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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2answers
22 views

In a box which has balls numbered 1..100 , 5 balls are drawn.

$X$- random variable that represents the largest number of the 5 drawn. Find the distribution of $X$. Now, it seems that this random variable is of discrete type. What I have trouble it defining it ...
0
votes
1answer
34 views

We write down the date of each person's birthday we meet (say Feb 29. doesn't exist).

Random Variable $X$ is the number on persons we met til we wrote down every date in a year. Find the expected value of $X$. Find $E(X)$- expected value. From this example I can definitely understand ...
-1
votes
3answers
35 views

probability about coins [on hold]

A gambler has two coins in his pocket, a fair coin and a two-headed coin. He picks one at random from his pocket, flips it and gets heads. What is the probability that he flipped the fair coin?
2
votes
1answer
36 views

Equation with mean of random variables

In a proof I found the following conversion $$E\left[|X|\mathbf{1}_{[a,b]}(Y)\right] = E\left[|X|P(a \le Y \le b)\right]$$ I understand, why $E\left[\mathbf{1}_{[a,b]}(Y)\right] = P(a \le Y \le b)$, ...
1
vote
2answers
29 views

Distances to the center of points uniformly distributed in a disk

We choose $n$ points at random from the surface of disk of radius $1$ (the points are chosen with equal probability). If we omit the point furthest from the center (from $n$ points), what is the ...
0
votes
0answers
10 views

Poisson Distribution Optimization Problem

A retailer buys $n$ cookies and has to pay $\zeta_1$ for each. He wants to sell them for a price of $\zeta_2$ (with $0$ < $\zeta_1$ < $\zeta_2$). Let X be a random variable which states, how ...
-1
votes
1answer
17 views

Find expected value of random process [on hold]

I found a problem, which I can't solve: Let's say that $\tau \sim Unif(0,1)$ distribution. Suppose that $X_t=(1\!\!1_{[0,\tau]}(t))^2, t\in[0,1]$. What is the $EX_t$ and $var(X_t)$? I don't know ...
-2
votes
0answers
205 views

Pareto distribution,fisher information, confidence interval [on hold]

Having a bit of problem at these questions, greatly appreciated if anyone can solve them. For the notation, k^ is k with a hat on top of it, don't know how to do that on a keyboard. The rest is ...
-1
votes
1answer
24 views

Need help with probability homework [on hold]

Alright so I could use some help with my homework, thank you in advance! Plura goes to the gym 15% of the days of the year. Carla goes to the gym 20% of the days of the year. a) what is the ...
2
votes
0answers
32 views

Expectation problems in probability.

Dan tosses a coin $n$ times independently, while the probability for a unique tail is $1\over 3$. For $1\le k\le n$, let us denote the number of sub-sequence of length $k$ of H's. For example, if n=5 ...
0
votes
2answers
27 views

During a night, each chameleon changes its colour to one of the other four colours with equal probability.

Five chameleons of all different colours meet one evening. During the night, each chameleon changes its colour to one of the other four colours with equal probability. Find the probability that the ...
0
votes
2answers
63 views

Counting problem: ways of opening stores in non-adjacent blocks?

A coffee company wants to set up stores along the main street of town, which has $n$ blocks. The company won’t open two stores in the same block, or in two adjacent blocks. Q: For this coffee shop, ...
1
vote
2answers
41 views

how to solve this conditional probability

Manufacture A and B produce one type of electrical element, given that the probability of produced element being faulty is $0.05$ for A and $0.01$ for B. If two of these elements has been picked, from ...
3
votes
1answer
33 views

Ito isometry for bounded Ito integral

Let $(W_t)_{t \in [0, T]}$ be a Brownian motion and $T$ be a finite time. If $\int^T_0 \beta_t d W_t$ is bounded and $\{ \beta_t \}_{t \in [0,T]}$ is locally integrable, I am curious whether the ...
2
votes
1answer
27 views

Probability of co-occurence

Of total $N$ people, $m$ people are good at mathematics and $c$ people are good at computer science. What is the expected number of people good at both mathematics and computer science? Or what is the ...
0
votes
1answer
28 views

proof of the convergence of confidence intervals

The confidence interval can be derived intuitively by replacing the standardized mean with the standard normal and variance with sample variance, but is there a formal limit? I'm trying to prove if ...
4
votes
4answers
181 views

Find the probability that the final score is 4 in a dice game with two throws

A game uses an unbiased die with faces numbered 1 to 6. The die is thrown once. If it shows 4 or 5 or 6 then this number is the final score. If it shows 1 or 2 or 3 then die is thrown again and the ...
1
vote
1answer
41 views

Probabilities: Meeting people

The probability of women meeting a man is $m$. Let's look at the perspective of a specific man. The probability of meeting him is $\tilde m$. Say women look twice for men. Then (assuming $\tilde m$ ...
2
votes
1answer
26 views

Convergence of random variables under different probability measures

I have a succession of random variables $X_n$ on $\Omega=[0,1]$ with $X_n=(1-\omega)^n$. I have to prove the convergence almost sure and/or in law in these case: $\mathbb P=\delta_{0}$ $\mathbb ...
2
votes
1answer
40 views

