Use this tag only if your question is about the modern theoretical footing for probability, for example probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use ...

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17 views

Cumulative beta density calculation. [duplicate]

The beta distribution of y, w.r.t $\alpha,\beta, min - a \text{ and } max - c $ is. $$f(y; \alpha, \beta, a, c) = \frac{ (y-a)^{\alpha-1} (c-y)^{\beta-1} }{(c-a)^{\alpha+\beta-1}B(\alpha, \beta)}$$ ...
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22 views

Martingale with bounded increments converges or diverges to $\pm \infty$ [duplicate]

Let $(M_n)$ be a martingale with $|M_n - M_{n-1}| \leq c$ for some fixed $c < \infty$. Check that the two disjoint events $$C:=\{M_n \text{ converges to a finite limit}\}, \; F:=\{\limsup M_n ...
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19 views

Probability calculation when the R.V. are the ratio of variables i.i.d. and exponentially distributed

Suppose to have two R.V. i.i.d., i.e. $Z$ and $T$, given by \begin{equation} Z = \frac{X}{1+Y}, \end{equation} and \begin{equation} T = \frac{U}{1+V} \end{equation} I have to calculate the ...
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1answer
38 views

Exercise about conditional expectation

I have to show that for sigma-algebras $\mathcal{F, G}$ with $\mathcal{F}\subseteq \mathcal{G}$ and $X, Y$ real random variables with $\Bbb E[X^2] < \infty$ the following holds: ...
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18 views

Show if $p^n(i,j)\rightarrow \pi(j)$ as $n\rightarrow\infty$ then $\pi(j)$ is a stationary measure..

Suppose $p(i,j)$ is a transition kernel on $S$ for a countable state markov chain $X_n$ with $$p^n(i,j)\rightarrow \pi(j)$$ as $n\rightarrow\infty$ for all $i,j\in S$. want to verify that $\pi$ is a ...
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1answer
31 views

Various modes of convergence of random variables

Let $\lbrace X_n \rbrace_n$ be a sequence of independent random variables such that $$P(\{X_n = \pm 1 \}) = \frac{1}{n}$$ $$P(\{X_n = 0 \}) = 1 - \frac{2}{n}$$ Is the sequence convergent: $1$) almost ...
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25 views

Have I used the Probability generating function of poisson point process correctly?

Let $v\in \mathcal{V}$ be measurable and let $\Phi$ be a Poisson Point Process with intensity $\lambda$ then the probability generating function (PGF) is $$\mathbb{E}\left( \prod_{x\in \Phi} ...
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2answers
55 views

Show that $V=\frac{Z_1}{\sqrt{(Z^2_1 + Z^2_2)/2}}$ has pdf $f(v) = 1 / (\pi \sqrt{2-v^2}),-\sqrt2<v<\sqrt2$

Let $Z_1, Z_2$ have independent standard normal distributions, $N(0,1)$. If the random variable in the numerator did not also appear in the denominator this would be a t distribution. Should ...
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51 views

Events $A_n\uparrow A$ meaning. $A_n\downarrow A$ meaning.

i simply do not understand the arrows in this context! :\
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134 views

Expectation of a discrete random variable: how to convert an integral to a sum?

According to wikipedia and all my textbooks, we define the expectation of a random variable on a probability space $(\Omega, \mathcal{F},P)$ : \begin{align} E(X) &= \int_{\Omega}XdP\\ \end{align} ...
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24 views

On why a limit of random variables implies $Z_k(w)\leq z+\frac{1}{m}$.

If $Z_1,Z_2,\ldots$ are random variables such that $\lim_{n \to \infty}Z_n(w)$ exist for all $w$ and $$Z(w)=\lim_{n \to \infty}Z_n(w)$$ and suppose $w\in\{Z^{-1 }\mid Z\leq z\}$, for $z \in \mathbb ...
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1answer
37 views

Does convergence in distribution of discrete random variables with same finite support imply convergence in probability?

Let ${X_m}\mathop \to \limits^D Y$ and ${\text{supp}}\left( {{X_m}} \right) = {\text{supp}}\left( Y \right) = \left\{ {0,1, \ldots ,n} \right\},\forall m \in \mathbb{N}$. Does ${X_m}\mathop \to ...
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1answer
39 views

Random variable related to binomial

The number of successes $A$ in $n$ independent trials with the probability of a success is $p$ for each trial is binomially-distributed. I am interested in a scenario that adds dependence to the ...
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1answer
27 views

A basic question on the existence of expectation?

