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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61 views

Joint Probability and Intersection Probability

Given two independent events A and B: $P(A \cap B)= P(A)*P(B)$ but then I saw somewhere that: $P(A \cap B)= P(A)*P(B)= P(A|B)*P(B) = P(B|A)*P(A)$ where for example $A$ is $X=x$ and $B$ is $Y=y$ ...
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1answer
17 views

Density function of $Y - Z$ if $Y,Z$ are exponentially distributed

Suppose $Y,Z \sim \mathrm{Exp}(\lambda)$. How do I work out the p.d.f of $Y-Z$? I have a feeling I need to do something like this but not sure how to do the integration: $$\mathbb{P}(Y-Z \leq x) = ...
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1answer
70 views

Elementary event in event space

I encountered a very basic question of probability. Consider the sample space Ω = {a,b,c,d} and assume that the only elementary events in the Event space F defined on Ω are {a} and {b}. Explicitly ...
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2answers
211 views

Three fair dice are rolled thrice: Probability of getting a “double 5” at least once?

Three fair dice are rolled thrice. What is the probability of getting a "double 5" at least once? I was thinking that it is $$\frac 16 \cdot \frac 16 \cdot \frac 56 \cdot 3$$
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3answers
65 views

profit from lottery ticket

Jessica is playing a game where there are 4 blue markers and 6 red markers in a box. She is going to pick 3 markers without replacement. If she picks all 3 red markers, she will win a total of 500 ...
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1answer
51 views

How to find $\mathbb P(XY<\frac{1}{2})$ and $\mathbb P(Y< X^2)$ without convolution?

Let $X$ and $Y$ be uniformly distributed on $[0,1]$ and independent random variables. Find $$\mathbb P\left(XY<\frac{1}{2}\right) \text{ and } \mathbb P\left(Y< X^2 \right).$$ Tip: one can do ...
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1answer
54 views

Drawing coloured balls from an urn

An urn contains 36 white, 1 red, 1 black, 1 green and 1 yellow ball. We draw one hundred balls with replacement. What is the probability that all of non-white balls will be drawn (each of them at ...
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1answer
63 views

Puzzle about poker and a goat

I am playing poker with three friends and from a well-shuffled deck we have each been dealt five cards. I have a hand consisting of the four kings and the two of hearts. Being a poker wizard I know ...
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1answer
32 views

Independent identical distributions and conditions for equality

Suppose $X,Y,Z$ are independent identical distribution, taking values in a finite set $x_1,...,x_n$. Is the following true? $$Pr\{X=Y: Y\neq Z\}\leq Pr\{X=Y\}$$ What do think about the condition for ...
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1answer
94 views

Variance zero => symmetry?

We have a random variable $X$ with variance zero. Does this imply that the distribution of $X$ is symmetric? I would say yes, but I'm not sure how to prove it. Any help would be appreciated.
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1answer
20 views

Distribution of numbers with limited information

I have a question regarding the ability to accurately predict the organization of numbers within a space. The specific example is as follows: Consider a square that is divided equally into 9 smaller ...
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1answer
85 views

In a symmetric distribution, Median=Mean?

Does the mean=median in a symmetric distribution?
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1answer
109 views

Weighted sum of large numbers

From the law of large numbers, if $X_1,X_2,...X_N$ are i.i.d random variable, then we have $$\lim_{N \rightarrow \infty} \frac{1}{N} (\Sigma_1^N X_i)=\mu$$ where $\mu$ is the mean of $X_i$. What I ...
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3answers
41 views

Probability of balls in the box.

How to do i.) and v.) ? and please my answer ii.), iii.) iv.) is correct or not, if not explain me where i wrong. Thank you.
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0answers
33 views

Comparing odds of winning

Started to play the national lottery and noticed I can play combinations of numbers instead of just plain 6 out of 49. Since my math knowledge is fairly basic I am wondering which has the better ...
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0answers
66 views

Is this a beta distribution?

I'm currently implementing something from a paper and at one point it describes using a beta distribution of the form: $Prob(p) = (n + 1)Bi(n, p; m)$ where $Bi$ is the binomial distribution, $n$ and ...
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2answers
114 views

In M/M/1 Markov process, why must entering and leaving the zero state be equal?

