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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7 views

Probablity: Is my way of thinking correct?

Problem Consider the model such as: The computer has not infected with any virus in the initial state. Every morning, the computer has infected with an new virus with a probability of $p$ ($0 < ...
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0answers
13 views

probability class 12

Three groups of children contain 3 girl and 1 boy;2 girls and 2 boys;and 1 girl and 3 boys. One child is selected at random from each group.find the chance that the three children selected comprise 1 ...
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0answers
14 views

Find MGF of random variable X

We are given rth raw moment i.e. $E(X^r)=(r+1)!* (2^r)$. We have to find MGF of random variable $X$.
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0answers
21 views

When calculating joint probabilities using double integrals…

When calculating joint probabilities using double integrals, do we use $dx dy$ or $dydx$ ? I thought it was the former, but then my book abruptly changes to using $dydx$ without an explanation ...
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0answers
11 views

Complement of Conditional Probability

I'm currently reading this paper Censored Exploration and the Dark Pool Problem and have difficulties in understanding the following simple equality: Let $S$ be a positive integer random variable. ...
0
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1answer
21 views

basic notions of measure theory: differences?

Could you help me differentiating the following notions of measure theory: law, probability, probability density, probability measure, probability distribution, distribution, distribution function. ...
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1answer
19 views

conditonal probability notation

Can someone shed some light on the conditional probabilities of P(A∪B|C) and P(A∩B|C) and how they can be performed? I've search many places but I might be confusing the two. (Also, Pr(C)>0) I know ...
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0answers
20 views

Interchange Order of Integrals

Can someone explain the last step in this process. Specifically, how do you get the new limits of integration? Expected Value Definition: $E[Y] = \int_0^\infty{P\{Y \ge y\} \, dy}$ Expand: $E[Y] = ...
2
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2answers
41 views

Probability of a group of people voting yes or no

I am in need of some explanation as for whatever reason I just can't wrap my head around a problem. The question basically breaks down like this: There are $8$ people on a jury ($3$ men and $5$ ...
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1answer
15 views

Characteristic Function and Convergence in Distribution of Sequence of R.V.

I am trying to solve the following: Let $X_1,X_2,...$ be a sequence of random variables with $P(X_n=\frac{k}{n})=\frac{1}{n}, k=0,1,2,...,n$. Find the characteristic function of $X_n$ and show that ...
1
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0answers
28 views

Mean-Square Ergodicity of Certain Quantities?

I apologize in advance for my lack of mathematical knowledge, especially in the field of stochastic processes, but I will try my best to formulate my question in a mathematical way. Is it possible ...
1
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1answer
25 views

find the expected error value

I want to calculate the expected error value in an n-bit number with the probability of bit flip $P_{bit}$. I will explain the calculation for a simple case in which two least significant bits might ...
1
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1answer
60 views

Solve c value in $c \cdot (x+2y) \cdot e^{x+y} $

Today I started to look at previous exam questions, but I can't figure out the solution of one the questions. I hope someone could help me. In this question I have to find the c value: $$ ...
2
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0answers
22 views

Stable distributions and equivalence of certain definitions

There are several definitions of stable distributions. The most ubiquitous is arguably that if $X, X_1, X_2, \ldots $ are i.i.d. random variables with probability distribution $F $ then, $F $ is ...
0
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1answer
34 views

Basic question concerning conditional expectation (of a non-mathematician)

Let $(X_i)_{i \geq 1}$ and $\tau \geq 1$ be independent random variables with $\mathbb{E}[X_i]=\mu$ for all $i \geq 1$. Moreover, let $S_k:= \sum_{i=1}^k X_i$. I want to show that ...
0
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0answers
28 views

Why is $\sum_{k=1}^{\infty}\mathbb{E}[\mathbb{1}(T=k)]=\sum_{k=0}^{\infty} k \mathbb{P}[T=k]$

Let $T$ be a non-negative random variable. Why is it true that $$\sum_{k=1}^{\infty}\mathbb{E}[\mathbb{1}(T=k)]=\sum_{k=0}^{\infty} k \mathbb{P}[T=k]$$ According to me it would make sense that ...
0
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0answers
9 views

Non-Linear System. Find the conditional expectation.

