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

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-1
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0answers
13 views

Finding distribution of $Y$ and $P(Y>1)$

$X$ is a random variable which has normal distribution with mean $4$, variance $9$ and $Y=3X-8$. What is the distribution of the random variable $Y$? How should I find $P(Y>1)$?
0
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1answer
17 views

Finding $a$ if $X$ is a random variable

$X$ is a normal random variable with mean $6$ and variance $4$. How should I find $a$ if $P(X>a)=0.4$?
0
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0answers
17 views

Equivalent definition of singular random variable

I'm taking an intermediate course in probability theory (that is without measure theory) and when defining singular random variables (after showing the devil's function), the book defines: $X$ is a ...
0
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1answer
29 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
25 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 ...
8
votes
1answer
138 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
55 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
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0answers
56 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
33 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
37 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
37 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
17 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
43 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
54 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
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0answers
22 views

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

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
votes
0answers
13 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
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0answers
22 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
89 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
63 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
vote
0answers
31 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
40 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
20 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
58 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
64 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
29 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
27 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
22 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 ...