Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

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Conditions on Poisson random variables to convergence in probability

Let $X_1,X_2,...$ denote iid random variables such that $X_j$ has a Poisson distribution with mean $\lambda t_j$ where $\lambda$ > 0 and $t_1, t_2,...$are known positive constants. a)Find conditions ...
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2answers
24 views

Die Probability Question + Basics of Conditional Probability

A die is rolled twice. What is the probability of observing: a) a four and a three P (obtaining a four and a three) or P(obtaining a three and a four) therefore P(obtaining a four)* P(obtaining a ...
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1answer
24 views

Absolutely continuous probability measures example

I was given the following definition: Then this example: It is said that $\mathbb P_1\ll\mathbb P_2$ , but I don't really see it.Please help.
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32 views

What should the contestant do? [duplicate]

Suppose there are three curtains. Behind one curtain there is a nice prize while behind the other two there are worthless prizes. A contestant selects one curtain at random, and then one of the other ...
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1answer
21 views

How to make this bet fair?

A person bets $1$ dollar to $b$ dollars that he can draw two cards from an ordinary deck of cards without replacement and that they will be of the same suit. How to find the value of $b$ so that the ...
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1answer
37 views

Invariant mesure of a reflected random walk

Let $(X_n), n \geq 0$ be a Reflected Random Walk defined by: $X_0 = 0$ and: $ X_{n+1}=\max( 0 , X_n + \xi )$ $\xi $ is a random variable such that $P(\xi=a)=\theta$ and $P(\xi=-b)=1-\theta$ for a ...
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1answer
21 views

How to compute of each player winning this sequence of games?

Players A and B play a sequence of independent games. Player A throws a die first and wins on a "six." If A fails, then player B throws and wins on a "five" or "six." If B fails, then A throws and ...
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3answers
38 views

How to compute these probabilities?

A pair of dice is cast until either the sum of seven or eight appears. How to compute the probability of a seven before an eight? Now, if this pair of dice is cast until a seven appears twice or ...
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2answers
47 views

How to compute this probability?

A drawer contains eight different pairs of socks. If six socks are drawn at random and without replacement, how to compute the probability that there is at least one matching pair among these six ...
0
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2answers
25 views

$ P[A] \leq P[A |\bar B] + P[B] $

How to prove that for any two events $A$ and $B$ $$ P[A] \leq P[A |\bar B] + P[B] $$ Can someone provide me hint how to proceed.
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19 views

How to establish the independence or otherwise of these compound events?

Suppose that $C_1$, $C_2$, $\ldots$, $C_n$ are mutually independent events in a sample space $S$. Then how to establish the independence or otherwise of these combinations of events? $C_1^c$ and ...
2
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2answers
46 views

Independence of two digits in a uniform$(0,1)$ random variable

I having troubles proving this: Let X be a Uniform$([0,1])$ distributed random variable. And let $X_n$ be the nth digit in the decimal expansion of $X$. Prove that if $n \neq m$ then $X_n$ and ...
2
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2answers
62 views

From distribution to Measure [duplicate]

I have been asked to create a new post with my question. So it is about starting from a distribution function and proving that we can always find a probability space. My attempt is this : So assume ...
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0answers
53 views

How to talk about a random variable which only exists on an event

Suppose I have some probability space, and some event $A$ which has probability close to, but not equal to $1$. Suppose that on this event, we can construct a random variable $X$. (For example, ...
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0answers
29 views

From distribution function to probability measure

So assume we have a probability space $(\Omega, \mathcal{F}, P)$ and a random variable $X : \Omega \rightarrow \mathbb{R}^*$. We can derive from this a distribution $P_X$, and a distribution function ...
1
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1answer
32 views

Measurability of a Borel function

I need some help on the following proof. The claim is: Suppose $f:\mathbb{R}^k \to \mathbb{R}$ and $f \in B(\mathbb{R}^k)/B(\mathbb{R})$. i.e. Borel measurable. Let $X_1$,...,$X_k$ be random ...
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3answers
49 views

Central Limit Theorem vs. Weak Law of Large Numbers

So, just to begin with I feel like this is a problem I am massively overthinking, and the solution is very simple. That said, it has been a while since I've taken a math class, and so some of my ...
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1answer
34 views

Law of total expectation: $\mathbb{E}( X | A )$ and $p_{A|B}(a|b)$

I've got a question regarding the law of total expectation and conditional probabilities. Assuming I know the EV: $\mathbb{E}(X | A)$ and the PDF $p_{A|B}(a|b)$ (A,B are not independet but binomially ...
2
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1answer
40 views

Convergence in distribution/Distribution of X

For each $n = 1, 2, ....$, suppose that $X_n$ is a discrete random variable with range $\{1/n, 2/n, ..., 1\}$ and $\hspace{15mm}\mathrm{Pr}(X_n = j/n) = \frac{2j}{n(n+1)}$, $j = 1,...,n$. Does ...
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1answer
49 views