I've had my test for this course and I think I failed it again. The hardest part for me is findig the correct distributions. This is a test exercise I couldn't figure out or at least, I probably ...
1
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1answer
36 views

Trouble finding the expected value of a random variable

Suppose that we have a procedure A that we run once and it returns as a result either success or ...
0
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0answers
12 views

process stochastics and branching process [duplicate]

Consider a discrete time branching process $X_{n}$ with $X_{0}=1.$ Establish the simple inequality $$P\{X_{n}>L\ \textrm{for some}\ 0\leq n\leq m\ |\ X_{m}=0 \}\leq [P\{X_{m}=0\}]^L$$ Note: This ...
1
vote
1answer
27 views

Deriving Probability Density Function from Probability Generating Function for Random Sum

I am trying to solve the following: Let $X_{i}$~$Geometric(q) i=1,2,...,N$ with $q=1-p, 0<p<1$. $N$~$Geometric(p)$. Define $Y=\sum_{i=1}^{N}X_i$ and assume each $X_i$ is i.i.d. and ...
1
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1answer
22 views

On the equality $p(A) = \int_{x} P(A|X=x)\ dF(x)$ in probability

I am trying to learn some probability, and I was reading something that I believe boils down to the following. Let $A$ be some event in a probability space, and let $X$ be a random variable with ...
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0answers
21 views

Clarifying Derivation of Entropy

I'm learning about probability from the book Pattern Recognition and Machine Learning by Christopher Bishop. It includes a justification for the definition of entropy that can be summarized as: let ...
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1answer
28 views

$E(f(|X_n|))$ property implies uniform integrability?

This is exercise 6.10 in Resnick's book "A Probability Path". We're given a sequence of random variables $(X_n)$ and an increasing function $f: [0, \infty) \rightarrow [0, \infty)$ such that $$ ...
0
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0answers
7 views

discontinuous Gaussian field

I am trying to build an example of a discontinuous Gaussian field. The simplest I could come up with is the following: Let $Y,Z$ be two independent brownian motions on $[0,1]$, and $T$ a uniform ...
2
votes
2answers
41 views

How to find out the probability of an event about which we have two informations

I would like to know how to find out the probability of an event about which we have two informations. Say we have $A$ and we know it is lower than $K$ but greater than $X$. How do you find the result ...
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0answers
18 views

Local martingale and integral condition

Suppose $M^i_t = X^i_t - X^i_0 - \int_0^t b_i(s,X)\, ds$ where $b_i:[0,\infty)\times \Omega \to \mathbb{R}$ is a progressively measurable functional and $X^i_t: C[0,\infty)^d \to \mathbb{R}$ ( ...
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votes
1answer
21 views

Find probability mass function from text

$5$ persons (each independent of the other) when in a good mood it opens the tap with probability $\frac{1}{2}$ or in a bad mood with probability $\frac{1}{2}$. When that person is in a good mood it ...
1
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2answers
34 views

Number of vectors over a finite field that are linearily independent to a subspace

let $S$ be a vector space over a finite field of size $q$ and let $T$ be a subspace of $S$. I am looking for a formula or an algorithm to compute the number of vectors from $S$ that are independent ...
2
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0answers
59 views

Why is this the solution?

I have this exercise with the solution. Let $X\sim N(0,1)$. Show that $P(2X = 3Y + 1) = 0$ if $Y\sim \text{Poisson}(\lambda)$. I have this solution $P(2X=3Y+1)= P(\bigcup_{k=0}^{\infty}(2X = 3Y+1, ...
3
votes
1answer
47 views

Conditional probability branching process

Consider a discrete time branching process $X_{n}$ with $X_{0}=1.$ Establish the simple inequality $$P\{X_{n}>L\ \textrm{for some}\ 0\leq n\leq m\ |\ X_{m}=0 \}\leq [P\{X_{m}=0\}]^L$$ Note: This ...
0
votes
1answer
30 views

Joint probability density for independent variables

Let $X_1$ and $X_2$ be two independent random variables each with probability density function $fX_i(x_i) = e^{-x_i}$, for $x_i > 0$ for $i = 1,2$ (a) Find the joint probability density function ...
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0answers
31 views

Not understanding the results of standard deviation

There are two scores from one sample from two measurements: $A_1=15.4$ and $A_2=16.6$ the standard deviation (in the range +/-1 $\sigma$) for the first ($A_1$) is 1.91, and for the second ($A_2$) is ...
0
votes
1answer
45 views

A prob =a random variable

When I read some proofs, some authors conclude that $P(A)=I_{A}$, where $A$ is an event and $I$ is the indicator function. They mean that $P(A)$ can take either $0$ or $1$. However, I do not ...
0
votes
2answers
36 views

How to find E(Y) given that the random variable X is exponentially distributed with lambda equal to 0.5?