$E\big[\sqrt(X)\big] <\infty \implies \sqrt(X) <\infty$ a.s $\implies X< \infty$ a.s $ \implies E[X] <\infty$ The expectation is computed wrt to the probability measure . So why the the ...
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2answers
453 views

Central Limit Theorem for exponential distribution

Suppose that $X_1$ ..... $X_n$ are a random sample from a population having an exponential distribution with rate parameter $\lambda$. Use the Central Limit Theorem to show that, for large n, ...
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1answer
32 views

Box-Muller Transform Normality

I'm trying to show that for uniformly distributed variables $X_1$ and $X_2$ that the vector $$ (\sqrt{-2\log X_1} \cos(2\pi X_2), \sqrt{-2 \log X_1} \sin(2\pi X_2)) $$ is two-dimensional normal. ...
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1answer
28 views

Show that if $X(\omega) = \infty$ then $EX = \infty$

I am trying to show that if $X(\omega) = \infty \space\forall \omega \in A$, $P(A) > 0$ and $X \ge 0$ then $EX = \infty$. The problem comes with a hint: $$EX = E\{X[I(A) + I(A^c)]\} = E[XI(A)] ...
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1answer
74 views

The expected revenue problem

Question : A travel agent company organizes a tour with ticket price $\$50$ and the ticket is non-refundable. The company has a bus with $20$ seats. The company knows that the participant might not ...
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2answers
45 views

I need to find the smallest lambda for which $P(X\ge 2)=0.99$ when $X\sim \text{Poisson}(λ)$

What I have tried to do is this: Since $P(X>=2)=0.99$, then $P(X<2)=0.01$, Hence $P(X=0)+P(X=1)=0.01$, so replacing in the pmf, I got $\exp(-λ)+λ\exp(-λ)=0.01$, $g(λ)=\exp(-λ)+λ\exp(-λ)-0.01$. ...
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1answer
108 views

Covariance of states of a finite Markov chain

I know it is possible to construct a covariance matrix for states of a Markov chain but I cannot seem to find a proper way to compute it. I will attach some theories I found from Kemeny and Snell's ...
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1answer
76 views

Almost sure convergence for measurability

So it is pretty obvious that the limit of measurable random variables is also measurable if they converge...but I'm not sure it's true of almost sure convergence. Suppose there is a probability space ...
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1answer
33 views

Identifying random variables with their generated distribution function - Necessity of countable additivity?

Let the state space $\Omega=[0,1]$ and $\lambda$ be the Lebesgue measure defined on the Borel $\sigma$-algebra on $[0,1]$. Consider measurable functions (random variables) $f:\Omega\to\mathbb{R}$ and ...
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44 views

Conditional Expectation: How to get $ \mathbb{E}( X | B ) $ from $ \mathbb{E}( X | A ) $ with $C=A+B$

I'd like to calculate some conditional expected values and I'm facing some problems. Here is what I've got: Known is: $ \mathbb{E}( X | A=a ) $ And I'd like to calculate: $\mathbb{E}( X | B=b )$ ...
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1answer
37 views

monotonicity of binomial coefficient

I am interested in $$f(x):={k-1 \choose x-1} p^{x} (1-p)^{k-x}.$$ How do I find out in which Domain this function is monotonically increasing, in which it is monotonically decreasing? For which $x$ ...
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42 views

What is the distribution of $Y_n$ and its convergency

Let $X_n$ be a iid sequence of Poisson random variables with parameter 1. Define $Y_0 = 1$ and $Y_n := X_nY_{n-1}$ for $n\geq 1$. How to show that $Y_n$ converges to $0$ almost surely, please? I think ...
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37 views

Distribution function inequality for a transformed random variable ?!

I'm stuck with the following problem. Let $A:=\{g: g:(0,1)\to\mathbb{R}^+, non-decreasing\}$ and $U\sim U[0,1]$ be uniformly distributed on the interval $[0,1]$ on the probability space $(\Omega, ...
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1answer
35 views

Solving this random variable problem

This is an earlier problem Proving this random variable problem but generalised, maybe you want to take a look at that one first? $X_1,X_2,X_3,\ldots$ are IID random variable taking values in ...
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1answer
93 views

Finding the MLE for $θ$ given a probability density function $f(y |\theta )$.

Let $Y_1$, $Y_2$, . . . , $Y_n$ denote a random sample from the probability density function $$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& ...
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1answer
99 views

Stopping time question $\sigma$

If $S$ and $T$ are stopping time, $S \vee T$ is $\max ({S,T})$, $F_S$ and $F_T$ are stopped sigma algebra, show that $F_{S \vee T} = \sigma(F_S,F_T)$. My thinking : I should take a set $A$ in $F_{S ...
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1answer
48 views

Quadratic Variation for $X_t= \int b_s ds$ where $b_s$ is an $F_t$ adapted process.

Let $b_S$ be an $F_t$ adapted process, Borel measurable in $t$ st $\int |b_s|^2ds < \infty$ (a.s). Setting $X_t=\int^t_0 b_sds$ and partitioning the interval $[0,t]$ i.e. $0=t^n_0<t^n_1... $ ...
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1answer
34 views

I. i. d distributions as best car offers

I was reading my probability text book and got stuck on what seemed to be a easy question: Let $X_0,X_1,\ldots,$ be i.i.d and imagine they are the offers you get for a car you are going to sell. Let ...
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1answer
82 views

Conditional expectation of a stochastic process in filtered space

It was suggested* to me that if we have a stochastic process with independent increments, and $T > t$, then $$ E(X_{T-t}| \mathcal{F}_t) = X_{T-t} $$ where $\mathcal{F}_t$ is the filtration at time ...
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38 views

Subadditivity of events

I do not get this exercise: Let $A_i \in \mathbb{A}$ be a sequence of events. Show that: $P(\cup^{n}_{i=1} A_i) \leq \sum^n_{i=1} P(A_i)$ The solution is: Set $B_1 = A_1$, $B_i = A_i \setminus \ ( ...
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47 views

The subspace which $X-E(X|\mathcal G)$ is orthogonal to, and the set of r.v.s generating the same $\mathcal G$.

