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

Prove that the probability of getting at least $k$ heads when using coins that skew more to heads is at least as good as using the other coin.

Let $0 \le p < p' \le 1$. Let $X_i$ be the Bernoulli random variable that takes the value of $1$ with probability of $p$ and zero otherwise. Similarly, let $X'_i$ be the Bernoulli random variable ...
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0answers
9 views

Does martingale model work for betting football matches?

Imagine I have 1 million USD and will be betting 1.000 USD on the win of FC Barcelona each time they play a match in La Liga (Spanish Tier 1 football league). If FC Barcelona loses or ties their last ...
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1answer
13 views

Jacod Protter “Probability Essentials” Problem 2.8

The question asks to show that a sigma-algebra $\mathcal A$ consisting of $A$ s.t. $A=f^{-1}(B)$, where $B$ is in $\mathcal B$ are Borel subsets of $R$ and $f$ is continuous, is contained in $\mathcal ...
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votes
1answer
621 views

Find unknown value in probability density function

"Suppose that a random variable $Y$ has a p.d.f. given by $f (y) = ky^3*e^{-y/2}$ when $y > 0$, and otherwise 0. Find the value of $k$ that makes $f (y)$ a density function." I found that $k=1.$ ...
1
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1answer
25 views

Length of pieces of stick broken at random

A stick of length 1 is broken at random. How much longer is the longer piece than the shorter piece, on average?
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1answer
15 views

Probability of drawing balls from an urn with variable composition

A coin is tossed $k$ times, with probability $p$ of heads. In an urn, as many white balls are introduced as the amount of heads obtained, and as many black balls are introduced as the amount of tails ...
1
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0answers
13 views

Periodicity of Markov chains under cartesian product

Suppose that you have a finite state Markov chain, with $n$ states and characterized by $p_{i,j}$ the probability of reaching state $j$ from state $i$. Consider the new Markov chain with $n^2$ states ...
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0answers
10 views

CDF of minimum of correlated and iid random variables

Consider two random variables $X_1=\min (W_1, W_2)$ and $ X_2=\min (W_3, W_4),$ where $W_1$, $W_2$,$W_3$ and $W_4$ are positive, identically distributed random variables. While $W_1$, $W_2$ are ...
2
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0answers
34 views

Chance of arriving first

I'd like to solve the following homework: A man and a woman agree to meet at a certain location about 8:30 p.m. If the man arrives at a time uniformly distributed between 8:15 and 8:45 and the woman ...
-1
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0answers
24 views

Determine the distribution [on hold]

Determine with justification the distribution (with parameters) of chimney fires in a large city over a week Assume the number of chimney fires over a year is 520.
2
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2answers
44 views

Is it true that E [ X | E [ X | Y] ] = Ex [ X | Y] ? Does this law have a name?

Let $X$ and $Y$ be two random variables (say real numbers, or vectors in some vector space). It seems to me that the following is true: E [ X | E [ X | Y ] ] = E [ X | Y] Note that E [ X | Y ] is a ...
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1answer
22 views

Probability question please guide

There are $5$ red and $3$ blue chips in a bowl. The red ones are numbered $1,2,3,4,5$ and the blue ones as $1,2,3$ respectively. if $2$ chips are drawn without repacement, find the probability that ...
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0answers
10 views

Proving criterion for a transient state in Markov Chain

Let $\{X_n\}_n$ be a homogenous Markov chain. Prove that if exist a connected subset of states (means set of states which exist positive probability to move between them), $S$ which is not closed, ...
7
votes
2answers
628 views

On average, how many times must I roll a dice until I get a 6?

On average, how many times must I roll a dice until I get a 6? I got this question from a book called Fifty Challenging Problems in Probability. The answer is 6, and I understand the solution the ...
0
votes
1answer
24 views

What's the probability of the game being cancelled due to players not showing up

There are two teams, and each team has 6 players. 4 players are required for the game to go on. The probability of a player not showing up is $10\%$. What's the probability of the game being ...
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0answers
9 views

Pushforward measure

We define $X := \{0,1\}, \mu := \frac{1}{2} (\delta_0 + \delta_1)$ and $(\Omega, \mathcal{F},\mathbb{P}) : = \bigotimes_{n=1}^{\infty} \left( X, 2^X,\mu \right)$. For $\omega \in \Omega$ we denote the ...
2
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0answers
28 views

Modified Doob's $L^1$ inequality

Let $X_n$ be a non-negative submartingale. Show that (1) for all $\lambda >0$ $$ P(\sup_{k\leq n} X_n \geq 2\lambda) \leq \frac{1}{\lambda} \int_{X_n \geq \lambda} X_n dP$$ (2) $\|\sup X_n\|_1 ...
1
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1answer
25 views

$Y_n = \sup_{k \geq n} E(X_k | F_n)$ is a martingale if $X_n$ is $L^1$ bounded non-negative submartingale