According to the image below, which I snipped from this article, the rate of leaving State 0 and the rate of arriving into ...
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1answer
93 views

An irreducible Markov chain is a martingale

Let $\{X_n\}$ be an irreducible Markov chain. Does exist example of such $\{X_n\}$ which is also a martingale given that: a. $\{X_n\}$ is recurrent with finite number of states (but bigger ...
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1answer
39 views

Expected value with probabilities

Joel owns a lawn care business and recently performed some research on the size of 50 lawns that he takes care of. Joel recalls that he is expected to take care of a total of $21$ acres of lawn for ...
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0answers
43 views

If $E(|X|\log|X|)<\infty$ then is $E\left[\frac{|S_n|}{n}\ \log\left(\frac{|S_n|}{n}\right)\right]<\infty$?

I am trying to finish a homework problem in my probability class. I think I am at the end of my problem if I can show that $$E(|X|\log|X|)<\infty$$ implies that $$E\left[\frac{|S_n|}{n}\ ...
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2answers
84 views

Verify the joint probability function

I had a question I was hoping for some help on: There are 8 similar chips in a bowl: 3 marked (0;0), 2 marked (1;0), 2 marked (0;1), and 1 marked (1;1). A player selects a chip at random and is given ...
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2answers
104 views

Conditioning on a continuous random variable

I have a random variable $N(a)$, which depends on a number $a$, having the property that for all $a \in \mathbb{R}$, $$P(N(a) \geq 1) = p $$ The example I have in mind is $N(a)$ is $T-a$ where $T$ the ...
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0answers
25 views

Proving the sample variance has a chi squared critical value

Let $X_1, . . . , X_n$ be independent normal observations with means $µ = 0$ and variances $σ^2$. For testing the null hypothesis $H_0 : σ^2 = 1$ versus the alternative $H_a : σ^2 > 1$ show that ...
2
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1answer
37 views

Independent events?

Say we toss a coin that is fair twice we have sample space $:[HH,TH,HT,TT]$. Let's say $A$ is the event with the first throw a head. $A=[HH,HT]$. $B$ is the event with the second throw is a head. ...
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1answer
22 views

Question about probabilities of independent events?

Consider a fair coin toss. Let $H$ be the event that a heads was thrown. Let $T$ be the event a tail was thrown. Given $H$ and $T$ are disjoint are then independent? I don't really know how to do ...
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1answer
53 views

Expected value of an expected value of a joint density function

I had a question I was hoping for some help on: Let $Y_1$ and $Y_2$ be continuous random variables with joint density function: $$f(x,y) = \begin{cases} 6(1-y_2) & \text{if $0 <= y_1 <= ...
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1answer
86 views

Show that if the sum of an diverges, no discrete probability space can contain independent events

Suppose that $0\leq p_n\leq 1$, and put $a_n= \min \{p_n, 1-p_n\}$. Show that if $\sum a_n$ diverges, then no discrete probability space can contain independent events $A_1, A_2, \ldots$ such that ...
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2answers
507 views

Find three events that are dependent but pairwise independent

Let $(\Omega, \mathcal F, P)$ denote the probability triple for the discrete uniform distribution on the set $\{1,2,3,4\}$. Q. Give an example of three dependent events with probabilities strictly ...
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0answers
19 views

Calculate $P(X_{16}=2|X_0=0)$

Given a Markov Chain with three states 0,1,2 with the following State Transition Probabilites: $$M = \left( \begin{array}{ccc} 0.3 & 0.3 & 0.4 \\ 0.2 & 0.7 & 0.1 \\ 0.2 & 0.3 ...
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2answers
216 views

Number of unordered cycles of length $3$ in a graph with $n$ vertices.

Consider an undirected random graph of eight vertices. The probability that there is an edge between a pair of vertices is $\frac{1}{2}$. What is the expected number of unordered cycles of length ...
2
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1answer
184 views

Does this argument suffice to show a “record” occurs at time n with probability 1/n?

I think it does, but, in addition to checking for correctness, I'd like to know what other argument we might use. Let $X_1, X_2,...X_n$ be be a sequence of independent identically distributed ...
2
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1answer
75 views

$X_n \stackrel{d}{\to}X$, $Y_n \stackrel{d}{\to} c \implies X_n+Y_n \stackrel{d}{\to} X+c$

Let $X_n\Rightarrow X$ and $Y_n\Rightarrow c$. Show that $X_n+Y_n\Rightarrow X+c$. Prove: There exists sequences of random variables $(X^{(*)}_n)$ and $(Y^{(*)}_n)$ such that $(X^{(*)}_n)$ and ...
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1answer
70 views

Bipartite graph with $2 \times 10^{6}$ vertices, I need help with removing edges from the graph.

