# Tagged Questions

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### How can I do a constructive proof of this:

Say Z is a non-negative R.V, and P(Z>0)>0. Then exists a a>0 and an b>0 such P(Z>a)>b. I am not sure how to start with the proof, I've been assigning numbers than can qualify for some CDFs but I don´t ...
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### Notation for image of a discrete random variable?

Suppose we have a discrete probability space $(\Omega,\Sigma,\mathbb{R})$ and a discrete random variable $X:\Omega \to \mathbb{R}$. A usual way to denote the set of values that $X$ takes is simply ...
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### Pdf of the product of an exponential r.v. and a beta r.v.

Let $X$ and $Y$ are 2 independent random variables, where $X$ has an exponential distribution with parameter $1$ and $Y$ is $\beta(a,b)$ distributed. What is the Pdf of $W=XY$ ? Thanks !
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### Exchangeability and pairwise exchangeability

Suppose we have $\{Y_{i}\}_{i=1}^{n}$ that is exchangeable so the joint distribution of this sequence is the same as $\{Y_{\sigma(i)}\}_{i=1}^{n}$, where $\sigma:\{1,...,n\}\rightarrow\{1,...,n\}.$ ...
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### Continuous and Discrete random variable distribution function

I have a very basic question in probability. It pertains to the difference between a continuous random variable distribution function and a discrete one. This question has confused me many times. ...
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### If $\mathbb P_{Y|X\in A}(B):=\mathbb P(Y\in B \ | \ X \in A)$ can we explicitly define the r.v. $(Y|X\in A)$?

When introducing conditional expectation, one can define $\mathbb P_{Y|X\in A}(B):=\mathbb P(Y\in B \ | \ X \in A)$ which is itself a law. I was wondering if there is a way to define a random variable ...
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### If $X$ is Poisson, find the expectation of $\frac{1}{a+X}$

If $X$ is a Poisson random variable with $\Pr(X=k)=e^{-\lambda}\frac{\lambda^k}{k!}$ and $a>0$ then find the expectation of $\frac{1}{a+X}$ If I make use of ...
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### finding out the probability density of a random process

I have to find out the probability density function of a random process with the following specifications:z(t)= xcos(wt)-ysin(wt) where x and y are two independent gaussian random variables. Now what ...
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### Why $E[X|\mathcal{G}]=X$ if $X$ is $\mathcal{G}$-measurable?

If $X$ is a $\mathcal{G}$-measurable random variable, why $E[X|\mathcal{G}] = X$? I know the intuition (basicly we're conditioning on the same informations on which $X$ is defined, $\sigma(X)$, we ...
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### Existence of density function for a sum of 2 Random Variables

Let's suppose that $Y$ is the normal distribution and that $X$ is another random variable whose density function may or may not exist. Does it follow that $Y+X$ has a density function? I am reading ...
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### $X = (X_1, X_2)$ is it not a multivariate random variable?

$X=(X_1,X_2,\ldots, X_P)$ is a $p$-dimensional random variable on $(\Omega, S, P)$ iff $X_i$'s are univariate random variables on the same probability space $(\Omega, S, P)$ ." We all know ...
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### Measurability of a Borel function

I need some help on the following proof. The claim is: Suppose $f:\mathbb{R}^k \to \mathbb{R}$ and $f \in B(\mathbb{R}^k)/B(\mathbb{R})$. i.e. Borel measurable. Let $X_1$,...,$X_k$ be random ...
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### Problem regarding Conditional probability

Let $\mathbf{X}$ be an $n-$ dimensional random variable. This variable can be written as $\mathbf{X} = \left[\mathbf{X}_1^T\hspace{5pt}\mathbf{X}_2^T\right]^T$. where, $\mathbf{X}_1$ is $m-$ ...
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### Prove that a.s.$\lim\limits_{t\to\infty}\frac{N_t}{t}=\frac{1}{\mu}$

Consider a diligent janitor who replaces a light bulb the instant it burns out. Suppose that the first bulb is put in at time zero and let $X_i$ be the lifetime of the i-th bulb. Suppose ...
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### random variables not independent but $\mathrm{E}[X|Y]=\mathrm{E}[X]$

I have to find two r.v. X,Y defined in a probability space ($\Omega, \mathcal{F}, \mathrm{P}$), which are not independent but for which $\mathrm{E}[X|Y]=\mathrm{E}[X]$ nonetheless, with ...
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### Random variable bounded by another random variable

How to find $\Pr(z<X<Y)$ if $X$ and $Y$ are independent exponential r.v.'s with parameters $\lambda$ and $\mu$ So $x$ is bounded by $z$ and $y$, and y must be go from $z$, (and not from ...
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### Proving independence of random variables

If $X$ and $Y$ are independent exponential random variables with parameter $\lambda$ and $\mu$. Let $Z=\min(X,Y)$, prove that $Z$ and $\mathbf 1_{\{X<Y\}}$ are independent. I don't know, how ...
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### Show that the shift map is measurable and measure-preserving

Show that the shift map $\theta$ of Definition 6.3 is measurable and measure-preserving. Not sure how to represent $\theta^{-1}$ which I believe is where I am stopped in solving this problem.
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### If $X_{i}$ are I.I.D and $n^{-1}\sum_{i=1}^{n}X_{i}$ converges a.s/in-distribution to a constant $a$ is it true that $a=\mathbb{E}[X_{i}]$?

The question itself is in the title. It is immediate by the strong law of large numbers that if $X_{i}$ had a finite first moment then we would have a.e convergence (and thus in probability and in ...
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### Finding tight upper/lower bounds for $\mathbb{E}[\frac{1}{1+X^{2}}]$ where $X$ is a RV with $\mathbb{E}[X]=0$ and $\mbox{Var}(X)=\nu<\infty$

The question is pretty much in the title. My first thought was using Jensen's inquality to get some sort of lower bound. Since $\frac{1}{1+x^{2}}$ is convex on ...
Let $\{ \xi _a \}_{a \in [0;1]}$ be a family of independent uniformly distributed on $[0;1]$ random variables on some probability space $(\Omega, \mathscr{F},P)$, indexed by a continuous parameter. ...
### Show that $\int_Ax^2\mu_X(dx)\le\frac{12}{11u^2}\{1-\Re(\Phi_X(u))\}$
Let $A=[-\frac1u,\frac1u]$, Show that $$\displaystyle\int_Ax^2\mu_X(dx)\le\frac{12}{11u^2}\{1-\Re(\Phi_X(u))\}$$ where $\Phi_X(u)$ is the characteristic function of the r.v. $X$ Hint: ...