Markov Chain Detailed Balance property

I am having a hard time to understand the concept of the detailed balance; mostly because of the intermingled notation most of the resources use; which involves constant usage of random and state ...
-1
votes
1answer
25 views

Derivative of a CDF [on hold]

Suppose that $X$ is a random variable whose mean is $m$. I need to show that $\frac{\partial}{\partial m} \text{Prob}\{X\geq x\} >0$. Intuitively, increasing the mean I'm shifting probability ...
3
votes
3answers
30 views

Property of cumulative distribution function

I was taking the course on random variables , where I faced below property of cumulative distribution function: $$\lim_{x\rightarrow a^+}F_X(x)=F_X(a^+)=F_X(a)\qquad\qquad ...
-4
votes
0answers
22 views

Prove: $E[E(X|X+Y))=E(X) ; E[X|X+Y] = n/(n+m)*(X+Y)$ [duplicate]

E[E(X|X+Y)]=E(X) is diferent of E[X|X+Y]. And in E[X|X+Y] I give the final result that is "n/(n+m)(X+Y)", and I am asking to demonstrate that E[X|X+Y] = n/(n+m)(X+Y). Let X and Y be independent and ...
1
vote
1answer
32 views

Probability distribution for a geometric distribution don't add up to 1

Say I'm rolling 2 dies,numbered 1 to 10. A successful outcome is considered rolling a multiple of 4. Therefore,probability of success=0.25 and prob of failure=0.75. This is an example of a geometric ...
6
votes
0answers
46 views

Probabilistic interpretation for representation of unity using the zeta function

There's a cute identity, I believe due to Borwein, Bradley and Crandall (Section 4): $$1=\sum_{n=2}^\infty (\zeta(n)-1).$$ There are some generalizations in the linked paper as well. Question: Is ...
0
votes
2answers
30 views

In a box there are $M_1$ balls numbered 1, $M_2$ numbered 2… $M_N$.

In a box there are $M_1$ balls numbered 1, $M_2$ numbered 2... $M_N$. From the box $n$ balls are drawn without returns. Find the mathematical expectation of the number of numbers that are not drawn. ...
0
votes
1answer
28 views

In a box there are 3 white, two black balls. Players A,B,C one by one draw balls from the box. Find the probability of winning each player.

White balls are returned to the box, black balls are kept. A player has won when he has drawn the last ball(black). And if $X$ is the random variable the represents the number of draws , find the ...
4
votes
2answers
51 views

Probability when cutting the stick twice

Given a stick of length $l$. We cut the stick twice. Let $X$ be the random variable defined by the length of the stick after the first cut, and $Y$ be the random variable defined by the length of ...
1
vote
1answer
36 views

What is the probability of arrive either A or B at starting point K?

There are two points which are $A$ and $B$. The distance between $A$ and $B$ is $50$ meter. One person goes to $A$ with probability $\frac{1}{6}$, he goes to $B$ with probability $\frac{3}{6}$. And he ...
1
vote
1answer
29 views

Coupon Collector's Problem — Expected Value of each item

So I guess my problem is based on the famous coupon collector's problem, which is, if you should not be familiar with it, the following: Given N different coupons from which coupons are being drawn ...
0
votes
0answers
39 views

Mastermind Probability Distribution [on hold]

I was thinking of the game MasterMind, a game with 6 different colored pegs and a 4 part code using those colors. There can be more than one peg of the same color in the code. For example ...
1
vote
1answer
39 views

Calculating complicated expectation

I need to calculate $\operatorname{E}( X_2 \mid X_1=x, Y=y)$, where $Y=\max\{X_2,X_3\}$ and joint density of $X_1$, $X_2$ and $X_3$ is given by: ...
1
vote
0answers
26 views

Is the almost surely limit of measurable functions measurable in probability spaces?

Suppose we have $(\Omega,\mathcal{F},\mathbb{P})$ and $\mathcal{F}_n$ a sub $\sigma$-algebra of $\mathcal{F}$. Let $(X_n)_{n=1}^\infty$ be a sequence of $\mathcal{F}_n$-measurable functions converging ...
2
votes
1answer
15 views

Box with balls of different colours. Probability of finding a specific colour.

A box has $10$ red balls and $5$ black balls. A ball is selected from the box. If the ball is red, it is returned to the box. If the ball is black, it and $2$ additional black balls are added to the ...
3
votes
1answer
166 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
15 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
2answers
20 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 ...
0
votes
3answers
43 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 ...
0
votes
1answer
26 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
26 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
2answers
30 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: ...
0
votes
1answer
54 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
2answers
299 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 ...
-6
votes
2answers
54 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 ...
3
votes
1answer
41 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 = ...
3
votes
1answer
30 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. ...
0
votes
3answers
36 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 ...
-1
votes
0answers
31 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 ...
1
vote
0answers
61 views
+50

Probability of absorption of a biased random walk when the starting point has binomial distribution

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, ...
1
vote
1answer
81 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 ...