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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Probability Venn Diagram Problem

I have this problem: We have two fruits: Apple and Banana 60% of people buy at least one of the two fruits. 30% of people who buy banana will subsequently buy an apple. Probability people will buy ...
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2answers
25 views

Bathtub Death Betting

A rather grim question if you ask me but the question goes as follows: Each year, about 300 Americans drown in their bathtubs. Which of the options should you choose: A: win \$100 if no bathtubs ...
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4 views

Concentration of the norm of a normal distribution given via this integral?

Let $X\sim N(0,\Sigma)$ be a multivariate Guassian distribution. What is the natural meaning of the integral: $C(\alpha)=\int_{[-a_\alpha,a_\alpha]} x^2 \bar{X}(dx)$, where $a_\alpha$ is defined by ...
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2answers
21 views

Counting Problem Concerning the Stars and Bars Technique

I need to distribute $k$ indistinguishable balls to $n$ distinguishable bins. Of course, this is plainly an example where the so-called stars-and-bars technique is helpful: this technique yields an ...
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0answers
7 views

Joint Limiting Distribution of Min and Max

Let $X_1,\ldots,X_n$ be iid from the uniform distribution $U(a,b)$. Let $X_{(1)}< cdots< X_{(n)}$ be the order statistics. Find the joint limiting distribution of $(n(X_{(1)}-a),n(b-X_{(n)}))$ ...
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Find an approximation for $P\{N=1\}$

I have the following question, I know I am somehow supposed to use binomial to Poisson approximation but can't figure out the trick. $X_1, X_2, ..., X_N$ are iid with $$P\{X_k = i\} = 1/10^6 , 0 ...
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8 views

Suggestion: good book on probability theory with emphasis on applications to other areas of mathematics and physics

On this website, there are many questions about books on probability theory, but I would like to ask if you can select (from all the references available on this website and elsewhere) a book ...
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1answer
24 views

probabilit on rolling die [on hold]

If a fair die is rolled 6 times, what is the probability that the number shown will be less than 4 exactly 2 times? I have tried to do this problem and I just can't get it. I am not even sure how to ...
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16 views

Expected Value of conditional expectation

Let $X_1 \in S_1 $and $X_2 \in S_2 =\mathbb{N} < \infty$ be finite random variables. $\mathbb{E}_{X_1}[X_2]$ = $\mathbb{E}[X_2 | X_1]$ denotes the conditional expected value. Show: ...
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2answers
36 views

What is the probability that a random 6-digit number will have at least one 0, at least one 1, and at least one 2?

Hi this a question from my textbook: A first course in probability, It doesn't have the solution so I'm curious as to what the answer is. This is the question: What is the probability that a random ...
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7 views

Rigorous Order Statistics with Indicator Functions

If three people are randomly placed along a 1 mile road, the probability that no two of them are less than $m$ miles apart for $m \leq \frac{1}{2}$ could be solved by using the density for the order ...
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1answer
14 views

Coupon Collecting Problem using Inclusion-Exclusion

I am very new to probability and combinatorics. I am trying to solve Coupon Collection Problem using Inclusion-Exclusion formula. I found a lot of references her a some: ...
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27 views

On what value of $f(X)$ minimizes $E[(Y-f(X))^2|X]$

$X$ and $Y$ are random variables. The question is: what value of $f(X)$ minimizes $E[(Y-f(X))^2|X]$. I am pretty sure I have found the solution to this problem by writing: $$E[(Y-f(X)-E[X|Y] +E[X|Y] ...
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0answers
13 views

Show the following definition does not give a $\sigma$-addtive measure pathwisely

Given the space of all square-integral functions over $[0,1]$: $L^2([0,1], \mathcal{B}([0,1]), m)$ and a Brownian motion $W_t$ defined on the probability space $(\Omega, \mathcal{F}, P)$, we define ...
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0answers
10 views

Proof an edge in a geometric graph

Suppose i take two random uniformly distributed points $X_{1},X_{2}$ in $[0,1]^{2}$. In addition i connect $X_{1}$ and $X_{2}$ by an edge if $||X_{1},X_{2}||_{\infty} \leq r$ where $0<r<1$ and ...
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2answers
34 views

Showing that $-\ln{X} \sim \exp{\alpha}$ for $X \sim Beta(\alpha, 1)$

The CDF for $X \sim Beta(\alpha,1)$ is given by: $$F(x) = \frac{\int_{0}^{x}t^{\alpha-1}dt}{\int_{0}^{1} t^{\alpha-1}dt}$$ I am given to understand that $-\ln{X} \sim \exp(\alpha)$ if $\alpha > ...
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11 views

