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
26 views

random variable follows

I have a random variable $x_1$ that follows the normal distribution with mean 0 and variance 1 with probability 0.6 and follows the normal distribution with mean 0 and variance 2 with probability 0.4. ...
2
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0answers
74 views

How can I prove it?! [duplicate]

let $X$ and $Y$ be two independent RV with the same distribution. Prove that: $$P\left(|X-Y|\le 2\right)\le 3P\left(|X-Y|\le 1\right)$$ I was wondering if you help me.
1
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1answer
25 views

If $(\xi_k), k \ge 0$ is a sequence of iid Gaussian variables, does it hold a.s. that $\sum \xi_k^2 = +\infty $?

I think that the probability is either 0 or 1 by Kolmogorov 0-1 law.
3
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0answers
68 views

Proving how to reduce a Brownian walk on a plane to a line (2D to 1D)

I have a Brownian motion on a plane and would like to find the time of when it is expected to hit a set of parallel lines, i.e the hitting time. In order to do so, I understand that I can reduce the ...
1
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2answers
61 views

Jar containing 80 balls. 40 red and 40 black.

The jar contains 80 balls. 40 are red and 40 are black. We pick 20 balls out of the jar without putting them back. What is the probability that 10 out of 20 balls will be black and the other 10 will ...
1
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0answers
14 views

Pearson correlation of neural responses with it's linear estimation

I am trying to anderstand the following fact: Suppose I have a linear estimation of a stimulus: $ \hat{s} = \mathbf{w}^T(\mathbf{r} - \mathbf{f}(s_0)) + s_0$ where $\mathbf{w}$ is a vector of ...
0
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0answers
25 views

Most visited vertex in a random walk with a place dependent drift

Consider the following Markov chain on $\mathbb{Z}$: $$ P(x,x+1)=1-P(x,x-1)=\frac{1}{2}+e^{-|x|}\cdot \mathbf{1}_{\{x\neq 0\}} $$ Do there exist constants $c,C>0$ such that $$ c\cdot P^t(z,z) \...
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1answer
43 views

A digit is drawn at random from the digits 0, 1, . . . , 9. [closed]

Let W denote the remainder of dividing the digit by 3. (a) Find the probability density function of W. (b) Find the cumulative distribution function of W. (c) Find the expected value of W.
1
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2answers
55 views

Variation on Vandermonde's identity

How can you show that $$ \binom{2n}{n}^2 = \sum_{m=0}^{n} \binom{2n}{2m} \binom{2m}m \binom{2n-2m}{n-m} $$? I was fooling around with random walks, and apparently both expressions are supposed to be ...
1
vote
1answer
25 views

Expected Number of Turns for a Game of Four Corners

Four Corners is a popular children's game, in which a person who is "it" is blindfolded in a four-corner room, with everyone else attempting to avoid being found. Whoever is "it" spins around, and ...
0
votes
2answers
84 views

Suppose that an urn contains 6 red and 3 green balls. Two are drawn randomly and without replacement. [closed]

The sample space is $S = \{(g_1, g_2),(g_1, r_2),(r_1, g_2),(r_1, r_2)\}$. (a) Fine $\Pr(E_i)$ for every singleton event $E_i ⊂ S$. (b) Let $X$ denote the number of red balls. Find the ...
2
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2answers
78 views

Application of Borel Cantelli Lemma

Let X be a non-negative random variable with finite expected value. Can we use Borel Cantelli Lemma to show that X is finite?
0
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1answer
33 views

probability - factor of a number [closed]

I have been struggling with the following problem. Can anyone help me with it? Thanks. What is the probability that a randomly chosen factor of 1590 is a multiple of 1565 ? Express your answer as a ...
3
votes
2answers
54 views

Card game probability with simultaneous drawing

From a usual deck of 52 cards we draw 10 cards simultaneously (thus no replacement). What is the probability to have 5 diamonds and a queen? My attempt: the cardinality of all 10 card possibilities ...
0
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2answers
18 views

compute probability using binomial distribution

Question: Of the customers purchasing laptops, 75% purchase a Dell model. Let X = number among the next 15 purchasers who select the Dell model. (a) Compute P(X>10), b) Compute P(6<= X <=10). Is ...
3
votes
2answers
86 views

Probability: Store opening time

Smith has a small booth where he sells lottery tickets. Customers arrive according to a Poisson process of rate $\lambda$= 1 per minute. He will close the shop on the 1st occasion that $a$ minutes ...
5
votes
1answer
76 views

Crossing a lane of traffic

A pedestrian wishes to cross a single lane of fast-moving tra c. Suppose the number of vehicles that have passed by time t is a Poisson process of rate , and suppose it takes time a to walk across the ...
1
vote
1answer
51 views

