This tag is for basic questions about probability and for questions in which one wants to calculate a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using measure theory), please ask under ...

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5 views

Interpretation of correlation (coefficient)

In an discussion we were confronted with a very special opinion about correlation in respect of financial assets. The widely used correlation coefficient is used here to give an idea about how ...
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0answers
5 views

Expected values of max(X,Y) and min(X,Y) for $N(\mu,\sigma^2)$ distributed $X$ and $Y$

Suppose that $X$ and $Y$ are independent and $N(\mu,\sigma^2)$ distributed. Then $E(\min(X,Y))=\mu-\frac{\sigma}{\sqrt{\pi}}$ and $E(\max(X,Y))=\mu+\frac{\sigma}{\sqrt{\pi}}$. I tried to ...
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1answer
270 views

Two different sequences of random variables each converge in distribution; does their sum?

My question is about basic probability. We have two sequences of random variables, $ \{ X_n \}$ and $\{ Y_n \}$, such that each converge in distribution - i.e. there exist random variables $X$ and ...
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2answers
25 views

Coin Toss Game - Probability of H when unequal number of coins tossed

Two gamblers are playing coin toss game: Gambler A has (n+1) coins and B has n coins. What is the probability that A will have more heads than B if both flip all their coins. Not sure how to go about ...
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1answer
16 views

Infinite Marbles in a Jar with Known Distribution

Let's say I have infinite number of marbles in a jar and $90\%$ of them are red and $10\%$ are green. If I pick $25$ out of the jars (with or without replacement probably doesn't matter because the ...
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1answer
36 views

Probabilities ant cube

I have attached a picture of the cube in the question. An ant moves along the edges of the cube always starting at $A$ and never repeating an edge. This defines a trail of edges. For example, ...
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0answers
6 views

Levy process measurable past

For a Levy-process $(X_t)_{t\geq 0}$ with stationary indepedent increments which is a markov process, we know that its law is defined by its one dimensional distribution. This is so because for its ...
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1answer
16 views

Logic of getting a full-house of cards.

Although I understand the correct solution of finding the total number of full houses in a 52-deck of cards (finding the number of ways of selecting the first value and then finding the amount of ways ...
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1answer
573 views

Greatest of three random variables

Assume that we have $3$ not equal random variables $(A, B, C)$. If we know that $$Pr(A>B)=x, \quad Pr(A>C)=y, \quad Pr(B>C)=z$$ What is $Pr(A$ is the greatest one)? I know that $Pr(A$ is ...
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1answer
14 views

From brownian bridge to brownian motion proof

Let $B_t$ be a brownian motion. and let $\{W_t=B_t-tB_1:0\le t\le 1\}$ be a brownian bridge. Now let $Y_t=(1+t)W_{t\over 1+t}$. Proof that $Y_t$ is a brownian motion in $[0, \infty)$ My attempt: 1) ...
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1answer
13 views

Running Query on Conditional Probability graph

I want to run the following query. P(Rain | WetGrass = True). What I know: Because we are given the child, Rain and Sprinkler are no longer conditionally independent. My first approach is to use ...
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1answer
35 views

Confusion related to calculating the probability distribution of a variable

I have this confusion related to calculating the probability distribution of a variable. If I have a variable $x_1$ which has a pdf $p(x_1)$.Lets assume that the distribution is gaussian with mean ...
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1answer
57 views

Integrate $\int x \frac{f'(x)}{f(x)} dx$

I need your advice in integrating $\int ln(f(x)) dx = \int x \frac{f'(x)}{f(x)} dx$, where $f(x)$ is a probability density function. So it is the same as $\int x \frac{F(x)}{f(x)} dx$. How can I ...
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3answers
44k views

Find $E(XY)$ assuming no independence with $E(X) = 4$, $E(Y) = 10$, $V(X) = 5$, $V(Y) = 3$, $V(X+Y) = 6$.

I am having trouble finding a way to solve this problem. I understand this would be simple if $X$ and $Y$ were independent however, I believe they are not since $V(X) + V(Y) \neq 8$. Find $E(XY)$ ...
3
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0answers
40 views

Hitting probabilities in a random walk on a graph

Consider a random walk $(X_n)$ on the graph below, where we jump from a given vertex to one of its adjacent vertices with equal probability. I want to find: the probability that we hit $A$ before ...
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0answers
20 views

CDF to PDF - Piecewise

So we have to find the CDF of the piece-wise function. I get every part of the conversion right but when I got to finding 2 is less than x which is less than 5 , the answer was 1/6 and I got ...
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1answer
37 views

5 red and 10 black balls in a bowl, with replacement

Problem A bowl contains $5$ red and $10$ black balls. A ball is picked randomly and the colour is noted. After every pick the ball is placed back, and an extra ball of the same color is added to the ...
2
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1answer
39 views

