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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98 views

Why does $n \choose r$ where $r = 1,n$ track $2^n$?

I bashed together a clunky ruby script to find the sum total of $n \choose r$ where $r = 1,n$ I wanted to determine how many lines of output I could expect from a script that produces all possible ...
2
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1answer
41 views

Ask for a question about independence of random variable from an event

Consider two independent tosses of a fair coin. Let random variable X take the value 0 if the first toss is a head and take the value 1 if the first toss is a tail. Let A be the event that the number ...
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4answers
373 views

Probability interview question

Suppose we have three positive integers $A, B, C$. We randomly choose an integer $a$ in the range $[0,A]$ and an integer $b$ in the range $[0,B]$. Find the probability that $a + b\leq C$. I am unable ...
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0answers
68 views

Proof that the sum does not depend on enumeration …

I've come across the following basic lemma in a basic probability book, and I can't seem to understand why the provided argument is enough to prove it. Lemma: Let $I$ be a countably infinite set and ...
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1answer
24 views

Recovering density parameters from distribution function

Let $X$ be a random variable with probability density function $g(x;\theta_1,\theta_2)$, where $g$ is parameterized by two real numbers $\theta_1$ and $\theta_2$. I'd like to specify that $$ P(a \leq ...
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1answer
75 views

Probability of getting the same vector result

This is part of a mathematical puzzle I was given to me by a friend a while ago and I can't work out how to solve it. Does anyone have any ideas? For a given vector $v \in \{-1,1\}^n$ we consider the ...
2
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0answers
31 views

Central limit theorem in multidimension with unknown covariance

Let $X_1,\dots,X_n$ be samples from a distribution on $\mathbb{R}^d$ that has a finite second moment. If $d=1$, $\bar{X}_n=1/n\sum_{i=1}^nX_i$ and $S_n=1/(n−1)\sum_{i=1}^n(X_i−\bar{X}_n)^2$ then $$\...
2
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1answer
191 views

Continuous and measurable in each variable $\implies$ product measurable?

Consider a metric space $A$ with a metric $d$, and consider the measurable space $(A,\mathcal{B}(A))$ with the Borel $\sigma$-algebra generated by $d$-open sets. Let $(\Omega,\mathcal{F})$ be a ...
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0answers
101 views

The law of large numbers - limits of $\max$ vs $\max$ of a limit.

Assume that $X_{1,1}, \dots , X_{1,n}, X_{2,1},\dots, X_{2,n}, \dots ,X_{n,1}, \dots , X_{n,n}$ are i.i.d. random variables, and that $\mathbb EX_{i,j}$ exists and is finite. From the strong law of ...
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0answers
22 views

Formula for running-time complexity

I'm regarding a stochastic process $(X_t)$of which the mean starts at $O(n)$ and is reduced by the factor $(1-r)$ in each step with $r = \Omega (1/n^9)$, so $$E(X_{t+1}) \leq E(X_t) (1-r) .$$ Now it ...
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1answer
22 views

computing weight from distance metric

I have a distance between two points in meters. I want to convert this distance into weight such that as distance increases the weight decreases. What are some good weighting function that can ...
2
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1answer
42 views

Probability in knockout games.

Suppose in a knockout tournament 32 players p1 , p2 .....p32 participate. In each round players are divided into pairs at random and winner goes to the next round. If p5 reaches semifinal what is ...
1
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1answer
63 views

Weak convergence of a double sequence of random variables

Consider two sequences of random variables, $\{X_n\}$ and $\{Y_n\}$. Let's assume $X_n\xrightarrow{D} X$, $Y_n \xrightarrow{D} Y$, and $\{X_n\}$ and $\{Y_n\}$ are independent of each other. It is well-...
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6answers
88 views

Finding $P(X < Y)$ where $X$ and $Y$ are independent uniform random variables

Suppose $X$ and $Y$ are two independent uniform variables in the intervals $(0,2)$ and $(1,3)$ respectively. I need to find $P(X < Y)$. I've tried in this way: $$ \begin{eqnarray} P(X < Y) &...
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0answers
26 views

Calculating Variance of payment in patterns of balls.

We have five different bags labeled from 1 to 5 and several colored balls. There are 9 different possible colors. We know how many balls of each color there are in each bag. We have a grid of 5x3 (...
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0answers
35 views

Distribution of sample skewness and kurtosis

I am working on my thesis right now and I'm almost done with it, but just on the last step I encountered some problems with a proof. I have an independent sample $X_{1}, ..., X_{n}$ that follows the ...
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2answers
60 views

A question on probability of choosing coins

Six identical-looking coins are in a box, of which five are unbiased, while the sixth comes up heads with probability $3 \over 4$ and tails with probability $1 \over 4$. Three coins are chosen from ...
1
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1answer
216 views

Distribution of the heads-tails difference after three coin tosses

Three fair coins are tossed and D is the positive difference between the number of heads and the number of tails obtained, so D takes the values 1 and 3. Tabulate the probability distribution of D and ...
2
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1answer
67 views

Error propagation, why use variences?

