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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2
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0answers
35 views

Need help showing $\mathbb E(5^x) = (1+4p)^n$

I have no idea where to even begin on this problem. The problem reads: Suppose $X$ is a binomial random variable with parameters $n$ and $p$. Show with a simple calculation (using the binomial ...
1
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1answer
57 views

Given two identically distributed random variables, must the probability that the first is larger be 1/2?

Let $(X_1 , X_2)$ be two real valued random variables with the same distribution, defined on the same probability space but not necessarily independent. Suppose that $X_1 \not=X_2$ almost surely. Must ...
1
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1answer
346 views

probability of arranging books on shelf

You have $n$ books on algebra, $k$ on probability and $l$ on calculus. The books are all different. If you place them on a shelf at random what is the probability that (a) Books on the same subject ...
2
votes
2answers
98 views

Strange deduction about relation of median and mean

On his blog T. Tao's proves the following concentration inequality, due to Talagrand. Let $K>0$, and let $X_{1},..., X_{n}$ be iid complex random variables all bounded by $K$. Let ...
1
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1answer
242 views

Distance between mean and median

I want to solve the following problem in T.Tao's random matrix theory book. Let $X$ be a random variable with finite second momment. A median $M(X)$ of $X$ saisfies ...
1
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1answer
207 views

$\max, \min$ relation for probability LP

Suppose we have 2 LPs; $$\text{maximize } c^T\mathbf{x}$$$$\text{subject to} \,A \mathbf{x}\geq0 $$ $$\sum \mathbf{x}=1$$ $$\mathbf{x}\geq 0$$ and the other is, $$\text{minimize } ...
2
votes
1answer
91 views

Creating a new random variable with the same distribution

Let $X,Y$ be two random variables taking real values. Suppose that $P(Y\leq t) \leq P(X\leq t)$ for any $t\in{\mathbb R}$. Is there always a random variable $Y^{*}$ with the same distribution as $Y$, ...
2
votes
1answer
98 views

bound on $P(X\geq Y)$ where $X$ and $Y$ are Poisson random variables

Let $X$ have Poisson$(\lambda)$ distribution and let $Y$ have Poisson$(2\lambda)$ distribution. Find constants $A<\infty, c>0$, not depending on $\lambda$, such that, without assuming ...
1
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1answer
96 views

Help with monotone convergence theorem

Prove that if $E|X_1| < \infty$ and $X_n\uparrow X$ a.s. then either $EX_n\uparrow EX<\infty$ or else $EX_n\uparrow \infty$ and $E|X|=\infty.$ I just don't see how to use the fact that $E|X_1| ...
0
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2answers
63 views

cookie among kids probability

I have 15 identical cookies and 4 friends. I am going to give cookies to my friends. In how many way can I do this if (a) I give all the cookies away and there are no other constraints on how many ...
1
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1answer
173 views

probability around a round table , PMF

A round table has n seats. n people are seated at random around the table. Fred dislikes two of the people. Let X be the number of neighbors of Fred whom he dislikes. Find the p.m.f. of X. (Note that ...
0
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1answer
71 views

$\lim_{n\rightarrow\infty}X1_{\{X>n\}}=0 \space\space\space\text{a.s.}$

$$\lim_{n\rightarrow\infty}X1_{\{X>n\}}=0 \space\space\space\text{a.s.}$$ This is a claim I need to use in part of a proof that involves the dominated convergence theorem. Since it holds almost ...
0
votes
1answer
44 views

Can a stationary distribution be zero vector

Suppose I have probabilities matrix between 3 states, for exampel we can take $P=\left(\begin{array}{ccc} \frac{1}{9} & \frac{8}{9} & 0\\ 0 & 0.3 & 0.7\\ 0 & 0 & 1 ...
0
votes
1answer
176 views

probability of arrangement of books on a shelf

Qn: You have s books on algebra, f on probability and v on calculus. The books are all different. If you place them on a shelf at random what is the probability that Books on the same subject are ...
0
votes
2answers
86 views

Expectation of nonnegative RV

I'm taking an intro course in Probability theory, and we have just defined expectation for a random variable as $E(X) = E(X^+) - E(X^-)$ if either of them is finite (extending the definition first ...
0
votes
1answer
392 views

Expected value of Roulette Game

The questions reads: There are 37 numbers starting from 0 to 36. Each number has a equal chance of turning up. Zero is green in color and odd numbers are in black and even numbers are in red. If you ...
1
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1answer
67 views

question on Kolmogorov's extension theorem

I'm reading Bernt Oksendal's "Stochasticc Differential Equations" and got confused at the Kolmogorov's extension theorem. In chapter 2, page 11 (sixth edition) it says: Theorem 2.1.5 (Kolmogorov's ...
9
votes
3answers
767 views

What is the motivation of Measure Theory when there is probability theory?

