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

learn more… | top users | synonyms

1
vote
1answer
66 views

Questions on finding expected value and variance on a Poisson distribution

FProblem: A student walks along a real line and tries to get to the origin. Each step he makes is random ; the larger the intended step, the greater the variance is of that step. When the student is ...
1
vote
2answers
37 views

Conditional variance of sum of two correlated random variables

Let $\theta\sim\mathcal{N}(\bar{\theta},1/\tau_\theta)$ be a normally distributed random variable, $\varepsilon\sim\mathcal{N}(0,1/\tau_\varepsilon)$ be a normally distributed noise term independent ...
1
vote
1answer
25 views

Find the distribution law of a function of a random variable

Let $X$ be a random variable with an exponential distribution $X\sim\operatorname{Exp}(\lambda)$, such that its expected value $\mathbb E[X] = 2$. Let $f$ be a function such that: $$f(x) ...
1
vote
2answers
34 views

Expected value of this continuous RV

I'm skimming through a basic introductory level stats-book and there's a problem which begins with: Let $X_n$ be a continuous random variable with a $PDF = f_n(x) = \frac {x^n}{n!} e^{-x}, ...
0
votes
0answers
33 views

Counter example of non continuity

I present in the following a variation of the problem described in Continuity of a deterministic function generated from a probability function. There, it has been proved that $g(x)$ is not ...
0
votes
1answer
28 views

Expected value of sum of uniformly distributed variables

Let $X_i$ be a uniformly distributed random variable on the interval $[-0.5, 0.5]$ that is: $X_i$ ~ $U(-0.5, 0.5)$, for $i \in [1, 1500]$ How can I calculate the expected value of the sum ...
3
votes
1answer
39 views

The expected value and standard deviation of $|X-Y|$ where $X$ and $Y$ are random variables

Suppose we have two independent random variables $X$ and $Y$, with expected values and standard deviations of $(\mu_X,\sigma_X)$ and $(\mu_Y,\sigma_Y)$, respectively. Can we say anything about the ...
2
votes
1answer
59 views

Continuity of a deterministic function generated from a probability function.

I am working on the proof of a specific proposition on probability theory whose particular case for two variables is presented in the following. Let $X_1$ and $X_2$ be different random variables ...
0
votes
1answer
25 views

Can the independence of random variables hold for their functions?

Suppose $X$ and $Y$ are two independent continuous random variables on $\mathbb{R}$. Define: $f:\mathbb{R}\mapsto\mathbb{R}$ as a $C^\infty$ map on $\mathbb{R}$. Then is it possible to find the ...
0
votes
0answers
12 views

Approximate CDF of arbitrarily aggregated random variable

I would like to know if my solution for the following is mathematically correct in general: I have a random variable $Z$ that is an arbitrary function of two other rvs $X$ and $Y$, so: $Z = f_{arb}(X, ...
-1
votes
3answers
42 views

Calculating the value of a binomial distribution

Suppose $X$ is a discrete random variable such that $X$~$B(100,0.028)$. What would be the fastest way to calculate something along the lines of: $P$ { $2 \le X \le 6 $ } ?
0
votes
0answers
14 views

Multivariate gaussian and average covariance matrix

Suppose we have a (possibly infinite) collection k-variate gaussian distributions $\{(\mathcal{N}(\mu_{\lambda}, \Sigma_{\lambda}))\}$ ($\lambda$ is just a label), and for each distribution $\mu \in ...
0
votes
1answer
34 views

Finding expected number of trials until we get head given density function?

Suppose we flip a coin with a random probability of Heads $P$ that has density $f(p) = 6p(1−p),\; p \in [0, 1]$. If we keep on flipping this coin until we get a single Heads, what is the expected ...
2
votes
1answer
38 views

Show that $a_n>0$ for all sufficiently large $n$

Let $F_n, G$ be distribution functions on $\mathbb R$. Suppose that $F_n(a_nx+b_n)\to G(x)$ as $n\to\infty$ for each $x\in c(G)$ where $c(G):=\{x\in\mathbb R:G(x)-G(x-)=0\}$. Here $a_n,b_n$ are ...
-1
votes
1answer
34 views

Question about finding expected value and variance of x given the mean.

Suppose Y is distributed as an exponential random variable with mean 0.5 and given Y = y, X is distributed as an exponential random variable with mean y. What is E(X) and Var(X)?
0
votes
1answer
27 views

Quick question concerning the sum of random number of random variables given mean and variance and average

$\DeclareMathOperator{\cov}{cov}$The problem is: Let $X_1, \ldots, X_n$ be independent random variables with mean $µ$ and variance $σ^2$. Let $X¯$ be the average of these n random variables. Find the ...
0
votes
1answer
23 views

Probability set function of the random variable $X$

Let a point be selected from the sample space $S = (0,10)$. Let $C \subset S$ and let the probability set function be: $$P(C) = \int_C \frac1{10}\ \mathsf dx$$ Define the random variable $X$ by: ...
0
votes
0answers
18 views