Let G be a bipartite graph. The number of vertices are equal to $2 \times 10^{6}$. Every node is of degree 10. We remove every edge with Probability $2^{-0,1}$. Show that the number of nodes after ...
2
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1answer
62 views

Fatou for weak convergence

I want to do exercise 3.2.4 from Rick Durett, Probability: Theory and Examples page 86. $$\text{Let } g\geq0 \text{ be continuous. If }X_n \Rightarrow X_{\infty} \text{ then } \liminf_{n\rightarrow ...
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0answers
77 views

Mean for seat allocation

There are a set of kids (let's say 30) asked to sit in a row of seats, leaving at least one empty seat between them until all seats are filled. At the end, how do I calculate mean of the fraction of ...
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2answers
60 views

Variance and expected values by dices, how does addition work?

I have read through some stuff and I am confused now. If we have a fair die and we just roll once, the expected value is going to be 3,5 and the variance is 2,916. Well, it is easy to count by one ...
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1answer
58 views

Question about sums of normal random variables

I have independent random variables $X_1$, $X_2$ such that $X_1 \sim N(1,1)$ and $X_2 \sim N(2,2)$, and I'm trying to find a constant $a$ such that $a(X_1 - X_2 + 1)^2$ has a chi-squared distribution. ...
3
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2answers
256 views

Probability involving chess board

if 2 cells are chosen at random on a chess board what is the probability that they will have a common side i tried solving the question by considering different cases for the cells on: 1. corner 2. ...
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0answers
310 views

Create the most 'stressful' tennis game ever!

Some games, such as tennis, use a complicated points system (point, game, set, match; with deuces and tie-breaks) for what would otherwise be an extremely simple and monotonous game. The main reason, ...
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1answer
310 views

Total Probability theorem and Bayes theorem

Two reinforced concrete buildings A and B are located in a seismic region. It is estimated that an impending earthquake in the region might be strong (S), moderate (M), or weak (W) with probabilities. ...
3
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1answer
44 views

Writing random variable formulas with set notations, What is the problem?

Is it wrong to write $\displaystyle P(X \mid Y) = \frac{P(X \cap Y)}{P(Y)}$ when $X$ and $Y$ are random variables? As I know a random variable is a function and therefore has a range and the two ...
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0answers
40 views

Estimate probablity: Chernoff bound

Two players $A$ and $B$ are playing following game: They throw cube. When thrown number $k$ and $k$ is even then player $A$ get $k$ points. When thrown number $k$ and $k$ is divisible by $3$ then ...
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1answer
38 views

What does it mean $\int_1^\infty\frac{F(y)}{y^2}\mathrm dy$?

Which type of functions will satisfy this? $$F: [1,\infty) \to [0,\infty)$$ $$\int_1^\infty \frac{F(y)}{y^2} dy \leq 1$$
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1answer
174 views

Find $E(|X-Y|^a)$ where $X$ and $Y$ are independent uniform on $(0,1)$

Let $X,Y$ be independent $Uniform(0,1)$ random variables. Find $E(|X-Y|^a)$ where $a>0$. My working: Define $W=1$ if $X>Y$ and $W=0$ if $X<Y$. We seek ...
2
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1answer
40 views

Conditional expectation of $Y_1$ given that $\sup Y_i=z$, for $(Y_i)$ i.i.d. uniform on $[0,\theta]$

Suppose that $Y_1,\ldots,Y_n$ are random variables independently and identically distributed as uniform on $[0,\theta]$ for some $\theta>0$. How do I find the conditional density of $Y_1$ given ...
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2answers
80 views

What is the logic behind the probability of getting 'four of a kind' in poker?

This hand ($5$ cards of $52$) has the pattern $AAAAB$ where $A$ and $B$ are from distinct kinds. The number of such hands is $\binom{13}{1} \binom{4}{4} \binom{12}{1} \binom{4}{1}$. The probability ...
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2answers
274 views

How can I reword this problem illustrating a scenario that needs Bayes Theorem to solve?

Taken from Stat Trek, an example explaining Bayes Theorm http://stattrek.com/probability/bayes-theorem.aspx Marie is getting married tomorrow, at an outdoor ceremony in the desert. In recent ...
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2answers
204 views

The square of a standard Normal random variable

I am having a bit of trouble with this: Let $U=Z^2$ where Z is a standard Normal random variable with pdf: $$f_z(z) = \frac{1}{\sqrt{2\pi}} e^{\frac{-z^2}{2}}$$ I want to use the inversion method ...
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0answers
49 views

Is this an easy conditional probability question?

Fifty-two percent of the students at a certain college are females. Five percent of the students are majoring in computer science. Two percent of the students are females majoring in computer science. ...
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1answer
55 views

Expected Value Coins Question

If I were to flip n coins and compute the product of the number of heads versus the number of tails what would be the expected value of this product? My logic: In n coin flips n/2 coins will be ...