Rejection region

I need help with the following problem: Let H0: p = 0.6 HA: p = 0.7 based on observing a binomial random variable with 10 trials. What is the rejection region for the most powerfil level sigma ...
0
votes
1answer
56 views

how to calculate cdf using binomial distribution

Question: A lab network consisting of $20$ computers was attacked by a computer virus, this virus enters each computer with probability $0.4$, independently of other computers. Find the probability ...
1
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2answers
69 views

Rock Paper Scissor - Probability Game

I heard this recently and it got me thinking. Let's say you and an opponent are playing rock/paper/scissors. You know that your opponent can't play rock. What is the optimal strategy to win? My ...
2
votes
5answers
89 views

Odds of two players meetings in an eight person single elimination tournament

There is an $8$ person tournament. The odds of winning are $50\%$ for each player. What is the probability of any $2$ players meetings at any point in the tournament? I understand that there are $7$ ...
2
votes
1answer
56 views

Does it pay to know what you know?

Let's play a game. I ask you question a yes/no question, and you answer. You don't answer with a yes or no though, you answer with a probability of it being yes ($P \in (0,1)$). For example, I might ...
0
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1answer
19 views

Find probability that a set contains 2 values from an outcome

For any outcome $\omega=(a_1,\dotsc,a_6)$, let $R(\omega)$ be the set $\{a_1,\dotsc,a_6\}$. This is the set of numbers that showed up at least once in the different rolls. For example, if $\omega=...
1
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1answer
22 views

Showing $X_n\rightarrow_P X$ and $X_n\rightarrow_P Y$ $\implies$ $P(X=Y)=1$

A sequence of random variables $\left\{ X_n \right\}$ $\textbf{converges in probability}$ to a random variable $X$, denoted by $X_n\rightarrow_P X$, if for every $\epsilon>0$, $$ P(|X-X_n|\geq \...
0
votes
1answer
26 views

Perfect secrecy of affine cipher

How can I show that the affine cipher has perfect secrecy if the key $(k,a)$, where $k\in\{1,3,5,7,9,11,15,17,19,21,23,25\}$? I know to show perfect secrecy I need to show that $Pr(Y)=\sum_{e_k(x)=y}...
2
votes
1answer
53 views

Choosing optimal strategy for real number assignment game

My opponent is dealt a real number $r\in [0, 1]$ uniformly at random and I know that with probability $1-r$ she chooses to discard that $r$ and be dealt a new number in $[0, 1]$ uniformly at random ...
2
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0answers
33 views

Conditional expectation on different probability measure

A random variable Y is defined on probability space $(\Omega , F,P)$. A new probability measure is defined $P_o$ on $(\Omega , F)$ by $P_o:=\int_A YdP $ .Take another random variable X on this new ...
2
votes
0answers
23 views

Probability distribution obtained by repeatedly sampling $S_x,S_y$ on a spin-$S$ system

While trying to rework an upcoming quiz problem for a quantum physics course, I came up with the following question which turned out to be harder than I expected. (Note: I take $\hbar =1$ in all ...
1
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1answer
304 views

Gambler's Ruin and Markov Chains

Suppose that on each play of a certain game, a person will either win one dollar with the probability of $\frac{2}{3}$ or lose one dollar with probability $\frac{1}{3}$. Suppose also that the person's ...
2
votes
1answer
35 views

Combine discrete uniform distributions to achieve a discrete uniform distribution of a larger range?

How can I effectively combine multiple discrete uniform distributions of a limited range to achieve a discrete uniform distribution of a large range? I.e. given $unif\{a,b\}$ generate $unif\{a,c\}, c&...
1
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1answer
51 views

How to become very good in probability and combinatory [closed]

How someone become very good in probability and combinatory? Is there is a set of methods used to solve problems in this field? If you can suggest any resources or tips to improve myself in this ...
1
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2answers
60 views

Approximating statistics for huge dataset

I'm investigating users accounts statistics for Vkontakte social network. There are $N\approx2 \cdot 10^8$ accounts that have different metrics along them – boolean, discrete and continuous. I found ...
1
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1answer
60 views

Markov DTMC and CTMC: How to build a $Q$ generating matrix using given $2$ state $q$ probabilities? Find Steady State Probabilities

I know that $$q_{(i,j)(i+1,j)}=\lambda_1$$ $$q_{(i,j)(i-1,j+1)}=\lambda_2$$ $$q_{(i,j)(i+1,j-1)}=0.5\lambda_3$$ $$q_{(i,j)(i,j-1)}=0.5\lambda_3$$ where $\lambda_1 , \lambda_2 , \lambda_3 $ are rates ...
0
votes
1answer
42 views

Conditional probability density

I have two independent Gaussian random variables $X$ and $Z$. Let $G = X-Z$, then $G$ is Gaussian with parameters $\mu_G= \mu_X - \mu_Z$ and $\sigma_G^2 = \sigma_X^2 + \sigma_Z^2$. I know that $Z=f(...
0
votes
1answer
39 views

Does $E[p_X(X)]$ have any significance? [closed]

I always find it very interesting when I find out stuff that looks superficially insignificant, or even silly, turns out to have deep implications in unexpected fields of mathematics. So I wondered ...
2
votes
1answer
48 views

Basic probability. Is the textbook wrong?

