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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0
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1answer
16 views

Conditional expectation of random variable

I have this home assignment in Introduction to Probability, and I'm not comfortable with definitions and heuristics. I really need someone to check if I'm even in the right direction. The question: ...
2
votes
1answer
29 views

How to prove the sign test

Please correct me if I'm wrong, but a version of the sign test assumes under $H_0$ that there is some distribution $F$ where $X_i \sim F, Y_i \sim F$ and $X_i, Y_i$ are iid. Then it states that $T = ...
0
votes
2answers
11 views

CDF of a Uniform probability density function

I want to find Cumulative distribution function (CDF) of the following density function: $ f(x)= \begin{cases} 3/20 & \text{for } 2 \leq x \leq 4 \\[8pt] 4/20 & \text{for }4 < x \leq ...
1
vote
0answers
13 views

Random Variables and Moment Generating functions

Let $(X_i)_{i∈\Bbb{N^+}}$ be a sequence of i.i.d random variables and for $n ∈ \Bbb{N^+}$ set $S_n := \sum _{i=1}^{n} X_i$ and $Y_n := max(X_1, . . . , X_n)$. Assume that the moment generating ...
1
vote
1answer
12 views

Convergence in probability

I need to prove that given the r.v. Xn with the same distribution functions, the sequence of r.v. Xn/n tends to 0 in probability. Following the definition i find: P(|Xn/n| > a) = P(|Xn| > na) for ...
0
votes
3answers
22 views

Probability of choosing two numbers so they differ by at least 2

A box has $10$ balls numbered $1,2, \dots, 10.$ A ball is picked at random and then a second ball is picked at random from the remaining nine balls. Find the probability that the numbers on the two ...
-5
votes
1answer
79 views

If $E(X)=0$, then $2E(|X|)\le\text{Var}(X)+1$ [closed]

If $E(X)=0$, $E\left(X^2\right)<\infty$, then $$2E(|X|)\le\text{Var}(X)+1.$$
0
votes
1answer
23 views

If mutiplication of probabilities of two events is equal to their intersection,then are the events always independent?

Here is an example , Let a ball be drawn from an urn containing four balls, numbered $1, 2, 3, 4$. Let $E = \{1, 2\}$, $F = \{1, 3\}$ If all four outcomes are assumed equally likely,then we have ...
2
votes
2answers
20 views

What is this conditional probability?

I have been doing some reading for a project on quantitive finance, and I have been seeing a lot of this kind of conditional probabilities on a "$\mathcal{F}_{t_i}$": $$\mathbb{P} ...
0
votes
1answer
321 views

finding unconditional distribution by integrating conditional distribution

Given $$ f_Y (y)= \begin{cases}\frac{1}{120} e^{-\frac{1}{120}y} &, y\ge 0 \\ 0, &, y< 0 \end{cases}$$ and $$f_{X|Y} (x|y) = \begin{cases}\frac{1}{y} &, x\in [0, y] \\0 &, ...
-1
votes
0answers
22 views

Convergence in law and probability

I have a succession of random variables $\{X_n\}$ with $P(X_n=3)=1/n^2$ and $P(X_n=4)=1-1/n^2$. It's defined $Y_n=nX_n$ and i have to prove the convergence almost surely,in law and in probability. I ...
5
votes
1answer
112 views

are elementary symmetric polynomials concave on probability distributions?

Let $S_{n,k}=\sum_{S\subset[n],|S|=k}\prod_{i\in S} x_i$ be the elementary symmetric polynomial of degree $k$ on $n$ variables. Consider this polynomial as a function, in particular a function on ...
1
vote
2answers
252 views

Average distance between two randomly chosen points in unit square (without calculus)

Imagine that you choose two random points within a 1 by 1 square. What is the average distance between those two points? Using a random number generator, I'm getting a value of ~0.521402... can ...
0
votes
0answers
11 views

Maximum likelihood estimator and confidence interval

Let $\theta$ be an unknown constant. Let $W_1,…,W_n$ be independent exponential random variables each with parameter $1$. Let $X_i=θ+W_i$. First, I need to find $\hat\theta _{ML}(x_1,\ldots ,x_ n)$. ...
1
vote
3answers
102 views

Showing independence of a limsup of an independent sequence

Let $\{X_n\}_{n \geq 1}$ be an independent sequence of random variables on $(\Omega, \mathcal{F}, \mathbb{P})$. Fix $n \geq 1$. I want to prove that $X_1, \ldots, X_n$ is independent of $\limsup X_n$. ...
-2
votes
3answers
83 views

Understand the steps in a Summation [closed]

I don't understand the following steps: \begin{align*} \sum_\limits{t=1}^{n-2} (-1)^{n-1} {n \choose t}a\cdot\tfrac{1}{5}\cdot(n - t)(n - 2t - 1) & = a\cdot\tfrac{1}{2}\cdot 2n(2n - 1) ...
-6
votes
2answers
45 views

Chances of this… [on hold]

9 people sat in a circle. They wrote their name on a piece of paper, folded it over and placed it in a hat. The hat was shuffled to mix up the pieces of paper. The first person picked out the name ...
1
vote
1answer
33 views

“Inverse” of nondecreasing, right-continuous function?

