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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16 views

Let $X_1$ and $X_2$ be two independent random variables each with probability density function $fX_i(x_i) = 1$, for $0 < xi < 1$ for $i = 1, 2$.

Find: (a) $E(X_1 X_2)$, and (b) $Var(X_1 X_2)$. Isn't (a) = zero, since this are independent? How do I go about (b)
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1answer
18 views

Best algorithm for finding permutations with constraint of average total value.

Let's assume I have a random number generator from 0-100 included (only integers) and I generate 5 numbers with it. I want to know the probability of hitting 80, 80, 80, 80, 80 with the constraint ...
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1answer
12 views

Optimize order of a list based on time to complete, probability of success

I'm a programmer participating in a coding challenge, but I'm not up on my advanced math. I'm currently working on a solution to a problem, and have a semi-functional program, but I'm still missing a ...
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1answer
15 views

Notation and a problem with Aleatory variables (Advanced Probability)

I am studying advanced probability and I have a question with notation. One exercise says: Let $(\Omega,B)$, show that $A \in B$ iff $1_A \in B$. But, $1_A$ is a function, what the book means with ...
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1answer
30 views

an exercise related central limit theorem

I'm working on the following problem in Durrett: Let $X_1, X_2, ...$ be i.i.d, nonnegative, $EX_i=1$ and $Var(X_i)=\sigma ^2$. Then we have $2(\sqrt{S_n}-\sqrt{n})$ converge to $\sigma \chi$ in ...
3
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1answer
11 views

Limit (in probability) of sequence of independent random variables

We have $\{X_n\}$ independent random variables convergent to $X$ in probability. I was aked to prove that X is constant, but I can only do that when some $X_n$ has finite variation or what is in fact ...
3
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1answer
22 views

Markov Chains - Strong Markov Property

I'm struck with an exercise. I tried, but the results don't seem to fit to those proposed. Exercise: Two players play the following game. The one who begins draws two cards from a deck of 40 cards ...
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1answer
27 views

Let X and Y be a random variables with $E(X) = 5$, $Var(X) = 30$, $E(Y ) = -􀀀5$, $Var(Y ) = 10$ and $Cov(X, Y ) = 7$

(a) Find $E(2X-3Y+1)$. (b) Find $E((X-2Y)^2)$. (c) Find $Var(3X-Y+pi)$ First I found $E(X^2)$ and $E(Y^2)$ using the given values for (a) I have $2E(X)-3E(Y)+1$ for (b) I come up with: ...
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0answers
8 views

Doob's submartingale stopping theorem in the context of the submartingale problem

Let $$X^\omega_f (t, w) = f(w(t)) - f(w(t \wedge \tau)) - \frac{1}{2} \int_{t \wedge \tau}^t \Delta f(w(s))\, ds$$ be a $P^\tau_\omega$-submartingale. 1) Why Doob's submartingale stopping theorem ...
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1answer
21 views

Law of Iterated Expectation with Probability?

I'm trying to follow a proof of the following proposition (source) Let X and Y be two independent random variables and denote by $F_X(x)$ and $F_Y(y)$ their distribution functions. Let $$Z=X+Y$$ ...
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2answers
41 views

probability and expected value

Hey I am not sure if I thinking correctly on this question? In a carnival, there is game which charges you $3$ dollars to play a game. You win $1$ dollar for every consecutive head you get and you you ...
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2answers
39 views

Showing that infinite product of random variables goes to zero: $\prod^\infty X_i \rightarrow 0 \text{ a.s.}$

I am doing the following exercise: Let $X$ be a strictly positive rv with $\mathbb E[X]=1$ but $X \neq 1$ almost surely. Let $X_1, X_2 \dots$ be iid with same distribution as $X$. Now let $M_0=1$ and ...
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3answers
64 views

How biased is this biased coin

Suppose that we have a coin that we suspect is biased, but that we don't know precisely how biased it is: all we know is that its probability p of landing heads is some fixed value between .4 and .6, ...
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0answers
16 views

How to solve following sequence equation?

Suppose I have a sequence ${p^*_j}$, $j=0,...,m$, satisfying relations $p^*_j=p^*_{j-1}p_{j-1,j}+p^*_{j}p_{jj}+p^*_{j+1}p_{j+1,j}$, with \begin{equation} p_{jj}=\frac{2j(m-j)}{m^2}, \end{equation} ...
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1answer
16 views

CDF and Convergence of Maximum of Sequence of i.i.d. R.V. of Random Length

Let $X_1,X_2,...$ be i.i.d random variable $U(0,1)$ distributed. Let $N_m$ be $Poisson(m)$ and independent of each $X_i$. i)Find the cumulative density function of ...
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3answers
32 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, ...
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1answer
33 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 ...
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1answer
21 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
9 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
18 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
21 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
38 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
49 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 ...
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1answer
38 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 ...
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1answer
17 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, ...
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2answers
17 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!
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0answers
13 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 ...
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1answer
41 views

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

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, ...
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13 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 $11$ ...
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0answers
39 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
20 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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1answer
33 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 $X$. so what is the simplest way to solve this problem.
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1answer
35 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 ...
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1answer
18 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. ...
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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
25 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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1answer
28 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] = ...
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2answers
42 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
25 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 ...
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0answers
34 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 ...
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1answer
26 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 ...
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1answer
63 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: $$ ...
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0answers
23 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 ...
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1answer
42 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 ...
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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 ...
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0answers
14 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 ...
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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 ...
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13 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 ...
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1answer
29 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 ...
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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 ...