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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3answers
34 views

probability of exactly one out of N events occuring

I have N events. Each "i" event has probability $P_i$. What is the probability of $n$ events occuring? I have seen this answered for two and three events, but not for an arbitrary N. In principle, ...
1
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1answer
34 views

Roulette with p=$\frac{2}{3}$. What is the probability of not going home?

I'm learning about the gamblers ruin. The problem is that I don't know how to calclate the formula. I got two exercise questions in my book. Both of the questions will be about a strange roulette ...
1
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1answer
24 views

Can any function of the second moment of a random variable be recovered from its quantile function?

Summary of question It is known that the expected value of a random variable can be obtained from integrating its survival function. This is easily restated in terms of the quantile function as: $$ ...
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0answers
13 views

Functions to manipulate (increase) probability exponentially or logaritmically?

Very simple. I want a function to manipulate a probability in order increase it without getting out of the range of 0 to 1. Basically a function similar to the blue lines in the following sketch: ...
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1answer
20 views

Condition of reversibility of Markov Chain [on hold]

Show that a Markov Chain is time reversible iff $\pi _{i} P_{ij}= \pi _{j} P_{j i}$
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1answer
25 views

Why do all steady state probabilities have the same denominator?

I have noted that the steady state probabilities of an irreducible Markov chain can be written as fractions that have the same denominator. Is there any result about this property? What does this ...
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1answer
44 views

Probability of winning consecutively [on hold]

India and USA play $7$ football matches. No match ends in a draw. Both the countries are of same strength. Find the probability that India wins at least $3$ consecutive matches.
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2answers
59 views

An exam`s points dilemma

On July 2 I have an exam, in this exam will be 40 questions in test with 5 variants of answer for each question. For each correct answer will be given +1 point. For each incorrect answer will be ...
0
votes
1answer
41 views

I don't understand how I can calculate this

"Let $X_1, X_2, ...$ independent random variables. $X_n\sim B(p_n)$ and $p_n = \frac{1}{n}$. Calculate $P(lim sup (X_n=0, X_ {n+1}=1,X_ {n+2}=0 ))$" I suppose that i can use the lemma of ...
1
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1answer
24 views

Convolution with one of the variables is mixed and the other continuous

Suppose $X$ and $Y$ are independent random variables with CDF $F$ and $G$ and nonnegative support. If $X$ has a point mass $p$ at $0$ and otherwise some "density" $f$ (that is, ...
1
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2answers
18 views

Given E(X) and Var(X) find the Expectation of $E[x-2(X-1)^2]$

Let X be a r.v. with $E(X) = 5$ and $Var(X) = 30$. Find $E[X-2(X-1)^2]$. I'm not sure as to how to approach this problem, any tips on how to approach it would be appreciated!
1
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0answers
18 views

limiting distribution of a function of joint normals

Let $Z_n=(X_{1,n},X_{2,n})\sim N(\mu,\Sigma_n)$ where $\mu=(0,0)'$ and $$\Sigma_n=\begin{bmatrix}a^2+\frac1n & ab \\ab & b^2+\frac1n\end{bmatrix}$$ Then where does ...
-1
votes
1answer
50 views

What are the odds of getting certain results on a six dice throw? [on hold]

You roll six dice and you can bet on the following: $$ \begin{array}{l} X, X, ?, ?, ?, ? \\ X, X, Y, Y, ?, ? \\ X, X, Y, Y, Z, Z \\ X, X, X, ?, ?, ? \\ X, X, X, Y, Y, ? \\ X, X, X, Y, Y, Y \\ X, X, X, ...
0
votes
0answers
19 views

Question about the conditional value-at-risk

I have a question about CVaR (Expected Shortfall) An investment who gives a certain amount of cash with a certain probability : A loss of $20$ millions with a probability of $0.0016$ A loss of $11$ ...
2
votes
1answer
54 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 < ...
-2
votes
0answers
22 views

probability class 12 [on hold]

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 ...
-1
votes
1answer
41 views

Find MGF of random variable X [on hold]

We are given rth raw moment i.e. $E(X^r)=(r+1)!* (2^r)$. We have to find MGF of random variable $X$. so what is the simplest way to solve this problem.
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1answer
38 views

When calculating joint probabilities using double integrals…

When calculating joint probabilities using double integrals, do we use $dx\ dy$ or $dy\ dx$ ? I thought it was the former, but then my book abruptly changes to using $dy\ dx$ without an explanation ...
0
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1answer
24 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
votes
1answer
22 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. ...
0
votes
1answer
32 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 ...
0
votes
1answer
30 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
votes
2answers
44 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 $5$ ...
0
votes
1answer
29 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
35 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
28 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 ...
2
votes
1answer
65 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
votes
0answers
24 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
votes
1answer
43 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
votes
0answers
31 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 ...
2
votes
1answer
31 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
39 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
14 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
35 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
29 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
23 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
29 views

$E(f(|X_n|))$ property implies uniform integrability? [on hold]

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
8 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
1answer
61 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 ...
0
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0answers
22 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
23 views

Find probability mass function from text [on hold]

$5$ 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
vote
2answers
38 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
64 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
57 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
35 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 ...
-1
votes
0answers
33 views

Not understanding the results of standard deviation [on hold]

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
46 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
37 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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votes
0answers
37 views

Math test: Probability [closed]

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
48 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 ...