5 views

Signing *change* of probability that one random variable is lower than another

Let $\tilde{z}_L \in [0,1]$ and $\tilde{z}_H \in [0,1]$ denote two random variables, with $F_L(z|\theta) := \Pr\{\tilde{z}_L \leq z|\theta\}$ and $F_H(z|\theta) := \Pr\{\tilde{z}_H \leq z|\theta\}$. ...
2
votes
2answers
19 views

zero covariance but not independent - normally distributed random variable $X$ and $X^2$

This is one of my homework question, which the answer sheet has already been given out. However, I still don't understand it. Exercise 1.1. It is well known that for two normal random variables, zero ...
1
vote
0answers
14 views

Cracking any linear congruential generator

I have a linear congruential generator $X_{n+1} = (aX_n + b) \bmod 2^k $with given arguments and number $Y$. The problem is to find the smallest $i$ that $X_i = Y$ or tell that there is no such $i$. ...
0
votes
2answers
24 views

IID variables in statistics and real-life assumptions

IID (Independent and Identically Distributed) Random variables are often used in statistics, where a truly random sample is assumed to be made of IID variables. I'm studying basics of statistics (as a ...