Random variable $X$ is exponentially distributed with the parameter $\lambda$ equal to $0.5$. Define also $Y = 1 - 2X$ Find $E(Y)$ , Var(Y) and the moment generating function of Y. I have $f_x(X)= ...
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0answers
33 views

Math test: Probability [on hold]

I have participated into a university course: Basics of statistics and probability. Online test with one specific question is giving headache: "4 out of 8 servers are required to provide cloud ...
2
votes
1answer
42 views

Convergence of a sequence

Let $X$ be a random variable with a distribution function such that $n^t P(|X|>n) \to 0$ as $n \to \infty$, for some $t>0$. Then, I know that for any $\epsilon>0$, there exists $n_0\in ...
0
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1answer
39 views

Showing Convergence in Distribution for Conditional Random Variable

I am trying to prove the following: Let $X$ and $Y$ be random variables such that $Y | X = x$ ~ $N(0, x)$ with $X$ ~ $Po(\lambda$). Show that $\frac{Y}{\sqrt{\lambda}} \to N(0,1)$ in distribution as ...
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0answers
25 views

Can someone show me how to answer this question? please [on hold]

Your socks drawer contains 10 pairs of white socks and 10 pairs of black socks. Suppose you are in the dark and need to get one sock at a time, without knowing which one. How many socks do you need to ...
0
votes
1answer
57 views

How many tickets sold so that $2$ people won the first prize in a lottery?

In some lottery one can buy a ticket by choosing seven distinct numbers each of them from numbers ${\{1, 2, \dots, 45}\}$ (so $1/(45379620)$ is the probability to win the first prize). Every week the ...
3
votes
1answer
37 views

Computing Conditional Characteristic Function

I am trying to compute the characteristic function of the following: Let $X$ and $Y$ be random variables such that $Y\mid X = x\sim N(0, x)$ with $X\sim\mathrm{Po}(\lambda)$. Find the characteristic ...
3
votes
1answer
42 views

Probability of True Positive of a random variable defined by an integral expression

$\newcommand{\Prob}{\operatorname{Prob}}$Let's assume that we have a random variable with the following pdf: \begin{equation} f_T(x) = \int_0^\infty f_T(x,g) \cdot f_{g}(g) \, dg = \int_0^\infty ...
4
votes
1answer
48 views

Probability of consecutive floors on an elevator with more people

Another user posted this question about elevator occupants, which made me curious about a harder question. In a $t$-story building (with no basement), $n$ people get on an elevator on the first ...
0
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1answer
23 views

Elementary probability question involving a 4-sided dice rolled twice

I'm beginning some probability courses so please explain your reasoning as if I were stupid. We have a 4-sided dice. Our experiment consists of rolling the dice twice: Let event $A = \{$maximum of ...
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0answers
104 views

Find measure such that…

I've a very concrete problem I can't solve. Consider the following function $k: [0,1]^2 \to \mathbb{R}:$ $$ k(x,y)=\begin{cases} 1 &\text{if } y > x \\ -1 &\text{if } x- \frac{1}{2} < ...
0
votes
2answers
110 views

Probability that after 10,000 steps (+-1) you'll end up at the origin. How to use Central Limit Theorem?

Starting at the origin and taking one step left or right with equal probability, what is the probability that you'll end up at 0 after 10,000 steps? I figured it'd be ...
3
votes
2answers
59 views

Probability that all colors are chosen

A box contains $5$ white, $4$ red, and $8$ blue balls. You randomly select $6$ balls, without replacement, what is the probability that all three colours are present. Most similar problems ask for ...
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votes
1answer
40 views

Let X and Y be two random variables with joint probability density function [on hold]

$f(x,y) = k(1+xy)$, $0<x<1$ and $0<y<1 $ (a) Find the value of $k$ such that $f(x,y)$ is a valid joint probability mass function. (b) Are $X$ and $Y$ independent? Justify your answer. ...
1
vote
1answer
127 views

$X_1$, $X_2$ i.i.d., prove that $E(X_1\mid X_1+X_2) = E(X_2\mid X_1+X_2)$

I got to the point where I only need to prove that for every $h$ Borel, $$E( h(X_1+X_2) (X_1-X_2) ) = 0$$ This is obvious when $h$ is the identity function, but I don't know what to do. Thanks!
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1answer
36 views

Find the expected value of a game [on hold]

It costs $\$10$ to play a game. You have a $15\%$ chance of winning. You collect $\$30$ if you win. Otherwise you lose your $\$10$. Find the expected value.
2
votes
1answer
37 views

How can I find the density of $E[X\mid Y]$ when $(X,Y)$ is gaussian

I was tying to prove the following: Given $(X,Y)$ a centered gaussian vector in $\mathbb{R}^2$ with the following covariance matrix $$ \Sigma = \begin{bmatrix} \sigma^2_x & \sigma_{x,y} \\ ...