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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2answers
24 views

n tasks assigned to n computers, what is the EX value of a computer getting 5 or more tasks?

Say a central server assigns each of n tasks uniformly and independently at random to n computers connected to it on a network. Say a computer is ‘overloaded’ if it receives 5 or more tasks. Q: ...
1
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0answers
5 views

Radon-Nikodym on a Process wrt to filtration

Given a probability space $(\Omega,\mathcal{F},P)$. Let $(X_t)_{t\geq0}$ be a stochastic process defined on it with cadlag paths, lets say on $(\mathcal{X},\mathcal{B}(X))$. Let be $\mathcal{F}_{t}$ ...
1
vote
1answer
562 views

Statistics question Conditional Probability

Question: Of three cards, one is painted red on both sides; one is painted black on both sides; and one is painted red on one side and black on the other. A card is randomly chosen and placed on a ...
1
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1answer
20 views

A certain store sells $31$ different flavors of ice cream. How many different $3$-scoop cones are possible if :

A certain store sells $31$ different flavors of ice cream .How many different $3$-scoop cones are possible if : a) each flavor must be different and the order of flavors is unimportant? ...
1
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0answers
14 views

Slutsky, Continuous mapping for uniform convergence

I have a question- suppose I have a function f(x,$\hat \theta$), $\hat \theta$ is a consistent estimate for $\theta$ and therefore it holds $\hat \theta \rightarrow \theta$ in probability. Suppose f ...
4
votes
5answers
457 views

Flipping a biased coin

Consider a hypothetical coin (with two sides: heads and tails) that has a $3/4$ probability of landing on the side it was before the flip (meaning, if I flip it starting heads-up, then it will have an ...
0
votes
1answer
25 views

Expected value of X and Y for a given problem

A couple decides to have children until they get a girl, but they agree to stop with a maximum of 5 children even if they haven't gotten a girl. If X and Y denote the number of children and number of ...
8
votes
3answers
631 views

Formally, why does a logical contradiction have probability zero?

In terms of formal probability theory, why does an event representing a logical contradiction (such as $A \wedge \neg A$) always have probability zero? I understand intuitively why this is the case, ...
0
votes
3answers
26 views

Value of a that minimizes $E([aX-\frac{1}{a}]^2)$

Suppose X is a random variable with mean $\mu$ and variance $\sigma^{2}$. For what value of a, where a > 0 is $E([aX-\frac{1}{a}]^2)$ minimized?
0
votes
1answer
31 views

What are some effective ways to go about this type of problem?

I need an efficient way to go about this problem, for practice for my problem solving test. This is not a part of the actual test. This is the type of question that I am struggling with There are two ...
0
votes
1answer
25 views

Expected area of a circle given pdf of radius

The measured radius of a circle, R, has probability density function \begin{equation} f(r) = \begin{cases} 6r(1-r) \;\;\;\; \textit{if 0 < r < 1}\\ 0 \;\;\;\;\;\;\;\;\;\;\;\;\;\;otherwise ...
-1
votes
0answers
12 views

Finding joint pdf from marginal pdf's

I have $N$ samples $(X_1,\cdots X_N)$ of exponential random variables with parameter 1. The samples are ordered such that $X_N \geq X_{N-1} \geq \cdots X_1$. I know the individual pdf's of $X_N$ and ...
1
vote
1answer
20 views

bivariate density function - components are not independent

I want to show that the random variables of the density function $$\begin{cases}π^{−1} & :\textsf{if }x^2 + y^2 < 1\\0 & :\textsf{elsewhere}\end{cases}$$ ... is dependent. However, I ...
0
votes
1answer
27 views

Randomly generated binary string [on hold]

Suppose we generate a sequence containing only 0's and 1's. We generate this binary string by randomly adding a 0 or 1 one at the time. We stop when the sequence contains k more 1's then 0's or k more ...
-1
votes
0answers
9 views

P irreducible transition matrix, why is vector $\mathbb{1}$ not in span of columns of P-I

Given irreducible transition matrix $P$, why is the column vector $\mathbb{1} \not\in$ the span of the columns of P-I? Put another way, why does the matrix $(P-I|\mathbb{1})$ have full rank?
2
votes
2answers
21 views

problem with Combination and Permutation

Four married couples have bought 8 seats in a row for a concert. In how many ways can they be seated: a)if each couple is to sit together? (8)(1)(6)(1)(4)(1)(2)(1) b)if all men sit together? ...
3
votes
1answer
94 views

Conditional probability generating function - Binomial

I'm working on the following problem: Y = $X{_1}+X{_2}+X{_3}+...+X{_N}$ $N\overset{d}{\sim}Bi(n,p) $ and $X_i\overset{d}{\sim}Bi(m,q)$ $N, X_1, X_2 $ are independent $a)$ Find ...
0
votes
1answer
15 views

Let $X$ have pdf $f(x) = e^{-x}$. Find the pdf of the integer part of $X$.

