Tagged Questions
1
vote
2answers
54 views
Must the sequence $X_n$ converge to $0$ in probability?
Let $X_1, X_2,\dots$ be a sequence of random variables with
$\lim_{n\to +\infty} E[|X_n|] = 0$.
Is it correct or wrong that the sequence $X_n$ must converge to $0$ in probability?
6
votes
2answers
243 views
Random sum of random variables
Say you sum i.i.d. variables $X_i$ a total of $Y$ times. If you know the distribution of random variables $Y$ and $X_i$, what is the calculation you have to do to get the distribution of the sum?
0
votes
1answer
76 views
Deriving the transformation function of a random variable from the original and the final distributions
Consider a random variable $X$ and consider that this variable can be either real or integral (so I would like to cover both cases: continuos and discrete random variables). Consider to transform this ...
3
votes
2answers
123 views
Prove the monotonicity of the expectation of a binary random variable function
Consider $R$ independent binary random variables $y^1, \ldots, y^R$ over the space $\{-1, +1\}$ such that $\Pr(y^j = 1) = p^j \geq 0.5$ and $\Pr(y^j = -1) = 1 - p^j$, $\forall j = 1,\ldots,R$. ...
2
votes
0answers
54 views
random walk with possibility to freeze
Consider a Random Walk on a one-dimensional lattice. The walker starts moving at time $0$ from $x=0$. At every step, the walker moves to the right with probability $p$, to the left with probability ...
2
votes
1answer
62 views
Where is the fallacy in this coupling argument of two Bernoulli variables?
With respect to the scenario introduced in Prove the monotonicity of the expectation of a binary random variable function, let us now suppose that the function:
$$\begin{align*}
f(\mathcal{J}) = ...
1
vote
1answer
49 views
Expected minimum distance of a random point with respect a set of random points on the plane
I need to estimate, or bound, the expected minimum distance of a random point with respect to a set of other random points, all of which are located inside of a bounded rectangle.
More specifically, ...
1
vote
2answers
411 views
Proof of analogue of the Cauchy-Schwarz inequality for random variables
The Cauchy-Schwarz inequality tells us that for two vectors $u$ and $v$ in an inner product space,
$$\lvert (u,v)\rvert \leq \lVert u\rVert \lVert v \rVert$$
with the equality holding iff one vector ...
1
vote
1answer
71 views
Is the relation of having positive covariance well behaved with respect to taking the inverse?
Let $X$ and $Y$ be two random variables, $X$ strictly positive. Assume that Cov$(X,Y)>0$. Does this imply that Cov$(1/X, Y)<0$?
I know that being positively correlated is not a transitive ...
0
votes
0answers
42 views
iterative transform of standard normal random variable
Given a discrete series of random variable $n(i)$ that each element follows the standard normal distribution $N(0,1)$, another series is defined iteratively as:
$$u(i+1)=au(i)+bn(i)$$
where ...
0
votes
0answers
40 views
a sequence of random variables that converge to a constant c in probability but fail to converge to c with probability 1?
Any example that a sequence of random variables that converge to a
constant c in probability but fail to converge to c with probability 1?
0
votes
1answer
66 views
correlation between two different variables
I am studying stochastic processes and found the next problem:
Let $A$ and $\Phi $ be two independent random variables such that $E(A) = 0$, $E(A^2) < \infty$, and $\Phi$ is uniformly distributed ...
0
votes
2answers
646 views
Joint PDF of two random variables and their sum
What is the joint PDF of two uniformly distributed random variables and their sum?
-2
votes
2answers
53 views
Expression of Discrete and Continuous random variables.
Let $X$ be a discrete random variable. And let $Y = cX$ for some constant $c$.
How can you express the distribution of $Y$ in terms of the distribution of $X$?
Let $X$ be a continuous random variable ...