Conditions for the existence of moments of the supremum of a random variable

let $X_1, X_2,\dots X_n$ denote a sequence of $n$ iid random variables with the first $k$ moments of $F_X$ exist. Under what conditions (if at all) do the first $k$ moments of the random variable ...
1
vote
0answers
28 views

Proof that Sum of $n$ Squared Errors ~ Chi Square with $n$ $df$

There is a youtube video dealing with the proof that the sums of the squares of normally distributed $n$ random errors, each one distributed as $\sim \chi^2(1\text{ df})$ follows a chi square ...
1
vote
1answer
85 views

Circuit probability question regarding sum of a random number of independent random variables

Suppose we have n circuits that operate in a home. Each one will live according to an exponential random variable with rate λ. If X denotes the time at which a circuit first dies (i.e. the first circuit ...
-1
votes
2answers
28 views

Conditional day distribution probability

Let $X$ be a random day of the week, coded so that Monday is 1, Tuesday is 2, etc. (so $X$ takes values 1, 2,..., 7, with equal probabilities). Let $Y$ be the next day after $X$ (again represented as ...
2
votes
1answer
36 views

Random variable to the power of minus one?

I have a definition, it goes as follows: $\Pr$ is probability. $X$ is a random variable. $x\in\mathbb{R}$ $$Pr(X = x) = \Pr(\{ \omega\in\Omega \mid X(\omega)=x\})$$ So for example for a dice of 6 ...
6
votes
1answer
60 views

Is this set of random variables a Hilbert space?

Consider a sequence of i.i.d. random variables $\left\{ {{\varepsilon _t}} \right\}_{t = 1}^\infty $ with $E\left( {{\varepsilon _t}} \right) = 0$ and $E\left( {\varepsilon _t^2} \right) = {\sigma ...
2
votes
3answers
50 views

Distribution of a fractional part of the sum of uniform RVs

I had a question in class not long ago which I couldn't solve. I've been digging into it for a few hours now but I can't find the right direction. So the question is: Let $ U_1,..,U_n$ be I.I.D ...
0
votes
1answer
31 views

Let $A$ be a random matrix with i.i.d entries, what can we say about $Ax$?

Assume $A$ is an $m\times n$ random matrix with i.i.d entries, and $x\in\mathbb{R}^n$ be a fixed vector with $\Vert x\Vert_2=1$. Then can we say something about $y:=Ax$? Does $y$ still have i.i.d ...
0
votes
1answer
40 views

sufficient conditions for a stochastic process to be wide sense stationary

From the page Stationary process, I have the following definition: WSS random processes only require that 1st moment and autocovariance do not vary with respect to time and from the page ...
0
votes
1answer
56 views

Find the CDF of a function of two random variables

The joint probability density function of two continuous random variables $X$ and $Y$ is: $$f(x,y) = \begin{cases} 6x,& 0\leqslant x\leqslant y,\ 0\leqslant y\leqslant 1\\ 0,& \text{ ...
1
vote
1answer
42 views

Why this process is nonergodic?

I am studying a tutorial on stochastic processes and there's an example in it which I don't understand anything of it. First of all there is this criterion for a mean-ergodic random process: For ...
3
votes
1answer
67 views

Covariance of 1-D random process is $n\times n$!!!!

I'm reading a tutorial on stochastic processes. There is an example in the tutorial as follows: General Moving Average random process given as $X[n]=\frac{(U[n]+U[n-1])}{2}$ where $E[U[n]]=\mu$ ...
1
vote
1answer
26 views

Determine $P(S_n\leq1)$ where $S_n=\sum_{k=1}^nX_k$

Suppose that $X_n$ are i.i.d. $Uniform(0,1)$ random variables. Let $S_n=\sum_{k=1}^nX_k$ with $S_0:=0$. Then, determine $P(S_n\leq1)$. I know that maybe by using Characteristic function of $S_n$ ...
2
votes
1answer
32 views

Is tossing a die in 10 consequent days an ergodic process?

IT maybe an elementary question but I'm totally new to the concept. In Wikipedia, ergodicity is defined as follows: In statistics, the term describes a random process for which the time ...
0
votes
1answer
25 views

what's the difference between variable and process from a statistical point of view?