$X$ is a random variable. I wonder if there are some or no relations between the subspace which $X - E(X|\mathcal G)$ is orthogonal to, which is the set of all random variables which are both ...
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2answers
135 views

Formal definition of a random variable

I'm not new to the concept of random variable and I know the measure theory. Anyway, I started reading the book "Stochastic Differential Equation" by B. Oksendal, and I'm having some problem in ...
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1answer
63 views

Probability Analysis of a Randomized Algorithm

A rock band has three sites A, B, and C that it needs to perform at. The band performs at site A, then randomly chooses between B and C as to where it performs next. The band keeps choosing one of the ...
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1answer
141 views

Calculating the probability of winning at least $128$ dollars in a lottery St. Petersburg Paradox

The St. Petersburg Paradox: Here is a lottery: A fair coin is flipped repeatedly until it produces "heads." If the first occurrence of heads is on the $nth$ toss, you are paid $2^{n-1}$. So for ...
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2answers
55 views

Inequality involving expectation and summation of probabilities

Prove that $$\sum_{n=1}^{\infty}P(|X|\geq n)\leq E(|X|)\leq 1+\sum_{n=1}^{\infty}P(|X|\geq n).$$ --I have that $$\sum_{n=1}^{\infty}P(|X|\geq n)=\sum_{n=1}^{\infty}nP(n\leq ...
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1answer
107 views

The probability distribution function of uniform random variables is as given

Given $U_1, U_2, \dots, U_n$ where each $U_i \sim U[0,1]$, then use uniqueness theorem to show probability distribution function of $X = U_1 + U_2 + \ldots +U_n$ (sum of independent uniform random ...
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1answer
66 views

How to multiply a standard normal RV times a uniform{-1.1} RV?

I'm not really sure how to even start this question. I know how to get the pdfs of both, but don't get what it equals when you multiply them together, especially since they have different ranges.
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1answer
414 views

Suppose that Z has a standard normal distribution. Find the density function of $U = Z^2$.

Suppose that Z has a standard normal distribution. Find the density function of $U = Z^2$. First of all, do I need to use probability function for normal distribution to find the cdf for U. If so, ...
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0answers
33 views

If $P_{\theta_0} (X_i \leq x) \leq P_{\theta_1}(X_i \leq x)$, then $P_{\theta_0} (\sum X_i \leq x) \leq P_{\theta_1}(\sum X_i \leq x)$

If $P_{\theta_0} (X_i \leq x) \leq P_{\theta_1}(X_i \leq x)$. Is it true that: $P_{\theta_0} (\sum_{i=1}^nX_i \leq x) \leq P_{\theta_1}(\sum_{i=1}^n X_i \leq x)$ I'm doing a long ...
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2answers
65 views

Show that $\sum_{n=1}^{\infty}\mathbb{P}\left(\frac{1}{n}\lvert X_n\rvert>\varepsilon\right)<\infty$.

Let $(X_n)$ be a sequence of identically distributed integrable random variables. My aim is to show for an arbitrary $\varepsilon>0$, that $$ ...
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1answer
47 views

Convergence in mean for the sequence of positive random variables

This is a follow-up to this question. Now let $(X_n)$ be a sequence of positive random variables. Suppose that the limit of expectation of this sequence ...
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38 views

How to define “compound entropy”

Entropy measures the "surprise" one experiences when uncovering a the actual value of a random variable as $$-\sum_i p_i \log_2 p_i$$ E.g., if we observe Red 8 ...
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1answer
182 views

Finding integration bounds for density of sum of two independent random variables

Let $X, Y$ be independent random variables, both uniformly distributed over the interval $(0,1)$. That is, $$f_{X}(a)=f_{Y}(a) = \begin{cases} 1 & \text{if $0 < a < 1$} \\ 0 & ...
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1answer
30 views

The limit of integer valued random variables must be integer valued?

I saw something like this: If $D_n$ are all integer valued random variables, and $D_n$ converges in distribution to $D$, then $D$ must also be integer valued. I am a little bit suspicious about ...
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1answer
260 views

Sample mean is uniformly integrable

Let $\{X_n\}$ be a sequence of iid random variables such that $E|X_1| = \mu < \infty$. I would like to show that $\{\bar{X}_n\}$ is uniformly integrable. Thoughts By the strong law of large ...
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1answer
1k views

Finding joint cdf and pdf of independent random variables

Let $X$ and $Y$ be independent random variables. Each has an exponential distribution with parameter $\lambda$. Define two new random variables by $W = \min({X,Y}) $ $Z = \max({X,Y})$ Find the ...
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1answer
91 views

A and complement A [closed]

So if A is independent, what can we say about A and complement A, are they independent or not, or undetermined? I think that when both happen then A and complement A intersect would be zero, then ...