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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1answer
375 views

Is there a symbol for “dependent”?

For random variables $A$ and $B$, $A \perp B$ is sometimes used to denote "A in independent of B". Is there a symbol that is commonly used to mean "A is not independent of B"?
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1answer
9 views

Distribution of the summation of k random variables and k is also variable

We have a set of positive random variables $\boldsymbol X=\{X_1,X_2,\ldots\}$, where $X_1,X_2,\ldots,$ are independent and identically distributed (i.i.d.). The CDF $F(x)$ and PDF $f(x)$ for Xi are ...
0
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1answer
92 views

Coupon collector problem

Let $T$ be the time to collect all $n$ coupons, and let $t_{i}$ be the time to collect the i-th coupon after $i − 1$ coupons have been collected. Think of $T$ and $t_{i}$ as random variables. ...
-1
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0answers
19 views

expected number of steps for chossing randomly each number between 1 to $n$ at least $k$ times

Assume the following game: Every step choose a number between 1 to $n$ randomly i.e. every integer between 1 to $n$ is chosen with probability $\frac{1}{n}$. Success is when every number has been ...
1
vote
1answer
18 views

the probability density function (PDF) of concatenation of two Gaussian variables

Gaussian variable $x$ follows from $N(u_x,\sigma_x^2)$ and $y$ follows from $N(u_y,\sigma_y^2)$. Assume we have the vector $\bf{z}=[x,y]^T\in R^2$, then it seems that no matter whether $x$ and $y$ are ...
1
vote
1answer
28 views

Is this a binomial or multinomial question?

You can donate to a company: $10$ dollars , $20$ dollars or nothing. In a mall there are $70$% young people and $30$ % old people. $50$% from the old people aren't donating anything. ...
2
votes
1answer
23 views

Non-Linear System. Find the conditional expectation.

I've had my test for this course and I think I failed it again. The hardest part for me is findig the correct distributions. This is a test exercise I couldn't figure out or at least, I probably ...
1
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2answers
32 views

Tail probability of a random variable

Here are two theorems about the "tail probability" of a random variable X Thm1: The expectation $E(|X|^\alpha) < \infty$ for some positive $\alpha$ if and only if $$\sum_{n=1}^\infty ...
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2answers
23 views

Understanding Conditional Probability Basics

In many online sources I've read a statement similar to: Probability of B happening given A is equal to the probability of A and B both happening divided by B happening or $p(A | B) = p(A \cap ...
3
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2answers
40 views

Dice Roll Probabilities

I'm trying to figure out the probabilities for the following casino game: You and the dealer each roll a pair of dice and the person with the highest individual die roll wins. If its a tie, you win. ...
1
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0answers
18 views

Is this a misuse of the term “probability space”?

Let me first state the definitions as I am using them. Do correct me if I am wrong here! A "probability space" is a triple $(\Omega, F \subseteq 2^{\Omega}, \mu : F \rightarrow [0,1])$. The ...
0
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2answers
72 views

Probability of an interval (A, B) being in (C, D) or vice versa

$S$ is the domain. $A, B, C, D \in S$. $A, B, C, D$ satisfy the condition $A \le B$ and $C \le D$ and hence $(A, B)$ and $(C, D)$ are intervals. All four are values picked from respectively $4$ ...
1
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2answers
34 views

Erin rolls 4 four-sided dice all at once, then can roll a subset of her choosing a 2nd time. What is the probability of getting all the same number?

Here's what I have so far: All 4 same on first try = (1/4)^4 * 4 3 same, then get 4th on 2nd roll = 4 * (1/4)^3 * (3/4) * (4!/3!) Here's where I'm confused: 2 same = 4 * (1/4)^2 * (3/4)(2/4 :to ...
1
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0answers
21 views

Probability of drawing >18 when drawing 3 cards

I am trying to calculate some probabilities for a card game. Players have to draw 3 cards each time and the cards must add up to a certain value for them to win - the value changes depending on the ...
0
votes
1answer
513 views

For each of the following, determine the constant c so that f(x) satisfies the conditions for being a p.m.f

For each of the following, determine the constant c so that f(x) satisfies the conditions for being a p.m.f. for a random variable X. c) f(x) = x/c, x = 1,2,...,n d) f(x) = c/(x+1)(x+2), x = ...
1
vote
1answer
24 views

Distribution of Summation of two discrete random variables

Here, $X_1$ and $X_2$ are independent discrete random variable and the support set of $\tilde{x}_1$ and $\tilde{x}_2$ respectively. We have mentioned the support sets below: $$ X_{1} = \{ 2,3,...,7 ...
3
votes
1answer
25 views

