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

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0
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1answer
23 views

What are the recent real life use or applications of the Cauchy Random Variable?

We have a short assignment on the described question and I already have gone through a lot of trash results from Google. I can't seem to find any. I don't know where else to post this question. ...
0
votes
1answer
20 views

Variance of Signum Function of Two Random Variables

Let $ X $ and $Y$ be two random variables with means $\mu_X$ and $\mu_Y$ respectively, as well as variances $\sigma_X$ and $\sigma_Y$ (all of which exist). I am interested in computing the following ...
5
votes
1answer
60 views

Is $\pi^k$ any closer to $[\pi^k]$ than expected?

Particular questions such as Why is $\pi$ so close to $3$? or Why is $\pi^2$ so close to $10$? may be regarded as the first two cases of the question sequence Why is $\pi^k$ so close to $[\pi^k]$? ...
2
votes
1answer
53 views

Number of inversions

Compute the sum of the number of inversions that appear in the elements of $S_n$. In other words find the total number of inversions that the elements of $S_n$ have combined. I mean how can we ...
1
vote
0answers
6 views

Centered Poisson, Scaled Poisson, Transformed Poisson

Given $y_1,y_2,\ldots,y_N$ with $y\sim \operatorname{Poisson}(\lambda)$. The question is, what is the distribution of $y_i-\bar{y}$ and $\frac{y_i-\bar{y}}{\bar{y}}$, where $\bar{y}=\sum_1^N y_i/N$. ...
3
votes
0answers
54 views

If $X_n\nearrow X$ then $E(X_n)\rightarrow E(X)$?

Let $(X_n)$ be an increasing sequence of real valued integrable rvs on a probability space $(\Omega,\mathcal{F},P)$, such that $(X_n)$ converges ae to some rv $X$. Is it true that $E(X_n)\rightarrow ...
0
votes
1answer
31 views

Find $c=c(n)$ so $T = c \sum_{i=1}^{n} |X_{i}|$ is an unbiased estimator.

I'm having some trouble trying to solve the following problem: Assuming that $X =(X_{1},\ldots,X_{n})$ is a random sample from the normal distribution with mean $0$ and unknown standard deviation ...
0
votes
1answer
14 views

Pseudo-inverse of the Cumulative Distribution Function of X

The goal of these calculations is to write a Python function that generates pseudo-random values with the distribution described below. This isn't relevant to the question (or even to this ...
0
votes
0answers
36 views

Joint probability distribution $X, Y$.

$f(x,y)= \frac{3}{2}(x²+y²)$, $\:\:0 \leq x,\: y \leq 1$ $0$, elsewhere Determine whether or not $X$ and $Y$ are independent. Independent characteristic when $f(x,y)=f(x)f(y)$ To find f(x) and ...
1
vote
0answers
35 views

Probability Random Variables Fall in an Interval

I've been trying to figure out a counting problem and can't wrap my head around how to calculate the probability. If we let $X_{1}, . . . , X_{10}$ be independent random variable with a uniform(0,1) ...
0
votes
0answers
16 views

Continuity of random variable as function of a random variable

Suppose, we are given a continuos random variable $X$ and a continuous and nondecreasing function $f$. Can it be shown that a second random variable $Y=f(X)$ is continuos on the support of $X$? What ...
1
vote
2answers
42 views

How to compute a “luck percentile” from a set of random numbers or die rolls

I think it's easiest if I start with my actual use-case: In a video game (XCOM), soldiers shoot at aliens. When they do, they have a % chance to hit. Hitting deals damage. I want to look at each ...
3
votes
1answer
51 views

Why a function in a measure space is random variable?

Let $(\Omega,\mathcal{F})$ be a measure space and $X$ mapping from $\Omega$ to $\mathbb{R}$. Assume that $X^{-1}((a,b])\in \mathcal{F}$ for all intervals. Prove that $X$ is a random variable. ...
-1
votes
0answers
22 views

How to find the max of two different random variables [on hold]

If I have 1 uniformly distributed RV and the other one constant with fixed value if the first one is please click here to see it FR(r)=r^2/c^2 and I have a constant length say r1. I want to write the ...
0
votes
1answer
16 views

Show that Uniform$(1,5)$ is neither singular nor absolutely continuous with respect to Uniform$(0,3)$.

Actually, I'm just studying singular continuity, absolute continuity.I know the definitions.And have solved few very basic sums. Now, in this problem, I'm not understanding what does this 'with ...
0
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0answers
12 views

Reduction of Two Independent Random Variables in Quadratic Form

Consider the $n \times 1$ random vector $\mathbf{x}$ and the $p \times 1$ random vector $\mathbf{y}$. The vectors are independent of each other, and $\mathbf{y}$ has an expected value of zero. I want ...
0
votes
0answers
8 views

Statistical distance between a multiplicative mask and a random number

Given $x \in \{1,\ldots,2^n\}$ and a uniform random $r \in \{1,\ldots,2^{n+k}\}$, then the statistical distance $\Delta(x + r\bmod q; r) < 2^{-k}$, for a $q > 2^{n+k+1}$. With addition this is ...
0
votes
1answer
23 views

Convergence of product of random variables with distribution $U(0, e)$

Let $X_1, X_2, \ldots$ be a sequence of independent random variables with uniform distribution on $[0, e]$. Let $R_n:=\prod_{k=1}^n X_k$. By Kolmogorov's zero–one law $(R_n)$ converges with ...
2
votes
3answers
83 views

Variance of the random sum of a Poisson?

We have that $N$ and $X_1, X_2, \dots$ are all independent. We also have $\operatorname{E} [X_j] = \mu$ and $\operatorname{Var}[X_j] = σ^2$. Then, we introduce an integer–valued random variable, $N$, ...
1
vote
0answers
21 views

Preserving independence of random variables

Suppose I have three random variables, $X,Y,Z$ with $X$ independent of $Z$, $Y$ independent of $Z$. Which transformation can I apply to $X,Y$ to that the result is again a random variable independent ...
4
votes
2answers
85 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 ...
0
votes
2answers
62 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
19 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
18 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
23 views

Finding mean and variance of $P = I^2 R$

Question given: The power, P in watt dissipated in an e.c with resistance, R in ohm. Given equation to be P = I^2 R, where I is current in ampere and R = 50 ohms. However, I is a random ...
0
votes
0answers
37 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
28 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
29 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
23 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
36 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
16 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
22 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
24 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
53 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
26 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
18 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
12 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
56 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 ...
5
votes
5answers
295 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
38 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
57 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
28 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 ...
2
votes
1answer
26 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
24 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
21 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
60 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
22 views

Expectation and Variance [closed]

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, ...