I have been reading up on error propagation and am slightly confused about something. We can the error in $c=f(a,b)$ as the: $$\sigma(c)= f_a \sigma_a+f_b \sigma _b$$ Firstly is this correct and am I ...
1
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1answer
374 views

How do I use interpolation with the Z table?

My textbook has an example of interpolation, but I am not sure how the book did it since it doesn't explain it. It says if we want $P(Z < 1.246)$ we must use interpolation and the steps given are: ...
3
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1answer
32 views

Median of waiting time for $k$-th ace from bridge cards

I can't figure out how to get median of a waiting time from the exercise 36 from W. Feller's book An Introduction to Probability Theory and Its Applications Vol.1 (bold in the quote): ...
3
votes
1answer
40 views

Show that $Y = \sum_{i=1}^n Y_i$ is distributed as $\chi _{2n}^2$.

The Statement of the Problem: Suppose that $X_1,\ldots, X_n$ is a random sample from the $U(0,1)$ distribution and $$ Y_i = -2\log X_i. $$ Show that $Y = \sum_{i=1}^n Y_i$ is distributed as $\chi ...
3
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3answers
6k views
2
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2answers
95 views

Probability returning to initial state

Let $P=\begin{bmatrix}0&\frac{1}{2}&\frac{1}{2}\\\frac{1}{2}&0&\frac{1}{2}\\\frac{1}{2}&\frac{1}{2}&0\end{bmatrix}$ and $P^{(n+1)}=P^{(n)}P.$ I know that if you start in any ...
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1answer
27 views

Probability involving a moment generating function

Suppose that X1 and X2 are independent and identically distributed discrete random variables. The moment generating function of X1 + X2 is: M(t)= 0.01e^(-2t) + 0.15e^(-t) +0.5925 + 0.225e^(t) + 0....
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1answer
55 views

Lottery probability with payout system

Assume we have a lottery which has following payouts 1,2,5,6,9,10,16. The organizer expects 4% profit from the lottery. I wrote ...
0
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1answer
200 views

How to find median from a probability distribution?

Having trouble on something that should be really, really easy. I need to find the median of the following probability distribution...but according to the website I linked below...I'm doing it ...
0
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1answer
46 views

An application of Jensen's Inequality for dependent random variables

Consider dependent and positive valued random variables $A,B$ and $X$. I want to prove that \begin{equation} E[X^2 A] E[B] \ge E[X A] E[X B]. \end{equation} If $A$ and $B$ were scalars, above would ...
1
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2answers
33 views

Show that the conditional pmf of $X_i$, given $T = t$, is $Binomial(t, \lambda_i/\lambda).$

The Statement of the Problem: The random variables $X_1, ..., X_n$ are independent and $X_i \sim Poisson(\lambda _i), i = 1, ..., n$. Set $$ T = \sum_{i=1}^n X_i \qquad \text{and} \qquad \lambda = \...
3
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1answer
73 views

Probability & statistics

Q: In a given town, $270$ days of the year have a winter-like weather and the other $95$ days have a summer-like weather. On a winter-like day the probability of a sunny day is $0.3$. On a summer-like ...
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0answers
103 views

Pure death processes

If $P_n (t)=\Pr (N (t)=n)$ and $N (0)=a$, how can I show that in a pure death process $$P_{(a-1)}(t)=a (e^{\mu t }-1)e^{-a \mu t}.$$ I showed that $P_a(t)=e^{-a \mu t}$. In fact I want to show $...
2
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2answers
110 views

Sum of remainders of $2^n$

Hints Only Let $R$ be the set of all possible remainders when a number of the form $2^n$, $n$ a nonnegative integer, is divided by $1000$. Let $S$ be the sum of all elements in $R$. Find the ...
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3answers
250 views

Monty Hall Problem

In the Monty Hall problem, when the host picks a door and reveals an goat, does it make any difference if he did not know which door the real car was behind, and he just happened to pick a door with a ...
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0answers
57 views

What is the PMF of the Hamming weight of a multinomial random variable?