Prove the inequality for PDF which non decreasing in interval

I am working on the following problem. Given a pdf $f$ that is non-decreasing in the interval $ a \leq x \leq b $, show that for any $s>0$ $\int^b_{a}{x^{2s}f(x)}dx \geq ...
4
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1answer
47 views

Central Limit Theorem exercise question

Let $ (X_n)_{n \in \mathbb{N}}$ be i.d.d. random variables with $E{X_1}=0$, $Var(X_1)=1$ and $ S_n = X_1 + X_2 +...+ X_n $. Calculate $ \lim_{n \to +\infty}\Pr(S_n>\sqrt{n})$. On the back ...
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2answers
29 views

Transforming discrete R.V. to uniform R.V.

Suppose that I can generate some random variable $X$ that is distributed according to the CDF $F$. If $F$ is continuous then $F(X)$ is uniform $[0,1]$ (can anyone explain this to me). My question is ...
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0answers
42 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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0answers
30 views

Reflected random walk

Suppose that $X_n$ is a reflected (in 0) random walk with parameter $\theta$. So $X_{n+1}-X_n = 1$ with probability $\theta$ , and -1 with probability $1-\theta$ when $X_n \geq 1$, if $X_n=0$ then ...
3
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1answer
48 views

Probability of intersection of non increasing sequence of events

Suppose that $(C_n)_{n\geqslant1}$ is a sequence of events such that $C_{n+1} \subset C_n$ for every $n\geqslant1$. Then how to prove the following probability statement? $$ \lim_{n \to \infty} ...
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4answers
50 views

What is the probability that at least one letter is in the correct envelope?

A secretary types three letters and the three corresponding envelopes. In a hurry, he places at random one letter in each envelope. What is the probability that at least one letter is in the correct ...
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2answers
28 views

How many bulbs should be inspected for probability to exceed $1/2$?

In a lot of $50$ lightbulbs, there are $2$ bad bulbs. How many bulbs should be examined so that the probability of finding at least $1$ bad bulb is at least $1/2$? My effort: Suppose $n$, where $0 ...
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0answers
40 views

Independence of two processes

Suppose $X_t$ is the solution of the SDE $$dX=a(X)dt+b_1(X)dW_1+b_2(X)dW_2$$ $Y_t$ is the solution of the following SDE $$dY=p(Y)dt+q_1(Y)dW_1+q_2(Y)dW_2$$ Here, $W_1$ and $W_2$ are independent ...
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1answer
36 views

Confidence interval for estimating probability of a biased coin

Suppose we have a coin with a probability $p$ of coming up heads and $q = 1-p$ of coming up tails on any given toss. (A coin is biased unless $p=0.5$). But we are not given what $p$ or $q$ are. We ...
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48 views

Problem regarding Conditional probability

Let $\mathbf{X}$ be an $n-$ dimensional random variable. This variable can be written as $\mathbf{X} = \left[\mathbf{X}_1^T\hspace{5pt}\mathbf{X}_2^T\right]^T$. where, $\mathbf{X}_1$ is $m-$ ...
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1answer
33 views

Putting a bound on some probability inequality

Assume that we have the following polynomial: $$ax^2 + bx =c$$ and a, b, c are i.i.d uniform random variables in [0, 1]. I'm trying to calculate the probability that the root is real, and that ...
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1answer
36 views

Product of a discrete and absolutely continuous random variable, part deux

I have the independent random variables $U\sim N(0,1)$, $V\sim N(0,1)$, $W\sim b(1,1/2)$ I define $X=WU + (1-W)(V+1)$. I need to determine that $X$ is absolutely continuous, and determine a density ...
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20 views

Combining classifiers

How can I compute $P$($X$ | $f_0$, $f_1$, $f_2$), given the following pdf's: P($X$|$f_0$), P($X$|$f_1$).P($X$|$f_2$). I am interested in the case of discrete probabilities, but I guess it will be ...
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1answer
17 views

Product of a discrete and absolutely continuous random variable

I have the independent random variables $U \sim e(1)$ and $V \sim po(2)$. We define the variable $Y=UV$. I want to prove that Y is not absolutely continuous. The only hint I have recieved is that I ...
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0answers
7 views

Sampling with an “oversampling” factor, in K-Means||

I'm trying to understand K-Means||, a scalable version of K-Means++, which itself is an "improved" version of the clustering algorithm K-Means. Please find here the link to K-Means|| paper ...
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1answer
58 views

Let F be a distribution function. Prove that X is a RV.