Conditional Probability for Exponential Random Variables

I'm working through a practice problem for an exam and I would like to verify that I've done it correctly. Additionally I'd like some insight on the intuition behind the numbers I'm getting. Problem ...
0
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0answers
14 views

Prove that $\tilde{X}_{\tilde{\theta}}(t)$ is a martingale

Let me introduce the objects: 0) $(\Omega, \mathcal{F},\Bbb{P})$ is a probability space 1)$S_N $ is the set of symmetric, non-negative definite $N\times N$ matrices 2)$a:[0, \infty) \times \Omega ...
1
vote
2answers
86 views

Probability of Warcraft

So I have this probability exercise from Khan Academy, which is about World of Warcraft ^^ Marvin lives in Stormwind city and works as an engineer in the city of Ironforge. In the morning, he ...
0
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0answers
24 views

Probability of Probabilities :)

So here is a tough one (or so i think). i have 15 games (30 teams). and only 2 options i can chose from (even / odd number of goals). I want to bet a ticket with each possible combination. How many ...
1
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1answer
17 views

Probability of picking all white marbles?

Consider that you have a drawer with n marbles of various colors. There are 5 white colored marbles. You grab k marbles from the drawer, where k <= n. What is the probability you find all 5 white ...
1
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1answer
16 views

Probability and Stats (loaded coin)

Smith is offered the following gamble: he is to choose a coin at random from a large collection of coins and toss it randomly.The proportion of the coins in the collection that are loaded towards a ...
0
votes
1answer
19 views

Confidence interval of a uniform distribution

I need some help with the following problem: I want to estimate $n$ of $X_i \sim U(1, n)$ with a $90\%$ confidence level. What is given is the sample size with $10$ and the maximum of the sample with ...
0
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0answers
29 views

Strategy for choosing lottery numbers when buying many tickets

In a given lottery a user must choose 5 out of 50 numbers, without replacement. Prizes are offered for matching at least 2 of the winning numbers. If a user can purchase multiple tickets (let's say ...
1
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1answer
285 views

One-tailed two-sample T-test OK?

I'm trying to conduct a one-sided hypothesis test between two random variables which are both asymptotically normally distributed with different variances. The variances are not known but have been ...
0
votes
1answer
18 views

Probability of Winning a Toss

I have an unfair coin with two sides 1 and 2. I have a problem and its constraints. The coin has to be tossed until I win; which happens when 1 shows up in a toss. Constraints: Since the coin in ...
3
votes
1answer
90 views
+50

Properties Least Mean Fourth Error

I am interested in whether a quantity \begin{align*} E[(X-E[X|Y])^4] \end{align*} has been studied in the literature before. I am not even sure if "least mean fourth error" is a correct name, since ...
0
votes
3answers
62 views

Cumulative distribution function of Cauchy distribution

Let X be a Cauchy distribution with X~Cauchy(1) (so a=1). Prove that Y=1/X has the same cumulative distrubtion as X. Now I've tried taking F_X(x) for a=1 combined with the identity ...
0
votes
1answer
15 views

probability of a proportion point estimate

I've got a problem where I'm supposed to find the probability of a point estimate but cannot see how my answer is differing from the given one. The problem is: Unknown to an experimenter, the ...
0
votes
2answers
39 views

12 six-sided dice are thrown. What is the probability of getting each number twice?

I got this: $\frac{6!12!}{6^{12}2!^6}$ but the answer is this: $\frac{{12!}}{6^{12}2!^6}$ Im not sure I understand why you wouldn't write the $6!$ because if the first die's value is #3 then you have ...
1
vote
1answer
35 views

Asymptotic Behavior of Binomial Distribution

I am considering the following problem: Given the following equation: \begin{equation*} c = \sum_{k=n}^{2n-1} \binom{2n-1}{k} p(c)^k (1-p(c))^{2n-k-1} \end{equation*} Which is the probability that ...
0
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0answers
12 views

Proving Properties of Discrete Time Markov Chain mathematically

I want to prove that the queue length at a store is not a Discrete Parameter Markov Chain (DPMC). Now I have the equation: $$Q_k = (Q_{k-1} - 1) + V_k$$ $Q_k$ is the queue length at time instant ...
1
vote
1answer
25 views

How to interpret a p-value that's significant from Fisher's Exact test

Given a binomial distribution with p=.03, n=902, the $.025$ and $.975$ quantiles are $17$ and $38$ respectively. I interpret ...
0
votes
1answer
15 views

Illegal lottery problem (Merging dependent bernoulli trials)

Suppose I am in a town that playing lottery is illegal. If I buy a ticket for 1 dollar, I will win the lottery with probability $p$. Each time I buy a ticket, the police may catch me and confiscate ...
-1
votes
1answer
12 views

Density of a distribution function at upper bound [on hold]