A continuous random variable has a pdf defined by $$f(x) = e^{-x} , x > 0.$$ The discrete random variable $Y$ is defined as the integer part of $X$, that is the largest integer less than or ...
2
votes
1answer
32 views

What is the probability that a given $ n $ event trains match the beginning of a Poisson process?

Here is my question with which I'm confusing myself: Assume that some event times $ \{\tau_i\}_{i \in \mathbb{N}} $ are a point process with rate $ \mu $ such that number of events that occurred ...
0
votes
1answer
30 views

What is the sample space of a dice labelled with 1,2,2,3,3,3 for the standard dice?

When we roll a dice labelled with 1,2,2,3,3,3 for the standard dice. What is the sample space of this activity? If someone argues the probability of getting 1 is $\frac{1}{3}$. Because the person ...
1
vote
1answer
19 views

Finding the mean given the probability

I'm doing some work on branching processes and would like to know where the process becomes extinct. If $X$ is the number of offspring of an individual, then the process goes extinct when ...
-1
votes
1answer
42 views

Finding the conditional probability

enter image description here Let $(X,Y)$ be a two-dimensional stochastic vector with density $$ f_{X,Y}(x,y) = \begin{cases} \dfrac{e^{-y}} y & \text{if } 0<x<y, \\[4pt] \,\,\,\, 0 & ...
3
votes
1answer
24 views

Expected Square Distance from Origin of Random Walk in $\mathbb{Z}^2$

I'm trying to find the expected value of the squared distance from the origin of a simple symmetric random walk in $\mathbb{Z}^2$ at time $n$. So far, I have calculated that if $(X,Y)$ is the ...
2
votes
0answers
21 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 ...
-2
votes
1answer
36 views

How can we prove this equation using marginalization and conditioning? [on hold]

I want to prove $$P(A|C) = \sum_{B} P(AB|C) $$ How can we prove this using marginalization and conditioning?
103
votes
15answers
17k views

Do men or women have more brothers?

Do men or women have more brothers? I think women have more as no man can be his own brother. But how one can prove it rigorously? I am going to suggest some reasonable background assumptions: ...
-1
votes
2answers
53 views

Conditional probability of a Joint distribution

Let $(X,Y)$ have joint density $f(x,y)=e^{-y}$ , for $0<x<y$, and $f(x,y)=0$ elsewhere. What is $f_{X\mid Y} (x,y)$ for $0<x<y$? I think that the answer is $1/y$, however, I am having ...
2
votes
0answers
79 views
+200

Tauberian theorem when limit is zero

Let $h \geq 0$ be a non-negative increasing function with Laplace transform $H$. Let $\rho \geq 0$ be a constant. A simple Tauberian theorem says that the following two statements are equivalent: I. ...
0
votes
1answer
33 views

Count the expected value

Random veriable $K$ has a uniform distribution on the interval $(1,5)$. Conditional probability distribution of $S_{N}=X_{1}+\dots+X_{N}$ given K has a compunded Poisson distribution. $N\approx ...
1
vote
1answer
26 views

why CCDF of x is equal to expectd value of probability of x given y

I found in an article a fact that is commonly used in scientific papers which mentions that: $\mathbb{P}(f(r)>T) = \mathbb{E}[\mathbb{P}(f(r)>T\,|\,r)]$ ($\mathbb{E}$ is with ...
2
votes
1answer
15 views

Divergence of asymmetric not-simple random walk

Consider a (not simple) random walk $S_n = \sum_{k=0}^n X_k$ where X_k are i.i.d and the mean $\overline{X}<0$. Is there is simple proof or a reference showing $P( \lim \limits_{k \to \infty} X_k = ...
1
vote
1answer
38 views

Probability function (p.f) of a random variable

If we have a Bernoulli random variable $W$ that is derived from a Variable $T$ (Poisson $\lambda$), by the following rules $W =$ (if $T=0$ then $W=1$ and if $T>0$ then $W=0$), I am having trouble ...
1
vote
2answers
690 views

Basic probability : the frog riddle - what are the chances?