Joint density function of exponential and gamma distribution

My problem is: $X_1,...,X_n$ are independent exponentially distributed random variables with $\lambda=1$ paremeters. I have to find the joint density funcitions of $ Y=\sum\limits_{i=1}^n{X_i}$ ...
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0answers
10 views

What is the variance of this random variable: number of items

Let us assume that we have a capacity $n$ which tends to infinity. We have an infinite number of random variables $X_1, X_2, \dotsc$, where each $X_i$ is independent and identically distributed with ...
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1answer
14 views

covariance matrix in bivariate distribution

I struggle to understand how exactly you get the covariance matrix in a bivariate normal distribution. The reason is probably that I have no idea how to obtain it at all. In the exercise I have I ...
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3answers
42 views

What's the probability of rolling the same number twice with a pair of dice?

There are a number of questions similar to this, but I'm asking about rolling two dice twice, not one die two times. Or I guess you could think of it as 4 dice total, with each pair distinguishable. ...
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1answer
26 views

Powers of random variables always well-defined?

Given the random variable $X$, is $X^{0}$ a random variable? Can we take the expectation $E(X^{0})$? Is $X^{0}$ or its expectation defined or undefined under any conditions (say, on the sample space ...
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1answer
5 views

How many base-out configurations would be possible in sleazeball?

Problem: In a baseball there are 24 different "base out" configurations (runner on first - two outs, bases loaded- none out, and so on). Suppose that a new game, sleazeball, is played where there are ...
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1answer
18 views

X and Y have same distribution. Is the distribution of X-Y symmetric to 0?

I met this question when I went through non parameter test. I know X-Y may not be 0. Like X follows standard normal. So does -X. But X-(-X)=2X. I just wonder is X-Y (the pdf, if exists) symmetric to ...
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0answers
13 views

Probability that 3 of 4 people are on 4 of 12 seats by allowing occupancy of seat for more than on person.

I asked a similar question here which is as follows: Four (identical) persons enter a train (section A has 4 seats, section B has 8 seats). What is the probability that exactly (not more or less) 3 ...
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0answers
14 views

Strong Markov property of continous time Markov process

In the book "Applied probability and queues" which is available here http://books.google.de/books?id=BeYaTxesKy0C&pg=PA32&hl=de&source=gbs_toc_r&cad=3#v=onepage&q&f=false , ...
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0answers
61 views

Expected deviation of a coin that obeys the gambler's fallacy

Suppose a magical coin $C$ comes up heads with probability $\frac12$ on the first flip, and thereafter comes up heads with probability $\frac t{h+t}$, where $h$ and $t$ are the number of heads and the ...
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2answers
22 views

Does inclusion-exclusion formula holds for coutanble index set?

Does inclusion-exclusion formula holds for countable index set? Here is the formula for index set of size 2. \begin{align} P(A \cup B)=P(A)+P(B)-P(A\cap B) \end{align}
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2answers
22 views

Thinking about a probability problem in terms of sets.

Here is the problem that I am working on: We have $A_1 \setminus A_4 = A_1 \cap A_4^c = \{(x, y) : x \leq 2, y \leq 4\}\cap\{(x, y) : x > 0, y > 1\} = \{(x, y) : 0 < x \leq 2, 1 < y ...
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1answer
20 views

Is there a result that the density function for $\chi^{2}$ must be related to the standard normal density?

Suppose we have a random variable $X$ with the property that $-X = X$ (in distribution) and $X^{2} = \chi^{2}(1)$? I want to be able to conclude that $X \sim N(0,1)$. The probability density function ...
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2answers
17 views

Expected value of a normal distribution over a range

Sorry if my question is very trivial. I want to calculate the expected value of a continuous random variable over a range. Specifically I want to get the expected value of a normal distribution in ...
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1answer
30 views

Select a random hyperplane

How can one generate a uniformly random $n$ dimensional hyperplane which passes through the origin? In 2d it seems one can pick an angle at random and draw a line from the origin at that angle. I am ...
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2answers
42 views

How many ways can SLUMGUILLION be arranged so all three L's precede all other consonants?

How many ways can the letters in the word SLUMGULLION be arranged so that the three L's precede all the other consonants. Attempt: There are 11 letters, and there are 3 Ls, 4 vowels: U U I O, and 4 ...
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1answer
22 views

Expected value and Variance calculation

Suppose $f$ is an uniformly distributed random variable with parameters $-1,1$ and $g$ is a Poisson-distributed random variable with parameter $\lambda >0$. We assume that $f$ and $g$ are ...
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2answers
36 views

How many hamburgers can be ordered, if there can be eight toppings?

