Articles of probability

Show $\psi$ and $\Delta$ are identifiable

Let $X_1$,…,$X_m$ be i.i.d. F, $Y_1$,…,$Y_n$ be i.i.d. G, where model {(F,G)} is described by $\hspace{20mm}$ $\psi$($X_1$) = $Z_1$, $\psi$($Y_1$)=$Z’_1$ + $\Delta$, where $\psi$ is an unknown strictly increasing differentiable map from R to R, $\psi$’ > 0, $\psi$($\pm$$\infty$) = $\pm$$\infty$ and $Z_1$ and $Z’_1$ are independent r.v.’s. (a) Suppose $Z_1$, $Z’_1$ have a $\mathcal{N}$(0,1) […]

Random walk problem in the plane

Let a particle in the plane $R^2$ executes random jumps at discrete times $t= 1, 2, …$. At each step, the particle jumps from the point it is a distance of lenght one. The angle of any new jump (say, with the $x$ axis) is uniformly distributed in $[0,2\pi]$. Question: If initially ($t=0$) the particle […]

$W_n = \frac{1}{n}\sum\log(X_i) – \log(X_{(1)})$ with Delta method

Note: $\log = \ln$. Suppose $X_1, \dots, X_n \sim \text{Pareto}(\alpha, \beta)$ with $n > \dfrac{2}{\beta}$ are independent. The Pareto$(\alpha, \beta)$ pdf is $$f(x) = \beta\alpha^{\beta}x^{-(\beta +1)}I(x > \alpha)\text{, } \alpha, \beta > 0\text{.}$$ Define $W_n = \dfrac{1}{n}\sum\log(X_i) – \log(X_{(1)})$, with $X_{(1)}$ being the first order statistic. I wish to show $$\sqrt{n}(W_{n}^{-1}-\beta)\overset{d}{\to}\mathcal{N}(0, v^2)$$ as $n \to […]

Prove that there are two frogs in one square.

A certain chessboard is infinite in size. There is a frog sitting in the center of every square. After a certain time, all the frogs jump such that They may jump to any possible square in the infinite chessboard They may jump and land at the same square again Prove that its possible for all […]

Equivalence of measures and $L^1$ functions

Suppose we have two probability measures $\mu$ and $\delta$ on $(X, \mathcal{B})$ such that $ \delta <<\mu << \delta $. How can I prove that $f \in L^1(X,\mathcal{B}, \mu)$ iff $f \in L^1(X,\mathcal{B}, \delta)$? My idea was to use that the Radon Nikodym theorem. So we know there exist $g$ $\mu$-measurable and $g^{-1}$ $\delta$-measurable such […]

Choosing randomly integers from $1$ to $10$

From Question 5 the practice book of the GRE math subject test: Sofia and Tess will each randomly choose one of the $10$ integers from $1$ to $10$. What is the probability that neither integer chosen will be the square of the other? Choices: (A) $0.64$, (B) $0.72$, (C) $0.81$, (D) $0.90$, (E) $0.95$ The […]

Computing the variance of hypergeometric distribution using indicator functions

I want to compute the variance of a random variable $X$ which has hypergeometric distribution $\mathrm{Hyp}(n,r,b)$, where $n$ is the total number of balls in the urn and $r$ and $b$ are the numbers of red/black balls, by using the representation $$X= I_{A_1} + \cdots + I_{A_n}$$ ($I_A$ is the indicator function of $A$ and […]

A problem of regular distribution

This is a exercise of Shiryaev’s Probability on page 233: Suppose that the random elements $(X, Y)$ are such that there is a regular distribution $P_x(B)=P(Y\in B\mid X=x)$. Show that if $E|g(X,Y)|<\infty$ then $$E[g(X,Y)\mid X=x]=\int g(x,y)P_x(dy) \text{ ($P_x$-a.s.)}$$ PS: I think there is a typo. Should we replace “$P_x$-a.s.” by “$P_X$-a.s.”? I firstly tried to […]

Distribution of the lifetime of a system consisting of two exponentially distributed components, one being backup

I have a system consisting of components $S_1$ and $S_2$ whose lifetimes $T_1$ and $T_2$ follow the exponential distribution with parameter $\lambda$. At time $t=0$ the component $S_1$ is switched on and $S_2$ is kept off until $S_1$ fails (and is immediately switched on). What is the distribution of the lifetime of the system? To […]

Probability distribution is symmetric at a point, distribution function $F(x) + F(-x) =1$

Show that if the distribution $P_X$ is symmetric at $m \in \mathbb{R}$, and there are no discontinuities of the distribution function, then the distribution function $F_X$ satisfies $F_X(t) + F_X(−t) = 1, t \in \mathbb{R}$ Could you help me with that? I can show this, straight from the equality: $F_X(t) = P(X \in (- \infty, […]