Suppose $F : \mathbb{R} \to \mathbb{R}$ is a nondecreasing and right-continuous function. Define $G : [\inf F,\sup F] \to \overline{\mathbb{R}}$ by $G(p)=\inf \{ x : F(x) \geq p \}$, with the ...
1
vote
0answers
18 views

Consider a random walk where $p \neq 1/2$, where the starting point is random and has a binom distn. Find the probability of absorption at $N$.

Consider a random walk $\{0,1, ... , N\}$ with up probability $p$ and down probability of $p-1$ where $p \neq 1/2$. Suppose there are absorbing barriers at $0$ and $N$ and that the starting point, ...
0
votes
3answers
31 views

Probability of a Rare Event Occurring within a Certain Number of Times

I'd like to know how to find the probability of an event occurring, given the probability of that event, within a certain number of chances for it to occur. For example, say that once every year ...
3
votes
1answer
27 views

probability question that just seems to easy to be the case

the game of mastermind starts in the following way: one player selects four pegs, each having six possible colors, places them in a line. the second player then tries to guess the sequence of colors. ...
3
votes
2answers
45 views

What is the intuition of why convergence in distribution does not imply convergence in probability

For me its very counter intuitive (that convergence in Probability and Distribution are not the same), because, conceptually, if two random variables have the same distribution, then they should be ...
0
votes
0answers
33 views

How to minimize the expectation?

Given random variables $X_0, X_1, \ldots, X_n$ with finite expectations $m_0, m_1, \ldots, m_n$ I want to prove that the numbers $a_i = \frac{\det \Lambda_{i0}}{{\det \Lambda_{00}}}$ minimise the ...
0
votes
1answer
26 views

Standard deviation: calculating how polarizing a question is

I'm trying to calculate how polarizing a question is. Let's say I have a question that has 3 possible choices. A certain percentage of people choose a specific answer. Answer a: $30\%$ Answer b: ...
5
votes
1answer
317 views

Probability that a given Poisson variable samples greater than its mean $\lambda$, provided $\lambda > D$

Given a random variable $X \sim \text{Poisson}(\lambda)$ such that $\lambda > D$, with $\lambda, D \in \mathbb{N}$, what is the probability that a sample obtained from $X$ is greater than ...
4
votes
2answers
48 views

Can some probability triple give rise to any probability distribution?

Suppose we have a probability triple $(\Omega,\mathcal{F},P)$ and random variable $X:\Omega\to(\mathbb{R},\mathcal{B})$ with $\mathcal{B}$ denoting the Borel $\sigma$-algebra. Then, the distribution ...
2
votes
1answer
527 views

Expected Value Problem (Q-function…inside a function)

I'm working through my textbook for a communications course I'm taking, and this problem is confusing me big time. Like always, the math questions give me the most problems. Maybe I should take the ...
4
votes
4answers
103 views

Casino turns 50% of your losses into “free play”, are odds in your favor?

As a limited-time promotion, if you gamble during your first week at this casino, and you suffer a net loss of money, the casino will give you half of your losses (up to a certain amount) as "free ...
1
vote
1answer
31 views

Independence of Random Variables

If $X$ and $Y$ are independent random variables so are also the random variables $f(X)$ and $g(Y)$ for $f$ and $g$ measurable and bounded functions. The independence of $X$ and $Y$ implies: ...
1
vote
1answer
28 views

Multiple examination of a result (probability)

A performs a task and submits the result to B and C for examination. B confirms the result. C thinks the result is wrong. The reliability of A is 0.7, for B is 0.8 and that of C is 0.9. (reliablity = ...
0
votes
2answers
53 views

Radon-Nikodym derivative of Measures [on hold]

Im having some trouble reconciling what I thought I learned about RN Derivatives as they relate to probability measures wikipedia,lecture notes with this blog post by John Baez mentioning it as it ...
0
votes
1answer
26 views

What is the probability that on a given day, the number of half gallon containers provided is enough?