Probability of picking marbles from a bag with only the ratio of marbles given

Here is a question that is puzzling me: A bag contains a large number of marbles; the numbers of the red, blue and yellow marbles are in the ratio $3:4:5$. Four marbles are randomly drawn ...
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1answer
27 views

Chevalier de mere paradox with game with three dice

Chevalier de Mere asked Blaise Pascal why in a game with three dice the sum $11$ is more favorable than $12$, when both sums have exactly the same possible combinations: For $11$ we have $(5,5,1), ...
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0answers
21 views

Brownian motion hitting time [on hold]

Let $B(t)$ be a linear Brownian motion and $a,b>0$. Show that $P(B(t)=a+bt \text{ for some } t>0)=e^{-2ab}$
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1answer
16 views

Probability of selection of $D$ if $C$ is elected unanimously

Five persons $A,B,C,D,E$ are contesting in an election in which $3$ persons are to be selected. If $C$ is elected unanimously, then find the probability that $D$ gets selected. I am not able to ...
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1answer
46 views

Is Brownian motion on $[0,b]$ bounded?

Is Brownian motion on $[0,b]$ bounded? Or at least bounded with probability one. Since Brownian motion is continuous with probability $1$, I guess the answer is YES.
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0answers
21 views

On the derivation of the Cauchy Distribution

I am currently studying from this video lecture series and the professor here goes over the derivation for the Cauchy distribution. I am able to follow most of it except for one minor part. Part of ...
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2answers
591 views

Elevator Probability Question

There are four people in an elevator, four floors in the building, and each person exits at random. Find the probability that: a) all exit at different floors b) all exit at the same floor c) two ...
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1answer
24 views

Find the PDF of $Y= \sin{(\pi X)}$, where $X \sim U[0,1]$

Let $X\sim U_{(0,1)}$ and lets define $Y= \sin{(\pi X)}$. I want to get the pdf of $Y$. My attempt: Clearly, $y\in(-1,1)\Rightarrow 1-y^2\ge0$, so $$ F_Y(y)=\Bbb P(Y\le y)=\Bbb P\big(\sin{(\pi X)}\le ...
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0answers
23 views

Is this a correct interpretation of maximum likelihood estimation?

Here is an excerpt from Pattern Recognition and Machine Learning by Christopher Bishop: This seems to be not quite right—"the probability of the data set", when the data set is drawn from a ...
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4answers
231 views

Probability of having a Girl

A and B are married. They have two kids. One of them is a girl. What is the probability that the other kid is also a girl? Someone says $\frac{1}{2}$, someone says $\frac{1}{3}$. Which is correct? ...
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2answers
19 views

How do I add multiple probabilistic results of a single experiment?

Let's say I've conducted an experiment that yields either a positive or negative result with a 50% probability of each. Three people attempt to determine the result of the experiment. They all only ...
4
votes
1answer
53 views

Conditional expectation maximum of sample

Find the conditional expectation $\mathbb{E}\left[\left.X_{1}\right|Y\right]$ if $X_1,..., X_n\sim\mathrm{Uniform}\left(0,1\right)$, where $Y=\max\left\{ X_{1},...,X_{n}\right\}$. MY ATTEMPT: We ...
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0answers
24 views

Trying to find RV's $C_1,\ldots,C_n$ satisfying $\mathbf{P}(Y \leq y) = \mathbf{P}(N=0)+\sum_{j=1}^n\mathbf{P}(C_j\leq y)\mathbf{P}(N=j).$

This is an assignment question that I just can't puzzle out; some hints or direction would be appreciated. We're given fixed parameters $p \in [0,1]$ and $m,\lambda \in \mathbb{R}_{>0}$. We also ...
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2answers
39 views

Probability of $m$ failed trades in series of $n$ trades

This is a trading problem: Let's say I have an automated trading system with a probability of success of $70\%$ on any individual trade. I run $100$ trades a year. What is the probability of ...
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1answer
200 views

Expected value and sum of independent variables.

EDIT: I've found my mistake. Flipped around the values because in my head I had them tails up at the start.. Not sure what to do with the question now... On a table there are three coins in a row, ...
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0answers
19 views

Is $B_{t\wedge H_a}$ bounded in $L^2$?

Let $a >0$, $(B_t)_{t\geq0}$ be a standard Brownian motion. Define the stopping time $$H_a := \inf\{t \geq 0 : B_t \geq a\}.$$ Then is the martingale $M_t$ where $M_t: = B_{t\wedge H_a}$ bounded ...
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1answer
28 views

What is the probability that an event happens an infinite amount of times in infinite trials?

For example, that in an infinite amount of coin flips, the event that the result are head k times in a row happens an infinite amount of times.
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0answers
28 views

Probability question - step function?