A and B are independent events such that P(A)=0.7, P(B)=k, P(A U B)=0.8 Find the value of k. Solution given: P(A U B) = P(A) + P(B) - P(A ∩ B).....(i) [Addition Rule] P(A ∩ B) = P(A).P(B).....(...
0
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0answers
149 views

Statistics hotel overbooking question

The question was: Suppose that a popular hotel for vacationers in Orlando, Florida, has a total of 300 identical rooms. As many major airline companies do, this hotel has adopted an overbooking ...
1
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1answer
43 views

Homogenous vs. Non-Homogenous Poisson Processes

I understand that at the main difference between a homogenous vs. non-homogenous Poisson process is that a homogenous Poisson process has a constant rate parameter $\lambda$ while a non-homogenous ...
0
votes
2answers
33 views

Find the moment generating function, mean, and variance of the piecewise function

Find the moment generating function, mean, and variance. $$F(x) = \begin{cases} {1 \over \phi} e^{-x/\phi}, & 0 \le x < \infty \\ 0, & x < 0\end{cases} $$ I'm just a little confused ...
0
votes
1answer
17 views

Discrete uniform RV proof

There are $10$ numbered toys in a box: $3$ trucks (numbered as $1$, $2$ and $3$), $1$ doll (numbered as $4$), $2$ cars (numbered as $5$ and $6$), $2$ cubes (numbered as $7$ and $8$) and $2$ balls (...
2
votes
2answers
72 views

Cutting a stick twice

Consider three situations - $1)$ A stick, placed at $[0,1]$ is cut at a point $x$ given by R.V $X$ uniformly distributed in $[0,1]$. Another cut is made in $[0,x]$ given by R.V $Y$, uniform in the ...
3
votes
1answer
37 views

Let $X_n$ be a one-dimensional random walk with $p \neq 0.5$. Show that $P(\lim_{n \rightarrow \infty} \frac{1}{n} X_n = 0)$ equals $0$.

Let $X_n$ be a one-dimensional random walk on the integers that starts at $0$ (as normally) but has $p \neq \frac{1}{2}$, i.e. so that it moves to its right neighbor with probability $p$ and left ...
1
vote
2answers
45 views

We flip a coin until a tail or five heads in a row occur. What is the number of expected flips?

We flip a coin until a taild or five heads in a row occur. What is the number of expected flips? I have tried to solve this by first defining 2 random variables: X ...
0
votes
0answers
27 views

Geometric Probability Function on real Dataset

Can anyone explain how to apply geometric distribution on a real dataset? I know there's a formula but I need to estimate the prob of success first. My problem is I want to build a model using such ...
0
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0answers
15 views

How to validate an error model for function approximation?

I have a model that i would like to experimentally validate. It is something like: $$\epsilon = \epsilon(q_1,...,q_n) = \epsilon(\vec{q})$$ This model describes the trend of the error bound of a ...
1
vote
1answer
25 views

Central Limit Theorem for Multinomial Trials to prove weak convergence to Brownian Bridge

I'm trying to demonstrate that if we define the empirical process by $X_t^n=\sqrt n (F_n(t)-t)$, where $$F_n(t)=\frac{1}{n}\sum_{i=1}^nI_{\xi_i\leq t},$$ and $\xi_i$ are independent uniform random ...
0
votes
0answers
40 views

renewal processes

Let $X_1, X_2,... $ be a discrete renewal process, in which $X_i$ denotes the time in between renewals with distribution: $Pr(X_i=1)=p$ and $Pr(X_i=2)=q=1-p. $ I want to show that the renewal ...
0
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0answers
27 views

Simplifying a probability distribution function using an exponential function

I have a pdf for a variable $r$ given two other variables $m, \kappa$ defined as follows: \begin{align} p(r|m,\kappa)=\frac{I_0(\kappa r)}{I_0(\kappa)^m}r\psi_m(r), \end{align} where $\psi_m(r)$ is ...
1
vote
0answers
35 views

Measure of the range of a function

Suppose that given a function $G:\mathbb{R}^m\times \mathbb{R}^n\to\mathbb{R}^d$, the goal is to compute or bound the probability measure of its range. Explicitly, I need to compute $$\int_{G(\mathbb{...
0
votes
2answers
56 views

Assume that $10$ students were randomly selected from this class. Find the probability that $2$ are $1$st year and $5$ are $3$rd year.

In a large university statistics class, it is known that $10\%$ are $1$st year, $25\%$ are $2$nd year, $40\%$ are $3$rd year, and $25\%$ are $4$th and special student. Assume that $10$ students were ...