In a grocery store 400 customers shop every day. The number of half gallons of nonfat milk bought by a randomly selected customer is a random variable X having P(X=0)=0.3, P(X=1)=0.5, and P(X=2)=0.2. ...
6
votes
2answers
61 views

what is the probability that the circumcircle of 3 point

Mary picks any three non-collinear points inside a given circle, what is the probability that the circumcircle of these 3 points will be covered by the original circle? This is from a test ...
0
votes
0answers
17 views

Separability of the Wasserstein space with respect to $W_2(\cdot,.) +|\phi(\cdot) - \phi(.)|$

I would be thankful, if someone could give me some short proof or reference for the following problem. Given a lower semi-continuous and geodesically convex functional $\phi$ on the Wasserstein ...
-6
votes
0answers
24 views

probability of getting lucky in exam? [on hold]

In an examination, you are given a choice to pick up a chit, which has a question, there are ten of those chits(randomly arranged), only half you have prepared(you know all the question but you're ...
2
votes
1answer
447 views

Sum of Wishart matrices

Considering two matrices, $H_1$ and $H_2$, that are independent of each other and follows complex wishart distributions as $\mathcal{CW} _m(n_1,\Sigma_1)$ and $\mathcal{CW} _m(n_2,\Sigma_2)$ ...
0
votes
1answer
40 views

Check My Work on a Poisson Process/Distribution Question

I'm just curious if my work is correct, and if not, where I made a mistake. My Task: Cars arrive according to a Poisson process with a rate of 12 per hour. (1) What is the probability that the ...
-1
votes
0answers
52 views

Expectation of an interval

Given $g(\theta) := Pr\{X\leq\theta\leq Y\}$ with $Y\geq X$, what is $E[Z]$ where $Z:= Y-X$ ? Also $X{\not\perp}Y$ Progress: $$X\leq\theta\leq Y\Rightarrow \{Z \geq \theta-X\}\cap \{Z\geq\ ...
0
votes
1answer
46 views

Distribution of Bernoulli and Uniform Random Variable

Here's a problem I am stuck on: Let $X$ and $Y$ be independent random variables such that $X$ is Bernoulli-distributed with $p=1/2$, and $Y$ is uniformly distributed on the interval $[0,1]$. Then: ...
1
vote
0answers
16 views

Show that a given sigma field is the smallest one containing the given class of sets

I've been trying to solve the following question from Leo Breiman, Probability but getting stuck in how to proceed and have few doubts as well. Define $\mathcal{B}^{(\infty)}$ as the smallest ...
0
votes
0answers
20 views

Combination of historical and current data in statistics

I have a general question about a statistical matter. Lets assume there exists a true and unique probability $p$ such that an event $X$ happens in the next 12 months. There is some information about ...
4
votes
1answer
34 views

Coin Flips and Hypothesis Tests

Here's a problem I thought of that I don't know how to approach: You have a fair coin that you keep on flipping. After every flip, you perform a hypothesis test based on all coin flips thus far, with ...
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votes
0answers
36 views

Four points inside a rectangle [on hold]

We randomly choose 4 points inside a rectangle.What is the probability that they lie in the same half ?
2
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3answers
33 views

2 restaurants located randomly

any help on following question will be much appreciated. Q. Suppose that $2$ restaurants are going to be located at a street that is $10$ km long. The location of each restaurant is chosen randomly. ...
4
votes
1answer
51 views

Two numbers are chosen at random over the interval $ [0,1]$

Two real numbers, $x$ and $y$ are chosen at random over the interval $ [0,1]$. What is the probability that the closest integer to $\frac{x}{y}$ will be even? Floor functions don't play nicely with ...
0
votes
1answer
13 views

Exercise on iid sequence of uniformly distributed random variables (and LLN).

I'm trying to solve following problem: Let $X_{1}, Y_{1}, X_{2}, Y_{2},\ldots$ - iid, from uniform distribution on $[0,1]$, $f\colon[0,1]\rightarrow[0,1]$ be measurable and $Z_{j} = ...
0
votes
0answers
22 views

Probability of a train journey

A trip from south east London to Southampton consists of three journeys: bus journey to Crystal Palace station, train journey from Crystal Palace to Clapham Junction, train journey from Clapham ...
1
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0answers
20 views

Easy Question from Application: Estimate for transition probabilities of random walk - finding a coupling

SHORT VERSION: Find appropriate Coupling Suppose we have a random walk on the natural numbers, where we go to the left with probability $p_L \geq \frac{1}{6}$, to the right with probability $p_R\leq ...
2
votes
3answers
25 views

Horse racing question probability

Been thinking about this for a while. Horse Campaign length: 10 starts Horse Runs this campaign: 5 Horse will is guaranteed to win 1 in 10 this campaign Question: what is the Probability of ...
2
votes
2answers
42 views

9 room probability and expected value

I got the following question: In a house with 9 rooms. There is 1 mouse that is looking for some food. This can be found in 2 rooms, but there are also 2 cats, these are in different rooms. When the ...