Assume that $X$ is a random variable following a multinomial distribution of parameters $n$ (number of trials) and $p=(p_1,\dots,p_k)$ (event probabilities). Hence, $$\Pr\{X=(x_1,\dots,x_k)\}=\frac{n!}...
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1answer
96 views

Given 5 colors to choose from, how many ways can we color the four unit squares of a $2\times 2$ board

Given 5 colors to choose from, how many ways can we color the four unit squares of a $2\times 2$ board, given that two colorings are considered the same if one is a rotation of the other?
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3answers
148 views

Given a 50 card deck with cards numbered from 1 through 10 in each of 5 suits, how many

Given a 50 card deck with cards numbered from 1 through 10 in each of 5 suits, how many 5 card hands are there that include exactly one pair of two cards that have the same numeric value?
3
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1answer
311 views

Intuition about Blumenthal's 0-1 law

I'm studying Brownian motion from Durrett. I'm trying to understand what Blumenthal's 0-1 law really says about what Durrett calls the germ field, $\mathcal{F}_0^+$. Let $\mathcal{F}_t^+ = \cap_{s &...
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2answers
44 views

Central Limit Theorem and understanding mean for a single object

The IQ of actuarial science majors is assumed to be normally distributed with mean 112 and standard deviation of 14. In a class of 19 students, find the probability that the mean IQ of all 19 students ...
2
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2answers
176 views

How to solve “ways of seating around a circular table”

Recently I asked a question about seating, here it is again: The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five ...
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1answer
39 views

Type I error in Normal distributions

Let $X_1,\dots , X_n \stackrel{iid}{\sim} N(\mu, \sigma^2 = 4)$ Test $H_0: \mu = 10$ vs $H_1: \mu > 10$ take a random sample of $n=16$ and reject $H_0$ if $\bar{x}>14$ Find $\alpha$ the type I ...
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1answer
78 views

Type I and type II errors

Let $X \sim uniform(0,\theta)$ we are testing $H_0: \theta = 1$ vs $H_1: \theta >1$ If we know that we reject $H_0$ if $X>0.9$ (1) find $\alpha$, the type I error (2)Suppose that $\theta=1.1$. ...
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1answer
36 views

getting intuition about a fact on probability

Let $X: \Omega \to \mathbb{R}$ be a random variable.let $L^1$ be the set of all random variables which have finite expectation. My books says that if $X^2 \in L^1$, then $|X| \in L^1$. My question is:...
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0answers
40 views

A finite field subset sum count

Given $d\in\Bbb N$, pick $N=2^{2d}$ distinct $a_j$ from $\big\{1,\dots,2^{d^2}-1\big\}$ and pick $i$ from $\big\{3,\dots,2^{d}\big\}$. On average how many of $i$-subsets in $\big\{\alpha^{a_j}\big\}_{...
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1answer
92 views

Integrals of indicator functions question

I have a result $\int_X \int_Y \mathbb{1}[h(x,y) < \mu]dP(y)dP(x) < a$ and I am trying to resolve the integral $\int_X \int_Y \mathbb{1}[|f(x) - g(x)| > \frac{\mu}{2}] \mathbb{1}[h(x,y) &...
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1answer
79 views

Probability of time between two events in a poisson process

Suppose people arrive at a certain place according to a poisson process with rate 10 per day. 1) What is the expected time until the arrival of 100 person. 2) What is the probability that ...
3
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0answers
54 views

Can Monotone Class Theorem be easier to check than $\pi$-$\lambda$ Theorem?

I've been working on problem 14.4 in Billingsley's "Probability and Measure", which says: "Let $C$ be the set of continuity points of $F$. Show that for every Borel set $A$, $P(F(X) \in A, X \in C)...
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1answer
43 views

How to derive formula for marginal probability of choosing nest in nested logit model?

I am trying to understand all the details of the nested logit and what confuses me is the formula for marginal probability of choosing the nest. In more details: the joint probability of individual n ...
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0answers
49 views

Probability of multiple wins in a series of evenly matched teams

24 evenly matched players are divided into 6 teams. (Evenly matched by virtue of golf handicaps). Team assignment is random. One player frequently wins, irrespective of which team he is assigned. I ...
2
votes
6answers
1k views

Find $ P(Z>X+Y)$ where $X,Y,Z \sim U(0,1)$

I'm trying to follow a line in a derivation for $P(Z>X+Y)$ where $X,Y,Z$ are independent continuous random variables distributed uniformly on $(0,1)$. I've already derived the pdf of $X+Y$ using ...
1
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2answers
58 views

Confusion regarding Burke's theorem

Arrivals occur at rate $\lambda$ according to a Poisson process the service time have an exponential distribution with parameter $1/\mu$ in an M/M/1 queue, where $\mu$ is the mean service rate where $\...