I'm reading a tutorial stochastic process: ergodicity and temporal averages and I'm totally confused. It is said that: Suppose an IID random process whose marginal PDF is Gaussian with mean ...
2
votes
2answers
93 views

Probability of Level Crossing

I am kind of stuck on how to proceed on this. $X_n$ is an IID process with $$f_{X_n}(y)= \frac\lambda2 e^{-\lambda |y|}$$ There is a stationary autoregressive process $Y_n$ defined as $$Y_n=\rho ...
1
vote
1answer
19 views

Expected Shortfall alternative definition

Define: $$q_\alpha(F_L)=F^{\leftarrow}(\alpha)=\inf\lbrace{x\in \mathbb{R}\mid F_L(x)\geq \alpha\rbrace}=VaR_\alpha(L)$$ I want to prove that: $$ES_\alpha = ...
1
vote
1answer
29 views

Expected Value of Two Random Variables

X is a random variable with a probability density function $f(x)$, g(x,y) is a function of two variables one of them is the random variable. I have \begin{equation} \int_{-\infty}^{\infty} ...
6
votes
0answers
103 views

Random variables that span copies of $\ell_p$

Consider the coin-toss measure $\mu$ on $\{0,1\}^\mathbb{N}$. Within this framework it is easy to construct a sequence of independent, symmetric Bernoulli random variables. Indeed the point-evaluation ...
3
votes
1answer
43 views

If ${a_i} \to 0$ and $\{ {X_i}\} _{i = 1}^\infty $ is a sequence of iid random variables with zero mean, does ${a_i}{X_i} \to 0$ almost surely?

My problem is slightly more specific than the title of this question: Let $0 < \beta < 1$ and let $\{ {X_i}\} _{i = 1}^\infty $ be a sequence of i.i.d. random variables with $E({X_i}) = 0$. In ...
0
votes
1answer
33 views

Seeking an example for Bayes estimator of two unknown parameters

I searched the web, taking advantage of several search approaches; however, due to redundancy of the existing information about Bayes estimator of one unknown parameter of random variables (either in ...
0
votes
0answers
20 views

Probability distribution of derivative of function of random variable

The calculation of probability distribution of a function of random variable is a well established theory and there are general rules on how to go from the distribution of r.v. to the distribution to ...
0
votes
1answer
34 views

Probability that one normal Random Variable will fall within a given range of another.

I'm struggling with the following problem: (ed: Don't be lazy. Just type it out. ) A certain small freight elevator has a max. capacity $C$, which is Normally distributed, with mean ...
4
votes
2answers
74 views

Lower bound for (function of) density of well-behaved random variable

Suppose we have a non-negative random variable $\tilde{\theta}$ such that $\mathbb{E}\tilde{\theta} = a > 0$, with finite variance $\sigma^2$. $\tilde{\theta}$ can take on values from $0$ to ...
3
votes
2answers
423 views

Probability of inequality between random variables

In order to prove a theorem in my research, I would like to use a lemma on basic probability theory, but I don't know if it is correct. For three random variables $X,Y$, and $Z$ not necessarily ...
1
vote
1answer
24 views

Rewriting probabilities as expectation

Consider the stopping time $\tau_a:=\lbrace{t>0| W_t >a\rbrace}$, where $W_t$ is a Brownian Motion. Define: $X_t:=W_{\tau_a+t}-W_{\tau_a}$. We have that $X_t$ is a Brownian Motion independent ...
0
votes
1answer
39 views

If independent r.v. converge in probability to a constant, do they converge almost surely?

I've seen several examples when a sequence of r.v. converge in probability but not almost surely, yet none of them had the sequence to be independent. Would additional conditions of independence and ...
1
vote
1answer
40 views

Continuous distribution and independence [closed]

Problem: In a room, there are 4 boys from high income families, 6 girls from high income families and 6 boys from low income families. How many girls from low income families also need to be present ...
0
votes
0answers
24 views

Is it possible to exchange a sum in a conditional expectation

Let $X_1, X_2, \ldots \geqslant 0$ and $Y$ be RVs over $\mathbb{R}^n$. Then is it true that $\mathbf{E} \left[ \sum_{i = 1}^{\infty} X_i \mid Y \right] = \sum_{i = 1}^{\infty} \mathbf{E} [X_i \mid ...
0
votes
2answers
50 views

Prove (or disprove) that $\mathbb{E}[X]\geq 0$ for positive random variable.

Let $X$ be a random variable such that $X\in[0,1]$. I was wondering if $\mathbb{E}[X]$ must be $\geq0$. Since $X$ is a positive random variable, we can apply the Markov-inequality: for each positive ...
0
votes
2answers
47 views

what is the expected value of $x^TAx$? [closed]

Assume $x\in \mathbb{R}^N$ is a random variable vector (like a noise sequence). You now want to calculate the following term: $E\{x^{T}Ax\}$, where $A$ is a constant matrix. How can this expression ...
0
votes
2answers
35 views

Eigenvalues of $\mathbb E\pmatrix{2X&X\\ 1-X&2X}$. [closed]

Let $X$ be a random variable between $0$ and $1$, such that: $\mathbb{E}[X]=\frac{1}{2}$. We have a matrix: $$A=\left( \begin{array}{cc} 2X & X \\ 1-X & 2X \\ \end{array} \right)$$ ...
0
votes
0answers
9 views

RBF transformation on a Normally Distributed Random Variable

I have a random vector $\mathbf{X} \sim \mathcal{N}(\mathbf{m,\Sigma})$ which is transformed by a Gaussian Radial Basis Function into the random variable $\mathbf{Y} = K(\mathbf X)$ where $K = ...