In my undergraduate studies, when probability was taught to me, it was taught to me starting from Probability Theory. However, when I go onto higher level of studies, probability gets taught using ...
1
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1answer
25 views

Prove that $\liminf_{n\to\infty}A_n\in\mathcal A$ and $\limsup_{n\to\infty}A_n\in\mathcal A$

Let $\mathcal A$ be a $\sigma$-a;gebra. Prove that if, for all $n \in \mathcal N, A_n \in \mathcal A,$ then $\liminf_{n\to\infty}A_n\in\mathcal A$ and $\limsup_{n\to\infty}A_n\in\mathcal A$
3
votes
1answer
26 views

$\displaystyle \cap_{\alpha\in A}\mathcal G_{\alpha}$ is a $\sigma$-algebra

Let $(\mathcal G_{\alpha})_{\alpha \in A}$ be an arbitrary family of $\sigma$-algebra defined on an abstract space $\Omega$. Show that $\displaystyle \cap_{\alpha\in A}\mathcal G_{\alpha}$ is also a ...
0
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1answer
86 views

show that for Pareto distribution mean deviation about mean cannot exceed the standard deviation

show that for Pareto distribution mean deviation about mean cannot exceed the standard deviation
5
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2answers
66 views

$X-Y$ equivalent in distribution to $0$?

If $X$ is equal to $Y$ in distribution, is it equivalent to $X-Y$ which is equivalent in distribution to $0$?
0
votes
1answer
133 views

Max function with probabilities

I have the following: $$p(Y<y) = p(\max(x_1, x_2, \ldots, x_t) < y)$$ Where $x_1, x_2, \ldots, x_t$ are independent(they come from a sample) why the following is true? $$p(\max(x_1, x_2, ...
2
votes
1answer
1k views

Does adding or/and dividing a random variable by a constant change its probability distribution?

Suppose that we have a random variable X with probability distribution $PDF_X(\mu_x,\sigma_x)$ Consider random variable $Y=\frac {X-a}b$ , I know that mean and variance of Y would be: $$\mu_y=\frac ...
4
votes
2answers
100 views

Prove Poissons' theorem

Let $(\Omega, \Sigma, \mathbb{P})$ be a probability space. If $A_1, A_2, \ldots$ are independent events and $\bar{p}_n$ and $N$ are defined as $$ \bar{p}_n=\frac{1}{n}\sum_{i=1}^n\mathbb{P}(A_i) \quad ...
0
votes
1answer
82 views

exit time and indicator function

let $D$ open set of $\mathbb{R}^{n}$ and $T_{D}=\inf\{t\geq 0 : X_{t}\notin D\} $ be the first exit time from the $D$ and $1_{A}$ is Indicator function of $A \subseteq \partial D$ $$ ...
4
votes
1answer
107 views

What is the right invariant $\sigma$-algebra for the Birkhoff ergodic theorem?

I have been reading stuff about ergodic theory, and I have encountered two versions of the involved "invariant sigma field". let the underlying probability space be $(\Omega,\mathcal{F},P)$, and let's ...
1
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1answer
62 views

Approximate the probability of choosing 4 of the most qualified people in a group of 15, in a committee of 10.

From a group of 15 mathematics graduate school applicants, 10 are selected at random. Let $P$ be the probability that 4 out of the 5 best applicants are included in the 10 selected. Which of the ...
0
votes
1answer
66 views

find the probability of at least two tails in $3T$ tosses, where $T$ is the expected number of tosses required to get the first Tails.

A biased coin is tossed repeatedly until the first "tail" occurs. The expected number of tosses required to produce the first tail is estimated as $T$. Assuming this is true, find the probability of ...
2
votes
1answer
100 views

A stochastic step function?

Is there a way to have a step function the heights and widths of whose constant portions are random? Said another way, can I define a "random" step function $f$ on $[0,1]$ using some parameters so ...
3
votes
0answers
140 views

Potentials in Probability Theory

Could someone give an intuitive interpretation of potentials in the field of probability theory. How do they link to the theory of stochastic processes. And maybe link this with SEP. References are ...
2
votes
3answers
1k views

10 children divide into 2 teams each. How many divisions possible?

In order to play a game of basketball, $10$ children at a playground divide themselves into $2$ teams of $5$ each $2$ teams of $x_1$ and $x_2$ How many different divisions are possible? This is ...
2
votes
1answer
219 views

Weak convergence of measures on a discrete probability space

What follows are two different question on weak convergence on a discretization of a probability space and applying some standard probability theory on such a discretization. Let ...
1
vote
1answer
260 views

Finding Probability of P(|X-Y| ≤ 0.5)?

