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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5
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2answers
478 views

Convergence of a sequence involving the maximum of i.i.d. Gaussian random variables

It's well known that, for a sequence of $n$ i.i.d. standard Gaussian random variables $X_1,\ldots,X_n$, where $X_\max=\max(X_1,\ldots,X_n)$, the following convergence result holds: ...
0
votes
1answer
53 views

sum of independent random variables where $N$ is a random variable

I want to show $E[S_N]=E[N]E[X_j]$ where: $X_1,X_2,\ldots$ is a sequence of independent random variables, and $N$ is a random variable independent of the sequence. $S_n=\sum_{i=1}^n X_i$, ...
1
vote
1answer
45 views

Probability: Random Variables

Let's $T_1$ be a random variable with pdf: $$f(t) = \frac{6+2t}{7}$$ and $T_2 \sim Exp(\frac{1}{3})$ Knowing that $T_1$ and $T_2$ are independent calculate $$P(T_1 + T_2 > 1) $$ During my ...
1
vote
0answers
37 views

Independent transformation of probability measures

I have a pair of dependent random variable $(\theta, X)$ where $\theta\in K$ for a compact subset $K\subset\mathbb{R}$ and $X\in\mathbb{R}^d$ with marginals $P_{\theta}$ and $P_X$. I want to ...
1
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0answers
112 views

Relatively compact sequence of Borel measures which is not asymptotically tight

I am trying to find a sequence of Borel measures on a metric space that is relatively compact but not asymptotically tight. Asymptotic tightness of a net (or sequence) of functions is defined in van ...
0
votes
1answer
44 views

Urn probability function

Suppose I have an urn with an infinite number of balls which can be either red or white. I do not know what the proportion of each colour is, but I do know it's a fixed proportion. After drawing $N$ ...
3
votes
4answers
714 views

Probability Problem on Divisibility of Sum by 3

From the 3-element subsets of $\{1, 2, 3, \ldots , 100\}$ (the set of the first 100 positive integers), a subset $(x, y, z)$ is picked randomly. What is the probability that $x + y + z$ is divisible ...
0
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0answers
38 views

Limits of expectation of a function

Suppose that $g(x,y)$ is a smooth function with respect to $x$ and $y$, and that it is bounded on the domain of interest: $a\leq g(x,y)\leq b$, with $a$ and $b$ being real constants. Now let $(X)_n$ ...
0
votes
1answer
62 views

What conditions are needed for a uniform bound on the deviation of a random variable from its expectation?

The title is quite clear, but a few specifics may help guide where I'm coming from. I have a probability space defined by a Poisson distribution over the non-negative integers with some mean $\mu$. I ...
1
vote
1answer
350 views

$X$ and $Y$ i.i.d., $X+Y$ and $X-Y$ independent, $\mathbb{E}(X)=0 $and $\mathbb{E}(X^2)=1$. Show $X \sim N(0,1)$

$X$ and $Y$ are independent and identically distribued (i.i.d.), $X+Y$ and $X-Y$ are independent, $\mathbb{E}(X)=0$ and $\mathbb{E}(X^2)=1$. Show that $X\sim N(0,1)$. We should use characteristic ...
3
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1answer
44 views

When does $\mathbb E_\mathbb P[X]=0$ imply $\mathbb E_\mathbb P[X\mid\mathcal E]= 0$ $\mathbb P$-a.s.

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, $\mathcal E\subseteq\mathcal F$ a sub-$\sigma$-algebra, $X:\Omega\rightarrow\mathbb R^d$ a $\mathcal F$-measurable map. My question is, when ...
1
vote
1answer
99 views

Betting strategy for coin flip

I am offered a game where there is a unfair coin $p > 0.5$ that heads comes up. I start with 1 dollar and I can bet fractional amounts. Payouts are 1 to 1. What is my optimal betting strategy if I ...
1
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1answer
113 views

Series of independent random variables are independent

In the proof of a theorem my lecturer seemed to have used this fact without first proving it: Let $(X_i)_{i \geq 1}$ be real-valued independent random variables on $(\Omega,\mathscr{F},\mathbb{P})$, ...
0
votes
1answer
46 views

Find PDF of $Z = X + Y$ where $X$ and $Y$ are jointly continuous random variables.

I want to check my solution. Let $X$ and $Y$ be jointly continuous random variables and $Z = X + Y$. For some $z$, $X = z - Y$. Thus $F_Z(z) = Pr(Z < z) = Pr(X < z - Y)= ...
0
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0answers
179 views

To find E(X|Y=y)?

I am having difficulty to solve the following problem to compute E(X|Y=y) Question: Let (X,Y) have joint mass function $$ \begin{align} p(k,n)= \frac{C.2^{-k}}{n} \\ \end{align} $$ for k=1,2,... ...
2
votes
1answer
134 views

Convergence of sum of independent gaussian random variables N(0,1/2^i)

Given $Z_i$ are independent gaussian random variables with $Z_i $~$N(0,\frac{1}{2^i})$, does $X_n=\sum_{j=1}^{n}Z_j$ converge in mean square sense. I know that it converges in distribution to ...
0
votes
1answer
62 views

Simple probability question with normal distributions?