Let $X_n$ be a $L^1$ bounded non-negative submartingale. Let $Y_n = \sup_{k \geq n} E(X_k | F_n)$. Show that (1) $Y_n$ is a martingale (2) $X_n \leq Y_n$ for all $n$ a.s. (3) $\sup \|X_n\|_1 = ...
0
votes
1answer
16 views

Probability generating function and a discrete random variable

A discrete random variable $X$ has probability generating function $G_X(t)$. If $Y=aX+b$ show that the probability generating function of $Y$ is given by $G_X(t)=t^bG_X(t^a)$. Hence prove that ...
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0answers
18 views

If $G(x)=P[X\geq x]$ then $X\geq c$ is equivalent to $G(X)\leq G(c)$ $P$-almost surely

Suppose $[\Omega,\mathcal{F},P]$ denotes a probability triplet and $X:\Omega\to\mathbb{R}$ is a real-valued random variable. Define $$ G(x)=P[X\geq x]. $$ My current reading material claims ...
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0answers
19 views

$\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$ is countably generated iff there is a random variable $X$ such that $\mathcal{G} = \sigma(X)$.

Where can I find a reference to the proof of the fact that a $\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$ is countably generated if and only if there is a random variable $X$ such that ...
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2answers
40 views

How to find $\lim_{n\to \infty} P(a≤(X_1X_2…X_n)^{-n/2}e^{n/2}≤b)$ where $X_1,X_2,…,X_n \sim U[0,1]$?

I am trying to calculate $$\lim_{n\to \infty} P(a≤(X_1X_2...X_n)^{-n/2}e^{n/2}≤b)$$ in terms of $a,b$, where $$X_1,X_2,...,X_n \sim U[0,1]\,\,\,\,\,\,\,(i.i.d.)$$ and $$0≤a<b$$ My attempt is to ...
0
votes
1answer
40 views

A Question on CDFs and PDFs (substitution/inverse?)

(a) So there has been an answer to the question. Can someone explain how the limits of integration were found? I don't know why the upper limit is going to $X$.
1
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1answer
272 views

Probabilities Pick 3 Lottery

In a Pick 3 lottery three numbers are drawn from three separate sets of numbers, between 0 and 9. Matching 2 winning numbers, one of them duplicated, can be done in 3 ways, according to all lotteries ...
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0answers
11 views

discrete values probabilities problem [on hold]

Can you help me with this?? An engineer is requested to design water supply and waste water removal systems in a new industrial park consisting of 5 independent buildings. Assume that the water ...
3
votes
2answers
31 views

Mathematical justification for incorporating a conditional event in expectation?

Let $X_1,X_2,\dots$ be independent and identically distributed random variables. Furthermore, consider the sum $$ Y = X_1 + X_2 + \dots + X_N $$ where the number of terms $N$ is itself a random ...
1
vote
1answer
521 views

Probability of drawing N unmatchable socks from a pile of N matched pairs

From the "sock-matching problem" over on SO, given a pile of $2N$ socks, each having one and only one matching sock in the pile, and drawing $N$ socks randomly and without replacement from this pile, ...
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3answers
232 views

Probability of passing this multiple choice exam [duplicate]

A multiple choice exam has 175 questions. Each question has 4 possible answers. Only 1 answer out of the 4 possible answers is correct. The pass rate for the exam is 70% (123 questions must be ...
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2answers
20 views

Probability that two items selected from a mixed bag will be of particular sorts [on hold]

You have bought a mixed case of soft drinks. It contains six bottles of Coke, four bottles of lemonade, two bottles of tonic water, and eight bottles of mineral water. What is the ...
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0answers
18 views

Expected time until beating an initial try

Consider the following problem: Let $X,X_1,X_2,...$ be i.i.d. random variables. We execute the following experiment. One samples $X$. Then, one samples $X_1$,$X_2$ and so on until the first time the ...
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0answers
6 views

interpreting the multivariate Kalman filter update equations

consider a multi-dimensional Kalman filter model with these state transition and measurement probabilities: $P(x_{t+1} | x_{t}) = Normal(Fx_{t}, \Sigma_{x})$ $P(z_{t} | x_{t}) = Normal(Hx_{t}, ...
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4answers
51 views

roulette with infinite money

Assumed I have infinite money and bet 10 on red or black and every time I lose I will double my bet until I win and then start again with 10 would I make profit? I did a bit code for that: ...
1
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1answer
22 views

$X_n \to X$ in $L_2$, show that $\lim_{n \to \infty}E[X_n^2]=E[X^2]$

$X,X_1,...$ are random variables, $X_n \to X$ in $L_2$. Show that $\lim_{n \to \infty}E[X_n^2]=E[X^2]$. My attempt: $X_n \to X$ in $L_2 \implies \lim_{n \to \infty} E[(X_n-X)^2]=0 \implies \lim_{n ...
2
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2answers
14 views

Weighted Average Proof

Been stuck on this for a while now, seems pretty straightforward but can't seem to prove it. Given $\mu$ is a weighted average of $\mu_1$ and $\mu_2$ such that $\mu = x_1\mu_1 + x_2\mu_2$ where $x_1$ ...
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0answers
14 views

Suppose that there are two cells in a parallel system. In order for the system to work, at least one of the two parallel subsystems must work.

