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 ...

learn more… | top users | synonyms (2)

0
votes
0answers
4 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 ...
0
votes
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 ...
2
votes
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 ...
1
vote
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 ...
5
votes
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, ...
1
vote
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 ...
1
vote
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) ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
0answers
21 views

Brownian motion hitting time [on hold]

Let $B(t)$ be a linear Brownian motion and $a,b>0$. Show that $P(B(t)=a+bt \text{ for some } t>0)=e^{-2ab}$
0
votes
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 ...
2
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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.
1
vote
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 ...
1
vote
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 ...
1
vote
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), ...
1
vote
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 ...
6
votes
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 ...
0
votes
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. ...
0
votes
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 ...
1
vote
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
votes
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
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." ...
-2
votes
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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
1answer
23 views

Deriving the the conditional PDF from Bayes' Rule

I am having trouble getting the conditional PDF from Bayes' Rule for the following problem: Fred wants to sell his car, after moving back to Blissville (where he is happy with the bus system). He ...
3
votes
0answers
24 views

N persons in K rooms, expected population of most crowded room

There are N people. Each of them randomly and independently select one of K rooms, with uniform distribution. Seems to me obvious that the expected population of each room will be N/K. What will be ...
5
votes
5answers
58 views

Difficult probability of choosing ball from bag with $7$ balls labelled from $1-7$

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with probability, which perhaps yields the shortest, simplest proofs, but other ...
1
vote
0answers
19 views

Probability in Chestnut Genetics [on hold]

This question concerns cross breeding American and Chinese chestnuts to transfer the blight resistance inherent in Chinese chestnut to the American chestnut. The chestnut genome contains approximately ...
0
votes
1answer
29 views

What is the probability of getting exactly one two and one three in a 5 card draw?

In a 52 cards deck, what is the probability of getting exactly one 2 and one 3 if 5 cards are drawn. I'm wondering what is the difference between doing it the following two ways. Intuitively I would ...
-2
votes
2answers
27 views

Statistics basics

Given that $X$ has mean $a$ and variance $b$. Then $E(X^2) = a^2 + b^2$. Why is this true? Please provide a proof alongside any other relevant information. Thanks in advance.
0
votes
0answers
17 views

Permuting a cycle when assignments exist.

There are $n$ agents total and $n$ objects total. Each agent ranks objects uniformly at random independent of other agents. Suppose I fix a subset of $k$ agents $a_1,..,a_k$, and $k$ objects ...
0
votes
0answers
18 views

why am I getting this result on a sample size calculation tool

I found an online calculator for sample size needed to perform an A/B test to a specific metric. Given a requested z-score of 1.96, a minimum detectable effect (MDE) of 5% and an estimated ...
1
vote
0answers
17 views

conditioning on a mixture random variable [on hold]

Suppose we have the conditional distribution $P(X,Y|Z)$ and $Z$ follows a mixture distribution such that $P(Z)=q*P(Z_1) + (1-q)*P(Z_2)$ where $q \in (0,1)$. Then, how can I write the conditional ...
1
vote
0answers
19 views

Brownian Motion maximum process intuition

I am studying the maximum value of a Brownian Motion (BM) on an interval of time (as explained here between boxes 28 and 40) and I am having an issue aligning intuition with the mathematical result. ...
1
vote
0answers
14 views

Doob's maximal inequality with stopping time

I have been searching for a version of Doob's maximal inequality with stopping time insides the time index, i.e. given $\Lambda_n$ is a positive sub-martingale and N is a stopping time is there any ...
1
vote
0answers
21 views

The probability that the d-dimensional symetric random walk returns to the origin - is this relatively short proof correct?

Let $p_n$ denote the probability of returning to the origin after n steps. If n is odd, $p_n = 0$. The main insight is that $\sum_{n=0}^{\infty}p_{2n}$ is asymptotically ~ $C \cdot \frac{1}{n^{d/2}}$ ...
1
vote
0answers
26 views

Probability problem related to Markov inequality

Problem Let $p$ be the probability of a person chosen at random to support Bernie Sanders. A sample is taken of $50$ persons chosen at random, each of them is asked if he or she would vote for ...
6
votes
1answer
182 views

Why was I wrong about the monster-gem riddler

Every week I like to do the fivethirtyeight.com Riddler, an interesting and pleasantly challenging (at least for me) weekly math puzzle which comes out Fridays, with the answer and explanation to the ...
0
votes
0answers
26 views

Notation: should Markov chains steps be noted by uppercase or lowercase letters?

I'm reading the chapter about perfect sampling of the "Monte Carlo Statistical Methods" by Robert and Casella, 2004. I've got an issue about notation, when they talk about random mappings, they say ...
2
votes
1answer
42 views

Probability Mass Function of infinitely re-rolled dice

I play a game called Shadowrun. It is a role-playing game that uses a dice pool mechanic. A player has a dice pool of $x$ six-sided, unbiased dice. Every 5 or 6 counts as a success. The more ...
-1
votes
2answers
39 views

Count the expected value from mimimum [on hold]

Random variable $S_{N}=X_{1}+\dots+X_{N}$ has a Poisson distribution(I assume that the author mean than $N$ has a Poisson distribution). wih $\lambda=5$. Random variable $X$ takes two values ...