A few days ago I was watching this video The frog riddle and I have been thinking a lot about this riddle. In this riddle you are poisoned and need to lick a female frog to survive. There are 2 frogs ...
0
votes
1answer
31 views

Probability function and random variables

Given a Bernoulli r.v., $W$, which is derived from r.v. $T$ (Poisson) (a) if $T=0$ then $W=1$ and (b) if $T>0$ then $W=0$. One has to show that the sample mean (the proportion of $0$s in the ...
0
votes
2answers
49 views

Probability, poor and good shooter

Imagine you have a duel with two opponents, Person $B$ is a poor shooter with probability of shooting his opponent equaling $1/3$, while person $A$ is a good shooter with probability of shooting his ...
0
votes
1answer
30 views

Heavy tailed discrete distribution infinite mean

I'm looking for an example of a discrete distribution with infinite mean $f_n = P(X = n)$ for $n=1,2..$ such that the sequence $r_n = \sum\limits_{k=n+1}^{\infty}f_k$ satisfies the relation $$r_n = ...
0
votes
0answers
42 views

Random walk visiting $k$ distinct points

I have a random walk on $\mathbb{Z}$ with starting point $0$ and with length $n$ and possible steps to right, left or stay where you are, all with the same probabilities. I am interested in exact ...
1
vote
0answers
40 views

Cumulative distribution function of two independent and uniform on $[0,1]$ random variables is a surjective map for $t\in [0,1]^2$?

I am trying to argue that the cumulative distribution function of two independent and uniform on $[0,1]$ random variables is a surjective map for $t\in [0,1]^2$. Below the argument I have developed. ...
1
vote
3answers
40 views

Calculate $E(X^{2n})$ where $X$ is normal (0,1)

I need help proving the following: Let $X$ be normally distributed with parameters $\sigma=0$ and $\mu=1$. Let $n$ be a positive integer. Show that: $$E(X^{2n})=\frac{(2n)!}{2^nn!}=:(2n-1)!!$$ I've ...
1
vote
1answer
25 views

What is the domain of a function of random variables?

Consider a random variable $X$ defined on the probability space $(\Omega, \mathcal{F}, P)$ $X:\Omega \rightarrow \mathcal{X}\subset \mathbb{R}$. Suppose $X$ has range (or image) $\mathcal{I}\subset ...
3
votes
0answers
33 views

Convergence of a sequence over supremum

Given a cadlag-process $X_{t}$ with stationary independent increments (Levy process) for which $E\left[\sup_{s\in[0,t]}\left|X_s\right|\right]<+\infty$ for all $t>0$. For $n\in \mathbb{N}$ the ...
3
votes
2answers
41 views

The probability of rolling 4 dice and getting a 6.

The probability of rolling 2 dice and getting a 6 on either one of the die or both is : 11/36 or about 0.305. Also I calculate the probability of rolling 4 dice and getting a 6 on either one, two, ...
0
votes
1answer
216 views

Gamblers ruin formula

Hello , I have been reading about gamblers ruin and I found this formula can anyone confirm its accuracy ? I assume they only bet one chip a time
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votes
1answer
25 views

Coefficient of correlation between linearly related random variables [on hold]

A random variable $X = -3Y+10$, where $Y$ is also a random variable and has zero mean. Are they correlated? What is the correlation coefficient in this case? I know that it equal ...
0
votes
1answer
29 views

Product of exponentially distributed and uniformly distributed random variables [on hold]

Let $X$ be an exponentially distributed random variable, and let $V$ be a uniformly distributed random variable on $\{-1,+1\}$ that is independent from $X$. Furthermore, let $Y = X \cdot V$. I want ...
1
vote
0answers
29 views

Probabilities, unfair coins and result [on hold]

I have the following question If there are two different coins ($A$ and $B$) with probabilities of tails equal to $p_A$ and $p_B$ and I toss coin $A$ $n$ times and coin $B$ $m$ times, what is the ...
0
votes
0answers
23 views

Difference of dependent central Chi-Square random variables with 2 degrees of freedom

Suppose we have $X$ and $Y$, both are dependent and complex Gaussian random variables with zero means and the same variance $\sigma^2$. The real and imaginary parts of $X$ and $Y$ are independent, ...
0
votes
0answers
12 views

Given an event field, is there a random variable generating it? [duplicate]

In probability space $(\mathsf{\Omega},\mathcal{F},\mathrm{P})$, for any event field $\mathcal{G}\subset\mathcal{F}$, there always exists a random variable $X$, such that $\sigma(X)=\mathcal{G}$? Is ...
9
votes
1answer
63 views

Hard Question in Stochastic processes - variance Martingales

I got some hard challenge to solve and I am looking for a small clue/help. My question goes like this: 10 Englishmen are trying to leave a pub in a rainy weather. They do it in the following ...
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votes
0answers
24 views

Heads or tails: Find the length of the longest subsequence that contained only heads, and the longest subsequence that contained only tails. [on hold]

If an unbiased coin is tossed, then the value of the coin (Tails) has the same probability of showing after falling as the other side that contains the coat of arms (Heads). Because of this decision ...