How would you figure out the probability function of this scenario: There are 9 pieces of paper with number 1-9 (one each) in a box. Select the paper 9 times (with replacement) then select a ...
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0answers
20 views

Probability if variable has $15\%$ CV

I have a relatively simple question, but I am not sure if I understand it right. I have estimated through my calculations the value $X$. $X$ depends on many things, but one of them is $Y$ and I know ...
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0answers
44 views

How to solve probability when sample space is infinite?

I came up with a random problem yesterday: Suppose that in a random trial, each point $(x,y)$ where $x,y \in \mathbb{R}$ and $0 \leq x,y \leq 1$ is assigned a value of $0$ with 50% chance and a ...
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0answers
48 views

Having rand2() function build rand5()

I was asked this question long time ago. Having a function $rand2()$ (in any computer language, "rand" means random) which returns $0$ or $1$ (two values only) with a uniform distribution, i.e. ...
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1answer
1k views

Uniformly Most Powerful Test and Rejection Region of Poisson Distribution

Let $X_1, \dots,X_n$ be a random sample from a Poisson$(\lambda)$ distribution where $\lambda > 0$. (1) Find the Uniformly Most Powerful (UMP) level $\alpha$ test for the following set of ...
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0answers
13 views

Tail bounds for functions of a Poisson point process

A Poisson point process consists of a sequence of points $0\leq t_1\leq t_2<\cdots$ where $t_i = t_{i-1} + X_i$ where $X_i$ is an exponentially distributed random variable with some rate parameter ...
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1answer
1k views

Conditional independence property: weak union

Let $(X,Y,W,Z)$ be disjoint sets of random variables each with finite space. Then prove that if $\Pr(X\mid W,Y \cup Z)=\Pr(X\mid W)$ then $\Pr(X\mid Y,Z \cup W) = \Pr(X\mid Z \cup W)$. This is ...
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2answers
1k views

The logic behind a sequence

I am trying to get the logic behind the sequence: for $n=2,3,\ldots$ $$\left(\frac{\log (2)}{\log \left(\frac{3}{2}\right)},\frac{\log (3)}{\log \left(\frac{17}{9}\right)},\frac{\log (4)}{\log ...
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1answer
634 views

Conditional expectation of the sum of three dice rolls given the sum of their maximum and product

Consider the random experiment in which three fair dice are rolled simultaneously (and independently). Let $X$ be the random variable defined as the sum of the values of these three dice. Let $Y_1$ ...
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5answers
3k views

Probability of being poisoned

You are playing a game in which you have $100$ jellybeans, $10$ of them are poisonous (You eat one, you die). Now you have to pick $10$ at random to eat. Question: What is the probability of ...
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0answers
10 views

calculate the $P(B(1)\leq 0,P(B(2)\leq 0))$, $B(t)$ is the standard brownian motion.

denote $W(1)$ by $(B(2)-B(1))$. then $P(B(1)\leq 0, B(2)\leq 0)$ = $P(B(1)\leq 0, B(1)+(B(2)-B(1))\leq 0)$ =$P(B(1)\leq 0, B(1)+W(1)\leq 0)$ =$P(B(1)\leq 0, W(1)\leq -B(1))$. by conditioning by ...
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0answers
24 views

Given X and Y ind. rv's, when is f(X,Y), g(X,Y) ind.?

I have to parallel questions. I was trying to solve this one: "Given two independent real-valued randomvariables X and Y defined on the same sample space, is it true that X and X+Y are independent." ...
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0answers
11 views

Cumulative distribution function and sum of random variables [on hold]

For two continuous (iid) random variables $X$ and $Y$, we have (ref): $P(X + Y \le c)=\int_{-\infty}^\infty \int_{-\infty}^{c−x} (f(x,y)dy)dx$ with $f$ being the joint density function. What is the ...
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1answer
30 views

Proving specific formula for stationary markov process [on hold]

In my probability class, right now we are dealing with Markov chains and I was stumbled by parts of this problem: Given a $ \{ X_n \}_{n=0}^{\infty} $ be a homogeneous Markov chain (the transition ...
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1answer
172 views

Verifying a standard Brownian Motion? [on hold]

Let $\{X_t, t\ge 0\}$ be a standard Brownian motion process. For a fixed positive number s and all $t\ge 0$, we define $Y_t = X_{t+s} - X_s$. Is $\{Y_t, t\ge0\}$ a standard Brownian motion? Attempt: ...
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1answer
720 views

Probability of dice roll (board games)

Assume that we have $n$ six-side dices. We will roll all the $n$ dices.What is a probability of getting at least $r$ ones, $s$ twos , $t$ threes and $u$ fours? Number $6$ can be used instead of any ...