Consider a strictly increasing continuously differentiable distribution F with support on $[a,b]$. Let $f$ be the pdf of $F$. What can we say about $f(b)$? Under what conditions is $f(b)>0$? ...
1
vote
2answers
31 views

One-One Correspondences

Adam the ant starts at $(0,0)$. Each minute, he flips a fair coin. If he flips heads, he moves $1$ unit up; if he flips tails, he moves $1$ unit right. Betty the beetle starts at $(2,4)$. Each ...
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1answer
28 views

On a 50 question multiple choice exam with 5 choices per questions, What are the odds that I get 100% if I were to Guess every answer? [on hold]

What would the odds be to get 100% on a multiple choice exam where I guessed the answer to all 50 of the multiple choice questions (5 choices per questions)? A 1 in how many chance?
0
votes
1answer
34 views

A detail on a proof of the strong Law of Large Numbers.

In the following blog post https://terrytao.wordpress.com/2008/06/18/the-strong-law-of-large-numbers/ one is presented with a nice account of the LLN. Suppose that I have shown that if $(n_j)$ is a ...
2
votes
1answer
27 views

Expected number of drawings to find collision

Consider a group $G$ consisting of $n$ distinct elements. Suppose we draw random elements of $G$ (one by one, replacing each element every time) until we draw an element that was drawn before (we say ...
2
votes
1answer
24 views

(Elementary) Markov property of the Brownian motion

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname{P})$, i.e. $B$ is a real-valued stochastic process with $B_0=0$ almost surely $B$ has independent ...
2
votes
2answers
41 views

Proving that the Poisson compound process has independet increments

Let $X_t=\sum_{i=1}^{N_t}J_i$ be a compound Poisson Process, where $J_i$ are independent and equidistributed. I have to prove that for every $0<t_1<t_2 \leq t_3<t_4$ : $X_{t_4}-X_{t_3}$ is ...
3
votes
1answer
416 views

The PMF of the larger of two numbers selected at random from $1,\dots,12$

Two balls are chosen at random from a box containing 12 balls, numbered 1;2; : : : ;12. Let X be the larger of the two numbers obtained. Compute the PMF of X, if the sampling is done (a) ...
8
votes
2answers
325 views

How far do I need to drive to find an empty parking spot?

A parking lot consists of an infinite row of bays. Cars arrive at random intervals (mean interval $T_a$) and stay for a random time (mean stay $T_s$). The time intervals are memoryless (negative ...
53
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9answers
9k views

If I flip a coin 1000 times in a row and it lands on heads all 1000 times, what is the probability that it's an unfair coin?

Consider a two-sided coin. If I flip it $1000$ times and it lands heads up for each flip, what is the probability that the coin is unfair, and how do we quantify that if it is unfair? Furthermore, ...
1
vote
0answers
16 views

A Question about the Kurtosis

Problem: Show that if a binomial distribution with $n = 100$ is symmetric, its coefficient of kurtosis is 2.9. Answer: First, I am interpreting the term symmetric to mean that $p = q = \frac{1}{2}$. ...
0
votes
0answers
13 views

An asymptotic ratio of samples

Let $S_n = \left\{X_1, \dots, X_n \right\}$ be a sample of idd random variables for all $n \in \mathbb{N}$. Consider two sequences of measurable sets $\left( A_n \right)$ and $\left(B_n \right)$ such ...
0
votes
0answers
20 views

Absolute value of a sum of non-identically distributed RVs

Let $X=\left|\sum _{i=1}^n Z_{i} \right|$ where random variables $(\textit{Z${}_{i}$})$ are independent but $not$ identically distributed, and, $Z_{i} =0$,$+1$ or$-1$, with probability $1-a_i$, ...
0
votes
2answers
24 views

The distribution of the product of Gaussian variable and Rademacher variable.

I have two independent variables: $X$ follows from standard Gaussian distribution $N(0,\sigma^2)$; $Y$ follows from Rademacher distribution, i.e., $Y$ can be either $-1$ or $1$ with the same ...
2
votes
1answer
29 views

How to compute $\mathbb{E}(\prod_{i=1}^n(1+X_i)\textbf{1}_{\prod_{i=1}^n(1+X_i)\leq M})$

I want to compute $\mathbb{E}(\prod_{i=1}^n(1+X_i)\textbf{1}_{\prod_{i=1}^n(1+X_i)\leq M})$, where $\textbf{1}$ is the indicator function and $X_i$ are continuous independent and equidistributed ...
0
votes
1answer
31 views

Bins and Balls problem several balls at once

I'm trying to calculate the expected value of the number of balls that i need to choose for fill all bins with at least one ball. There are $N$ empty bins labeled from 1 to $N$, and infinitely many ...