Consider a particular lifetime value $t_0$, and suppose we want to determine the probability that the system lifetime exceeds $t_0$. Let $A_i$ denote the event that the lifetime of cell $i$ exceeds. ...
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0answers
8 views

sum of two Gaussian random variables conditioned on their sum

I have two independent standard normal R.V.s X and Y, and their sum is Z = X + Y. I am trying to calculate the PDF of X conditioned on Z taking the value z. I know that this is the joint PDF of X and ...
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0answers
20 views

Finding a random variable.

Let $X,Y$ be two non negative random variables such that density of $Y$ is the same as the survival function of $X$. Is there any way we can find $Y$? Thank you for your time and help.
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1answer
23 views

There are 2 red, 3 pink, 4 orange, and 5 yellow jelly beans in the pocket. how many different ways can you choose at least one jelly bean?

Thanks a lot!There are 2 red, 3 pink, 4 orange, and 5 yellow jelly beans in the pocket. how many different ways can you choose at least one jelly bean?
3
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2answers
31 views

Poker odds: Chances of a straight flush, given H4,H5

I'm trying to learn Bayes's formula, and am coming up with some poker problems to learn this. My problem is as following: given a $H4,H5$ ($4$ of hearts, $5$ of hearts) hand, what are the odds that ...
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2answers
38 views

There is 25% Rob will wear blue shirt, 60% he will have dinner with his girl. the probability he wear a shirt but not having dinner

Event $A$ is that Rob will wear a blue shirt. Event $B$ is that Rob will have dinner with his girlfriend. What is $P(A \cap B')$? $P(A)$=$0.25$, $P(B)$=$0.6$ The events are independent. Please ...
2
votes
1answer
489 views

Bonferroni's Principle

I am working with following Principle (My Approach At Bottom) : 1.2.3 An Example of Bonferroni’s Principle Suppose there are believed to be some “evil-doers” out there, and we want to detect them. ...
2
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0answers
34 views

$X = E(Y | \sigma(X)) $ and $Y = E(X | \sigma(Y))$

Suppose $X, Y$ are random variables in $L^2$ such that $$X = E(Y | \sigma(X)) $$ $$Y = E(X | \sigma(Y))$$ Then I want to show that $X=Y$ almost everywhere. What I've done: By conditional Jensen ...
2
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1answer
48 views
+200

Correlation of a vector generated and its one-period lag, both generated using AR(1) data

Suppose that $C_0$ is $100$ and $\{e_t\}_{t\geq 1}$ is a sequence of i.i.d. standard normal random variables. We generate $C_t=C_{t-1}+e_t$ for $t\geq 1$ and set $$ x_t=C_t^2-C^2_{t-1},\quad ...
0
votes
1answer
25 views

estimates Gaussian moments

Let $X_i \sim N(0,\sigma_i^2)$. Let $k\geq0$ be a fixed integer. I would like to compute $$A:=E[|X_1-X_2|^k|X_2|^k]$$ My idea was \begin{align*} A=&\int_{\mathbb{R}^2}|x_1-x_2|^k |x_2|^k ...
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1answer
33 views

what could 1/probability represent?? [on hold]

I was working on a concept in probability theory with a friend and we came across 1/probability. Does the inverse of probability appear anywhere in mathematics and what are its applications?
2
votes
1answer
289 views

Probability question involving sets of cards

I have an infinite deck built out of sets of 10 cards (in other words 10*n cards). The sets are identical so one '2' is identical to another '2'. A player draws 6 cards. If he draws: any '1' AND a ...
0
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1answer
17 views

probability of 2 coin tossedsimultaneously [on hold]

Two coins are tossed simultaneously. what is the probability that the second coin would show a tail given that the first coin has shown a head ?
0
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1answer
34 views

If $F(a) - F(a^{-})$ is continous then $F(a)$ is continous [on hold]

Suppose $F$ is a distribution function and, $$H(a) = F(a) - F(a^{-})$$ is continous for all $a \in \mathbb{R} $, where $$F(a^{-}) = \lim_{\epsilon \to 0^+}F(a-\epsilon)$$ How to prove that $F$ is ...
-2
votes
0answers
23 views

Probability (Please see picture below) [closed]

On a scratch card you win if you find a sun in the first square you scratch off. Here are the scratch cards before the suns are covered. $$\mathbf{A}\ \ \ \ \ \ \ ...