Let F be a distribution function. On $(\Omega, \mathfrak{F}, P)=((0,1), \mathfrak{B}(0,1),\lambda)$ where $\lambda$ denotes Lebesgue measure. Define X: $\Omega \ to \mathbb{R}$ by $X(\omega) = sup(y ...
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2answers
62 views

If a probability space has no measurable subsets with $P$ strictly between $0$ and $1$, then every random variable is constant a.s.

Let $(\Omega, \mathfrak{F}, P)$ be a probability space such that $\forall F \in \mathfrak{F}, P(F) = 0 \ or \ 1$. Show that for all random variables X on $(\Omega, \mathfrak{F}, P)$, $\exists \ c ...
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0answers
41 views

Most important results from pure math in applied probability?

I'm taking a course next semester at my university on applied probability (with relevance to signal and information theory). Although the nature of probability is mostly problem solving and applying ...
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2answers
57 views

If X and Y are random variables with the same distribution, prove that f(X) and f(Y) are random variables that have the same distribution.

Suppose X is a RV on $(\Omega, \mathfrak{F}, P)$. Let f be Borel-measurable on $(\mathbb{R}, \mathfrak{B})$. 1 Show that f(X) is also a RV on $(\Omega, \mathfrak{F}, P)$. 2 Let Y ba RV on ...
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0answers
51 views

Azuma's inequality: basic question [closed]

In the statement of Azuma's inequality, it is assumed that random variables $X_n$ are martingales (and therefore $X_n - X_{n-1}$ are martingale differences). But what is the essential step in the ...
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2answers
43 views

How to show: $\sum_{k=1}^{\infty}\sum_{n=1}^{k}P(X=k)=\sum_{n=1}^{\infty}\sum_{k=n}^{\infty}P(X=k)$

Can anyone explain why this equation is true. $$\sum_{k=1}^{\infty}\sum_{n=1}^{k}P(X=k)=\sum_{n=1}^{\infty}\sum_{k=n}^{\infty}P(X=k).$$
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74 views

properties of characteristic function

Let $X,Y$ be two independent random variables having the same distribution, centred and with variance 1, $\phi$ is the characteristic function of $X$ and $Y$. If $X+Y$ and $X-Y$ are independent, show ...
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1answer
35 views

Does the Lindeberg condition hold for this random variable as $n\to\infty$?

Suppose we have a sequence of independent random variables $\xi_n$, with $P(\xi_n = 1) = P(\xi_n = -1) = \frac{1}{2}\left(1-\frac{1}{n^2}\right)$, and $P(\xi_n = \sqrt{n}) = P(\xi_n = ...
2
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1answer
31 views

Independence of increments of a pair of independent Brownian motions

Suppose I have two Brownian motions $X$ and $Y$, which are independent. In other words, for any finite set of times $0 < t_1 < t_2 < \cdots < t_n$ the random vectors $(X(t_1),\ldots , ...
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1answer
45 views

How to compute the integrals in inverse formula?

I have following characteristic function for certain random variable X: $$\Phi (t) = \frac{\beta_1\beta_2}{\eta_1}\frac{\eta_1 - it}{(\beta_1 - it)(\beta_2 - it)}$$ where $\eta_1 > 0, \quad\beta_1 ...
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1answer
36 views

Variance of sum of multiplication of independent random variables

Suppose that we have $Z=\sum_{i=1}^n (a_i+b_iX_i)(c_i+d_iY_i)$. Where $a_i,b_i,c_i$ and $d_i$ are real numbers and $X_i$s and $Y_i$s are all independent random variables. How can I find the variance ...
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Is the plane created by ($\int f_0^u f_1^{1-u},\int f_1^u f_0^{1-u})$ continuous?

$f_0$ and $f_1$ are two continuous density functions on $\mathbb{R}$. I wonder if $$(x,y):=\Bigg(\int f_0^u f_1^{1-u},\int f_1^u f_0^{1-u}\Bigg)$$ for all $f_0$ and $f_1$ is complete for some ...
0
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1answer
32 views

Compute expectation using tower property

Imagine I want to compute $E[f(X,Y)]$ where $f$ is a suitable function and $X,Y$ random variables. If $X$ and $Y$ are correlated, let us say $X=g(Y)$ or even in my case $X_t$ and $Y_t$ are stochastic ...
2
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2answers
80 views

Formal approach to (countable) prisoners and hats problem.

I've found this nice puzzle about AC (I'm referring to the countable infinite case, with two colors). The puzzle has been discussed before on math.SE, but I can't find any description of what is ...
2
votes
1answer
62 views

Showing $E(X|Y) = E(X|Y,Z)$

Let $(X,Y)$ and $Z$ be independent random variables. Show $E(X|Y) = E(X|Y,Z)$. No other information is given. First off, if $(X,Y)$ and $Z$ are independent, are $X$ and $Z$ (and $Y$ and $Z$) also ...