If $A,B$ are random variables that are normally distributed around zero, what is $P(A+B \ge 0)$? My thought is $P(A+B \ge 0) = P(A \ge -b | b \ge 0) P(b \ge 0)+P(A \ge -b | b \le 0) P(b \le 0)$. Let ...
1
vote
1answer
49 views

Can someone explain the relation between random variables and continuity?

For example, if we look at distributions with $P(T<5)$ and $P(T\le5)$, the two notations equal due to continuity. I don't understand how continuity is related to the reason these two notations are ...
0
votes
2answers
63 views

Expectation of $(X|Z)$ where $Z=X+Y$

I need help to solve the following problem. Assume that $X$ and $Y$ have geometric distribution with parameter $p$. Compute $E(X|X+Y=k)$ for all $k = 2,3,4,...$ This is how i attempted so far. Let ...
3
votes
1answer
143 views

Markov Chain with 7 sided polygon and 2 balls

7 men stand at the vertices of a 7-sided polygon and play a game with 2 balls. Initially (at time 0), the men at vertices A and D hold a ball each. The game ends when any man receives the two balls ...
1
vote
1answer
190 views

Is showing almost surely convergence equivalent to lim sup = lim inf on a set with probability 1?

I know there are a lot of questions and answers concerning a.s. convergence on StackExchange, but I didn't find any addressing this in particular. What I am wondering is if you are given a problem of ...
2
votes
0answers
156 views

Explanation of Radon-Nikodym derivates wrt to probabilities

I am currently working in communications, where a lot of work is done via probability calculations (densities and such). As I am not a mathematician, I do have a quite hard time understanding one ...
2
votes
1answer
81 views

How to model a stochastic process, continuous in stepsize, which converges against a simple random walk?

I want to compute the probability distribution for a stochastic process with discrete number of steps, where each real value has a nonvanishing probability to be the next stepsize. And I want to ...
1
vote
1answer
98 views

Stochastic differential for general semimartingale

By using the canonical representation of a semimartingale in Eberlein, Glau and Papapantoleon: "$H = B + H^c + h(x) \ast (\mu − \nu) + (x − h(x)) \ast μ$ where $h = h(x)$ is a truncation ...
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0answers
85 views

If the prior distribution of $\lambda$ is a gamma distribution with mean 1.4 and std dev of .5 find the appropriate values and $\alpha$ and $\beta$

If the prior distribution of $\lambda$ is a gamma distribution with mean of 1.4 and standard deviation of .5 with the form $\pi(\lambda | \alpha, \beta)=\frac{\beta^{\alpha}}{\Gamma(\alpha)} ...
1
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1answer
61 views

A measure having no point masses.

What does it mean for a measure to have no point masses? Is this sort if like saying that individual points have measure zero?
0
votes
2answers
406 views

Compute the probability mass function of $X -Y$

I'd be very thankful if someone help me guide how to solve the following problem. Let $X$ and $Y$ be independent and geometrically distributed with the same parameter $p$. 1) Compute the ...
0
votes
1answer
288 views

The meaning of almost surely convergence

Consider the space of sequences of tosses of a fair coin. In particular let $X_{nH}$ be the number of times that a coin lands on heads in a sequence of $n$ coin flips. Consider statement $S$ below. ...
0
votes
1answer
84 views

Is monotonicity condition not required in this short derivation?

For given density functions $p_1(x)$ and $p_0(x)$ ($x\in\mathbb{R}$) the following equation is to be satisfied: $$(1-\epsilon_1)\{P_1[p_1/p_0>c] +cP_0[p_1/p_0\leq c]\}=1$$ where $c\in\mathbb{R}^+$ ...
1
vote
1answer
105 views

Conditional expectation w.r.t. discrete measure

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, $\mathcal E\subseteq\mathcal F$ a sub-$\sigma$-algebra. Assume that $$\mathbb P=\sum_{i=1}^\infty a_i\delta_{\omega_i}$$ where $a_i\in ...
0
votes
1answer
431 views

Find Bayes estimator of $\theta$

I've got this exercise, which I'm trying to work off using an example, but the example seems very different so I'm not sure if what I'm really doing. I've got a loss distribution for $\theta$: ...
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2answers
200 views

Calculating integral with standard normal distribution.

I have a problem to solving this, Because I think that for solving this problem, I need to calculate cdf of standard normal distribution and plug Y value and calculate. However, at the bottom I ...
1
vote
1answer
552 views

How to check distributional equality of conditional expectations

Let's say $X$ is a bounded real-valued RV, and that $Y, Z$ are RVs with values in the same, but arbitrary measure space. (Take for example $\mathbb{R}^\mathbb{N}$ if specificity is required to make ...
2
votes
2answers
86 views

Does the weak law of large numbers imply that the ratio $\frac{\frac{1}{n}\sum_{i=1}^nX_i}{\mu}\rightarrow 1$ in probability?

