# Questions tagged [random-variables]

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

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### $\limsup_{n\to\infty}S_n=+\infty\ \mathrm{a.s.}$ and $\liminf_{n\to\infty}S_n=-\infty\ \mathrm{a.s.}$ for the random walk $(S_n)$?

Let $(X_n)_{n\geq1}$ be i.i.d. real-valued random variables such that $\mathbb{E}(X_1)=0$ and $\mathrm{Var}(X_1)=\mathbb{E}({X_1}^2)>0$, where the variance $\mathrm{Var}(X_1)$ can be $+\infty$. Let ...
1 vote
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### Expectation of a function of three random variables

I have three random variables $X, Y,$ and $Z$. $X=a$ whenever $Y \geq b$, and $X=0$ whenever $Y<b$, where $a$ and $b$ are constants. I have some function $g(X,Y,Z)$ of all three variables. I know ...
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### ballistic problem

A projectile is launched at an angle $θ$ with respect to the surface with velocity $v_0$ (deterministic). If the angle of inclination is a uniform random variable on $[0,π/2]$, calculate the function ...
32 views

### How to prove finite absolute value of expectation for $Z(t)=W(t)^2t$ when $W(t)$ is a standard Wiener Process? [closed]

The question lies in the title. I am trying to show that $Z(t)$ is a Martingale. The martingale property associated with the filtration I have computed.. I tried using the Hölder's Inequality to prove ...
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### Random variable $f(N_\lambda)$ with same law as Poisson $N_\lambda$

Set $S_n(p)$ random variable which follows the binomial distribution of parameters $n$ and $p$, that is $$P(S_n(p) =k) =\begin{pmatrix} n\\k\end{pmatrix}p^k(1-p)^k.$$ It is clear that $n-S_n(1-p)$ ...
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### Uniform random triangles in the square.

Select $3$ points in the unit square $[0,1]^2$ randomly using the uniform distribution. That gives you a triangle. What is the expected distribution of edge lengths and internal angles? You could ask ...
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1 vote
### A general moment made up from a pair of independent Gaussian random variables $A$ and $B$ with identical variance factorizes?
Let $A$ and $B$ be real-valued, centered and independent Gaussian random variables such that \begin{equation} Cov(A,A)=Cov(B,B):=C \end{equation} Since they are independent, it is clear that \begin{...