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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3
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2answers
1k views

Grad degree that mainly deals with probability/game theory/optimization?

I'm currently working but am going to take classes as a non-degree student to beef up the math part of my background. I've only taken calc 1-3, ODEs, linear algebra, logic, and decision theory so my ...
11
votes
2answers
436 views

What is the distribution of gaps?

Randomly select $n$ numbers from the universe $\{1,2\dots,m\}$ with or without replacement, and sort the numbers in ascending order. We can get a list of number $\{a_1,a_2,\dots,a_n\}$, and then we ...
2
votes
2answers
87 views

is there a solution to the following maximization problem such that $a = b$?

Let $X = (X_1,...,X_n)$ be a vector of $n$ random variables. Consider the following maximization problem: $\max\limits_{a,b} \;\mathrm{Cov}(a\cdot X, b \cdot X)$ under the constraint that $\|a\|_2 = ...
3
votes
0answers
150 views

About bound based on r-th central moment

We know that Chebyshev's inequality is: $\Pr[|X-\mathbb E(X)| \ge t] \le \frac{\mathrm Var (X)}{t^2}$ And for all $r>0$ and $t>0$, $\Pr[|X-\mathbb E(X)| \ge t]= \Pr[|X-\mathbb E(X)|^r \ge ...
1
vote
1answer
84 views

What percentage of employees fail the test? Percentage of those fired were innocent?

To reduce theft among employees, a company subjects all employees to lie-detector tests and then res all employees failing this test. In the past, the test has been proven to correctly identify guilty ...
0
votes
1answer
168 views

How do determine probability related to Möbius function

If you pick a number at random from $1 \ldots 10^k$ where $k$ is large, call this number $Z$. $M(Z)$ is defined as 0 if $Z$ is not divisible by two repeated primes. Find the probability that $M(Z) = ...
0
votes
1answer
782 views

Use Bayes' methods to find the probability that the cab involved in the accident was actually Blue

A cab driver was involved in a deadly hit-and-run accident at night. Two cab companies, the Green and the Blue, operate the city; 15% of the cabs are Green and 85% are Blue. A witness identies the cab ...
1
vote
2answers
154 views

For a random permutation, what's the probability that each half of the elements keep relative order?

If we take a random permutation of a sequence of $2k$ elements, $X_1, X_2, \ldots X_k, X_{k+1}, \ldots, X_{2k}$. What's the probability that $X_1, X_2, .. X_k$ and $X_{k+1}, \ldots, X_{2k}$ both ...
2
votes
2answers
63 views

Is this argument about probability correct?

Let's say, we have $i$ continuous IID random variables $X_1, X_2, \ldots,X_i$ whose domain is $\mathbb R$. This sequence divides the real axis into $i + 1$ intervals. Now, if we have another random ...
4
votes
1answer
580 views

Relationship between median and mean of a pmf

If you consider a distribution that has many medians: $$P(X=x) = \{(1, 0.25)(2,0.25),(3,0.25),(4,0.25)\} ,$$ we know that this distribution has multiple medians between $2$ and $3$ if we define a ...
0
votes
2answers
90 views

Having problem in coin tossing problem

I was going through this question on a site. It stated as below, One more problem that is tying me up..I think because it is so much simpler than what I have been working on. Any help would be ...
1
vote
1answer
105 views

Derivatives with respect to a symmetric matrix, with an application to maximum likelihood

I am quite unsure about this whole matter of differentiation with respect to a matrix. First, I'd like a good (online hopefully) reference for getting up to speed on the theory - as opposed to a bunch ...
0
votes
1answer
101 views

Understanding Events

So I have this problem before me: Jane is taking two books on her holiday vacation. With probability 0.5, she will like the first book; with probability 0.4 she will like the second book; and ...
4
votes
1answer
4k views

how to find the expected number of boxes with no balls

If you have 10 balls and 5 boxes what is the expected number of boxes with no balls. The probability that each ball goes independently into box $i$ is $p_i$ with the $\sum_{i=1}^5 p_i =1$. Also, what ...
2
votes
1answer
97 views

Proof $\frac{1}{(\frac{n}{3})!}=2^{-\Omega(n \log n)}$

I saw this in Wegener(2003), Methods for the Analysis of Evolutionary Algorithms as a upper bound on the probability. After applying Stirling approximation to $(\frac{n}{3})!$ I still keep getting ...
7
votes
1answer
362 views

A multinomial problem (balls, bins, etc.)