Suppose there is a sequence of $n$ bounded i.i.d. random variables $X_1,\ldots,X_n$, i.e. for all $i$, $a<X_i<b$ with $a$ and $b$ real constants. Denote the mean of these random variables by ...
0
votes
1answer
32 views

Even numbered moments of N(0,1) using characteristic functions

Let $X$ be $N(0,1)$. Show that $E\{X^{2n+1}\}=0$ (Easy - calculate it directly using the definition of expectation, and you're taking the integral of an odd function over a symmetric interval, so =0), ...
1
vote
1answer
107 views

Characteristic function of an r.v. with finite variance and zero mean.

Suppose $E{|X|^{2}}<\infty$ and $E{X}=0$. Show that Var(X)$= \sigma^{2}<\infty$ (done), and that $\varphi_{X}(u)=1-\frac{1}{2}u^{2}\sigma^{2}+o(u^{2})$ (what I can't figure out how to find, ...
1
vote
1answer
233 views

Probability Theory: On weak convergence of distribution functions.

On the top of page 97 in Durrett's text on Probability, it says that a sequence of distribution functions $F_n$ converges weakly to some limit function $F$ if it converges to $F$ for all continuity ...
1
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2answers
53 views

$E[W_{t/3}W_{t/2} \mid \mathcal{F}_{t/5}]$ where W is a Brownian Motion and $\mathcal{F}$ is the natural filtration?

I am unsure how to go about finding this value. $\mathrm{E}[W(t/3)*W(t/2)|$ $\mathrm{F}(t/5)]$ I assume the trick invovles an additional conditional expectation, but I am not sure how to go about ...
0
votes
1answer
45 views

How is this cumulative distribution formed?

I have a probability density function that equals $f(x)=\begin{cases}.1\quad \text{for $0\le x\lt 2$}\\.2\quad \text{for $2\le x \lt 4.5$}\\.3\quad \text{for $4.5\le x\lt5.5$}\\ 0\quad ...
0
votes
3answers
72 views

Prove not mutually independent

We are throwing two dice. Let $A$ be the event that the first die is odd. $B$ be the event that second die is odd. $C$ be the event that sum of outcomes of two dice is odd. Show that those three ...
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0answers
83 views

Can posterior distribution for a continuous variable be greater than one?

This might sound a dumb question but I am really confused about it. According to Bayes' rule we do have the following: $$p(\theta|X)=\frac{p(\theta)p(X|\theta)}{\int{p(\theta)p(X|\theta)d\theta}}$$ I ...
0
votes
2answers
428 views

Product of two random variables

How can one show that the product $X \cdot Y$ of two real-valued random variables $X,Y$ is again a random variable? We can fix some set generating the Borel sigma algebra on the real line, then take ...
0
votes
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 ...
0
votes
1answer
47 views

When does “positive expected value” imply “positive conditional expectation with positive probability”?

Let $(\Omega,\mathcal G,\mathbb P)$ be a probability space, $\mathcal F\subseteq\mathcal G$ a sub-$\sigma$-algebra and $X:\Omega\rightarrow\mathbb R$ $\mathcal G$-measurable map. Assume that $\mathbb ...
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vote
0answers
151 views

Maximum of a sequence of almost-identical independent normal random variables

Take a sequence $X_1,\ldots,X_n$ where each $X_i\sim\mathcal{N}(\mu,\sigma^2)$ is an i.i.d. normal random variable. Denote by $X_\max$ the maximum of this sequence. A well-known fact about ...
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4answers
186 views

Showing the expectation of the third moment of a sum = the sum of the expectation of the third moment

Let $X_{1},\cdots,X_{n}$ be independent, each with mean 0, and each with finite third moments. Show that $E\left\{\left( \sum_{i=1}^{n}X_{i}\right)^{3}\right\} = \sum_{i=1}^{n}E\left\{ X_{i}^{3} ...
1
vote
4answers
116 views

What is an example for an algorithm which makes use the power of randomness?

Can someone give a (most simple) example for an algorithm on a machine, which has access to random numbers, and which is faster than any other known algorithm for the same task? My actual motivation ...
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vote
0answers
438 views

Characteristic Function of a Double Exponential (Laplace) Distribution

Let X have the double exponential (or Laplace) distribution with $\alpha =0$, $\beta = 1$: $f_{X}(x)=\frac{1}{2}e^{-|x|}$, $-\infty < x < \infty$. Show that $\varphi _{X}(u)=\frac{1}{1+u^{2}}$. ...
0
votes
1answer
121 views

Find joint probability P(X=0, Y=0)

I have this problem where I'm not too sure on how to proceed. I need to calculate $Pr(X=0 $ and $ Y=0)$ using the following information: The conditional distributions $f(x|\theta)$ and ...
1
vote
0answers
97 views

what is relative entropy between to random binary string with length of $L_1$ & $L_2$?

I want calculate relative entropy between two strings of binary such as: $L_1:11000100011101001$ $L_2:00101110110111001$ It is primarily when the lengths of two strings is same and in general when ...