A fast food restaurant offers customer a choice of eight toppings that can be added to a hamburger. How many different hamburgers can be ordered? Attempt: I don't know if this is correct 8!? I think ...
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0answers
51 views

Expected size of the connected component containing a randomly selected node

Given an Erdős–Rényi random graph with n nodes and edge probability p, what is the expected number of nodes in the connected component containing a randomly selected node? In other words, if I ...
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1answer
21 views

Which random variable distribution can be scaled towards zero mean and unit variance?

can any random variable, not necessarily normally distributed, scaled and shifted in such a way that the new mean is 0 and the new variance is 1? If not, which can? I remember hearing about ...
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0answers
9 views

Relation between Bernoulli RV, binomial RV, geometric RV and Poisson RV [on hold]

what is the relation between Bernoulli RV, binomial RV, geometric RV and Poisson RV? And how we represent them?
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0answers
22 views

Trouble with Conditional probability and expectation [on hold]

I have a few questions in probability that have been bothering me. The first is this: Why is $$E(T-t | T \ge t) = \int_t^\infty \frac{(s-t)f(s)~ds}{P(T\ge t)}. $$ The second is this: How does one ...
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1answer
21 views

Probability problem with average value

First day of Math101 Course have average of 4% absent usually. There are 32 student assigned for Mat101 of year 2015. So teacher prepared 30 student class for first day of this year. What's the ...
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3answers
24 views

How to determine a probability?

A coin is tossed and a head or a tail observed. If a head results, the coin is tossed second time. If a tail results on the first toss, a die is rolled. a) Draw a tree diagram and list the sample ...
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0answers
36 views

How to prove the independence of two random variables. [on hold]

If $X,Y$ are two random variables.for every $ x$ in $\mathbb R$: $$\mathbb E\left[ e^{ixX}\mid Y\right]=\mathbb E\left[e^{ixX}\right]$$ How to prove that $X,Y$ are independent?
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1answer
23 views

What is the probability that the lot will NOT be passed by the Inspector? [on hold]

Batteries for torch lights are packed in boxes of 10 and a lot contains 10 boxes. A quality inspector randomly chooses a box and then checks two batteries selected randomly without replacement from ...
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1answer
25 views

Lower bound on a conditional probability

Let $X_i$, $i=1, \dots, n$, be identically distributed and suppose they satisfy $X_i\in \{0,1\}$. Without assuming any independence, is it true that $P(X_1=1|\sum_{i=1}^n X_i \neq ...
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1answer
38 views

number of ways to seat 6 students (3 females and 3 males)

we have 3 female and 3 male students traveling to spain. On the plane we have 3 seats on each side of the gangway, kinda like this: |A B C | |D E F| a) Suppose At this point that one female sits to ...
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1answer
31 views

Probability paradox: Mario's dice game

Consider the following: Mario invites an infinite number of friends to a party, and challenges them at a game. First, Mario randomly splits his infinite friends in groups of incremental size, the ...
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1answer
18 views

How many straight lines can be drawn between five points (A, B, C, D, and E), no three of which are colinear?

How many straight lines can be drawn between five points (A, B, C, D, and E), no three of which are colinear? Attempt: Given 5 points, a line consist always of 2 points. Thus the total number of ...
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2answers
40 views

Expected number of people getting their own hat given that at least one of them gets his hat.

Suppose there are $N$ people at a party. Their hats get mixed and when leaving they grab a hat at random. Let $\displaystyle X_i=I(\text{$i$th person gets his own hat})$ and ...
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1answer
20 views

How many zip codes are as large as 6000-0000, are even numbers, and have a 7 as their third digit?

When they were first introduced, postal zip codes were five digit numbers, theoretically ranging from $00000$ to $99999$. (In reality, the lowest zip code was $00601$ for San Juan, Puerto Rico; the ...
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1answer
23 views

the conditions for a measurable function to be the uniform limit of simple functions

In our homework we are asked to prove that, on a measurable space $(\Omega,\mathcal{F})$, every function $f:\Omega \rightarrow R, f\geq 0$ can be written as the uniform limit of an increasing limit of ...
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3answers
29 views

In how many ways can the word ELEEMOSYNARY be arranged.

In how many ways can be the letters of the word ELEEMOSYNARY be arranged so that the S is always immediately followed by a Y? Attempt: There are 3 Es, and 2 Ys, and and then all letters appear once ...