Consider the well known multinomial setting: there are L balls, thrown at random at n bins so that the probability that a ball falls in bin i is $p_i$, independent of the other balls (the $p_i$’s are ...
1
vote
2answers
186 views

Showing that $g(x)$ is a valid PDF

Having a difficult proving that $g(x)= f(x)/(1-F(x_0))$, $x \geq x_0$ and 0 otherwise is a valid PDF. I have shown the first to criteria for it to be a PDF, in which that all values $x \leq x_0$ are ...
4
votes
2answers
630 views

Maximizing growth rate in betting on multiple events

Suppose we have $n$ independent events. We know their probabilities $p_i,\dotsc,p_n$. We are given betting odds $c_1,\dotsc,c_n$. We can make bets to any of the events, and also any combination of ...
2
votes
1answer
348 views

Probability cards question

What is the probability that the first card is spades given that the second and third cards are spades? Attempt: Use Bayes' rule to calculate $P(E_1|E_2E_3)$ $E_1$ = first card is spade $E_2$ = ...
2
votes
1answer
260 views

Finding the probability that red ball is among the $10$ balls

A box contains $20$ balls all of different colors including the red color. If we select $10$ balls randomly without replacement, what is the probability that the red ball will be among these $10$ ...
2
votes
1answer
174 views

Independence of discrete random variables

Suppose $X,Y$ are uncorrelated random variables, $\mathbb{E}(XY)=\mathbb{E}(X)\mathbb{E}(Y)$, taking on two values $m,n\in\mathbb{R}$, that is, $P(X\in \{m,n\})=P(Y\in \{m,n\})=1$. How should I go ...
6
votes
1answer
102 views

The number of functions with a certain property

Let $f$ be chosen uniformly at random from all functions $f:\{1,\ldots,n\}\rightarrow\{1,\ldots,n\}$ such that $f(k)\in\{1,\ldots,k\}$ for $1\leq k\leq n$. What is the probability that $f$ is ...
4
votes
2answers
99 views

If $X, Y \sim N(0,1)$, find the CDF of $\alpha X + \beta Y$ [duplicate]

Possible Duplicate: Proof that the sum of two Gaussian variables is another Gaussian Let $X,Y$ be independent normally distributed $N(0,1)$ random variable, and $\alpha,\beta\in ...
3
votes
0answers
235 views

Level of connectivity in a random graph

If you take a random $k$-regular graph, with high probability (tending to 1 as the size of the graph grows) almost all of the nodes will be in one giant connected component. How well connected is this ...
7
votes
4answers
582 views

Why does this expected value simplify as shown?

I was reading about the german tank problem and they say that in a sample of size $k$, from a population of integers from $1,\ldots,N$ the probability that the sample maximum equals $m$ is: ...
36
votes
6answers
2k views

A variant of the Monty Hall problem

Everybody knows the famous Monty Hall problem; way too much ink has been spilled over it already. Let's take it as a given and consider the following variant of the problem that I thought up this ...
5
votes
2answers
944 views

Showing $\cos(t^2)$ is not a Characteristic Function

Usually when we try to show a function is not a characteristic function, we would prove it is not uniformly continuous. I am wondering if there is any other way to show $\cos(t^2)$ is not a ...
1
vote
1answer
84 views

Multinomial coefficient not coming out right

I'm trying to find the coefficient of $ x^{36} $ in the expansion of $ (2 - x + x^2)^{21} $ So I found all possible combinations of $ \displaystyle 2^x, x^y, (x^2)^z $ that yield 36 and $ x + y + z = ...
1
vote
1answer
63 views

What is this distribution called?

$F_i=n {m-i \choose n-1}$ where $m \ge n, 1 \le i \le m-n+1$ For instance, if $m=10, n=5$, I can draw a line graph.
3
votes
1answer
220 views

Sequence of characteristic functions

Let $\eta_k(t)$ be the characteristic function of a random variable $X_k$, for $k=1,2,...$ Consider a sequence of positive real numbers $c_1,c_2,...$ Take a function $g(t)=\sum\limits_{k=1} ^\infty ...
4
votes
1answer
49 views

Are these conditions sufficient to calculate this expectation?

We have \begin{aligned} E(Z_1) = A \\ \Pr \{ Z_2 = Z_1 + 1 \} = \frac 1 2 \\ \Pr \{ Z_2 = Z_1 - 1 \} = \frac 1 2 \end{aligned} Are these conditions enough to get $E(Z_2)$?
1
vote
4answers
287 views

probability question- expectation

A bowl contains 10 chips , of which 8 are marked 2 dollars each and 2 are marked 5 dollars each. Let a person choose at random and without replacement, 3 from this bowl. If the person is to receive ...
8
votes
2answers
915 views

Biased Random Walk and PDF of Time of First Return

I have a random walk process where each step the probability of $+1$ is $p$ and $-1$ is $q$, with $p+q=1$. $p$ may not equal $q$. The walker starts at zero. I want to know the probability that the ...
0
votes
1answer
107 views