The joint density of X and Y is given by f(x,y) = (x + y); 0 < x,y < 1 = 0; otherwise Find ...
4
votes
2answers
187 views

conditional probability question from sheldon ross

In any given year a male automobile policyholder will make a claim with probability $p_{m}$, and a female policyholder will make a claim with probability $p_{f}$, where $p_{f} \neq p_{m}$. The ...
1
vote
1answer
134 views

tightness under a certain uniform integrability condition

Let $\Omega$ be a subset of $\mathbb{R}_+^d$. $X$ should be the canonical process, i.e. $X_k(\omega)=\omega_k$, where $k\in \{1,\dots,d\}$ and $\omega_k$ is just the $k$-th component of ...
1
vote
1answer
62 views

Value of a and b in f(x) = a sin bx in order to define density function

I have a density function f(x) = a sin(bx) when 0<=x<=1 . I have to define the values of a and b so that this becomes a density function . How can I find the values from this function ? Is ...
2
votes
1answer
790 views

Probability of finding P(X=k)?

A factory produces 10 glass containers daily. It may be assumed that there is a constant probability p=0.1 of producing a defective container. Before these containers are stored they are inspected and ...
0
votes
1answer
73 views

If $E[X|\mathscr{F}] = E[Y|\mathscr{G}]$ then$E[Y|\mathscr{G}] = E[X|\mathscr{F}]$

Suppose we show that $E[X|\mathscr{F}] = E[Y|\mathscr{G}]$, that is that for any $Z$ such that $Z$ is a version of $E[X|\mathscr{F}]$, $Z$ is a version of $E[Y|\mathscr{G}]$. Does it follow that for ...
1
vote
1answer
2k views

Probability density of Continuous uniform distribution over the unit circle

If we want to chose a point $(x,y)$ uniformly at random from a unit circle in a plane, why is the joint probability density of the random variable $f(x,y) = \frac{1}{\pi}$ for $x^2+y^2\leq1$? The ...
5
votes
1answer
266 views

Girsanov: Change of drift, that depends on the process

Known: If I am looking at an SDE like: $dX_t = b(t,\omega) dt + dW_t$ with $W_t$ a Brownian motion under a measure $P$. I know that I can change the drift by using Girsanov to $dX_t = ...
2
votes
2answers
110 views

Expression for Expectation of expectation

Suppose we have two random variables: $X$ is a continuous r.v.; $Y$ is a discrete r.v. taking values $0$ and $1$. Is the following expression true ? $E[(E[X|Y])^{2}]= [(E[X|Y=1])^{2}]\times P(Y=1)+ ...
2
votes
1answer
19 views

A question about distributions/densities

Given two random variables $X,Y$ how to show that $P(X\leq Y+x)=\int F_X(y+x)f_Y(y)dy$? I know that $f_Y(y) = \int f_{XY}(x,y)dx$, but have no idea how to go with the previous equation.
4
votes
1answer
132 views

Kolmogorov continuity theorem for Banach space valued random processes

I am interested in the Kolmogorov continuity theorem. I would like to know if this theorem holds for Banach space valued random processes (probably separable Banach space). I cannot find a paper or a ...
2
votes
2answers
101 views

Why is this standard deviation $20$?

If I have two random variables $X_1$ and $X_2$ with $X_1\sim N(520,10)$ and $X_2\sim N(500,10)$, and $X_1$, $X_2$ are both speeds of airplanes where the first one is 10 km ahead of the second one. I'm ...
3
votes
2answers
85 views

Separability almost everywhere

Let $(X,d)$ be a metric space and $\mu$ a Borel probability measure. Suppose that for every $\epsilon>0$ we have that $\mu(B_{\epsilon }(x))=c_{\epsilon}>0$ a.e. Is this enough to show that ...
0
votes
2answers
429 views

How to convert a histogram to a PDF

I know this may be an easy question, but due to lack of math knowledge I do not know the answer. Would you please explain to me with a simple example that how can I find PDF from a histogram. Thank ...
3
votes
1answer
40 views

Independent sets in subfield

Let $X$ be a set, $\mathcal F$ a $\sigma$-field of subsets of $X$, and $\mu$ a probability measure on $\mathcal F$. Suppose that $A_1,\ldots,A_n$ are independent sets belonging to $\mathcal F$. Let ...
3
votes
2answers
33 views

Expectation of the squared error with regards to a sub sigma field

I am totally stuck. Given a probability space $(\Omega, \mathcal F, \mathbf P)$ and a random variable $X$. Let $\mathcal A$ be a sub-$\sigma$-field of $\mathcal F$. Let $Y$ run over all $\mathcal ...
4
votes
3answers
1k views

Showing that ${\rm E}[X]=\sum_{k=0}^\infty P(X>k)$ for a discrete random variable

Let $X$ be a discrete random variable whose range is $0,1,2,3,\ldots$. Prove that $$ {\rm E}[X]=\sum_{k=0}^\infty P(X>k). $$ How to prove this? I tried a bit but unable to post due to formatting ...