Unbiased (random?) selection algorithm

Let say we have the following set $S = \{x_1, x_2, x_3, ..., x_n\}$ where $x_i$ is a real number between $0$ and $1$. Now I want to find an algorithm that randomly generates a subset of $S$, free to ...
6
votes
1answer
569 views

The definition of independent discrete random variables

In probability books, the definition of independent discrete random variables are often given as The random variables $X$ and $Y$ are said to be independent if $\mathbb P(X \leq x, Y \leq y) = ...
4
votes
1answer
115 views

Extension of $3\sigma$ rule

For the normally distributed r.v. $\xi$ there is a rule of $3\sigma$ which says that $$ \mathsf P\{\xi\in (\mu-3\sigma,\mu+3\sigma)\}\geq 0.99. $$ Clearly, this rule not necessary holds for other ...
2
votes
1answer
547 views

Normalizing a conditional probability to within range of a Sigmoid function

Given the following scenario from another post of mine where we are building a matrix that expresses the probability of first order transitions from one character to another in an english text. We ...
-2
votes
1answer
326 views

One drunk receptionist and four brilliant mathematicians

Suppose that there is one hotel with nine floors (first floor = ground floor + 1) where the math seminar takes place, four brilliant mathematicians who are guests of the hotel, one drunk receptionist ...
0
votes
1answer
60 views

Trying to show a few facts about expectations of $e$ if $y = x\beta + e$

I have a few questions where I'm trying to show if things are true or false. I'll say upfront that these are homework so I'd rather not get the entire answer just someone to point me in the right ...
6
votes
1answer
539 views

Expected Value of a Continuous Random Variable

I've been reviewing my probability and statistics book and just got up to continuous distributions. The book defines the expected value of a continuous random variable as: $E[H(X)] = ...
1
vote
1answer
110 views

What does $\stackrel{\mathcal L}{=} $ mean?

I guess $ A \stackrel{\mathcal L}{=} B $ means $A$ and $B$ hand side has the same probability distribution. Is that right?
4
votes
5answers
592 views

Probability with my Facebook friends

If I have 5000 Facebook friends, what is the probability that a day with no one having that birthday exists? I assume there are 365 days in a year, and a uniform distribution of the dates of birth. I ...
2
votes
1answer
190 views

Probability with Intersections, Unions, and Complements

I just want to make sure I'm doing this correctly. Here is the problem: Let $A$ and $B$ be sets such that $P(A \cap B)=\frac{1}{4}, P(\tilde{A})=\frac{1}{3},$ and $P(B)=\frac{1}{2}$. What is $P(A ...
1
vote
1answer
121 views

Showing $\bar{y} \rightarrow \mu$ Have I done enough? Converges in probability

From the title I'm supposed to show $\bar{y} \rightarrow \mu$ (converges in probability) where $$y_t = \mu + u_t$$ $$ u_t = \rho u_{t-2} + \epsilon_t$$$$ E(\epsilon) = 0, E(\epsilon^2) = \sigma^2, ...
5
votes
1answer
1k views

Probability that two random numbers are coprime

This is a really natural question for which I know a stunning solution. So I admit I have a solution, however I would like to see if anybody will come up with something different. The question is ...
5
votes
1answer
146 views

Prove ranks are uniformly distributed

We have n IID random variables $X_1, X_2, \ldots, X_n$. Let $R_i$ be $X_i$'s rank in the set $\{X_1, X_2, \ldots, X_3 \}$ when we order from large to small. How to prove $R_i, \forall i \in \{1, 2, ...
4
votes
1answer
216 views

A Boolean function with total influence 1 must be a dictatorship

Let $f:\{-1,1\}^n\to\{-1,1\}$ be a boolean function. Define the influence of the $i$'th coordinate of $f$ as follows: $$\operatorname{Inf}_i(f)=\Pr_{x}[f(x)\neq f(\hat x_i)]$$ where $x$ is uniformly ...
1
vote
1answer
1k views

Question regarding counting poker dice

Problem Poker dice is played by simultaneously rolling 5 dice. How many ways can we form "1 pair", "2 pairs"? For one pair, I got the answer right away. First I consider there are 5 spots for 5 ...
2
votes
0answers
86 views

Random walks - two questions

Let us suppose that a person are throwing a coin. He'll get one dollar if he win, and he'll pays one dollar if he loses. I understand that the winning will trend to zero in the case of unlimited ...
2
votes
1answer
652 views

Birthday attack/problem, calculate exact numbers? [duplicate]

Possible Duplicate: Birthday-coverage problem An example of what I wish to do is the following: http://stackoverflow.com/questions/4681913/substr-md5-collision/4785456#4785456 How would I ...