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
5 views

Understanding different definitions of bayes theorem

I am taking course on probability and reading about bayes theorem. In Sheldon Ross' book, it given as $$P(E) = P(E|F)P(F) + P(E|F^C)P(F^C)$$ with note: Equation above states that the probability of ...
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0answers
11 views

Inequality for the derivative of a density of a random variable convolved with a normal r.v.

I have a question about the following proof. The statement is: Let $X$ be a random variable and $Z_\tau \sim N(0,\tau)$ be an independent random variable. Then $Y_\tau := X + Z_\tau$ has a ...
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1answer
21 views

Random variable with all higher order moments zero?

Is there a random variable with finite first and second moment but all higher order (non-central) moments are zero?
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1answer
47 views

Why does this expectation integrate to 1

Let $p(y|\theta )$ be our likelihood, and $p_{N}(\hat{y}|\theta)$ be an unbiased estimator of our likelihood. Let $z=\ln p_{N}(\hat{y}|\theta) - \ln p(y|\theta )$, and $g_{N}(z|\theta)$ be the ...
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1answer
12 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 ...
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0answers
29 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 ...
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1answer
19 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 ...
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1answer
30 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. ...
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2answers
24 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 ...
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1answer
25 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 ...
3
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1answer
25 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 ...
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2answers
37 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 ...
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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 ...
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0answers
25 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 ...
3
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1answer
27 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 ...
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1answer
18 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 ...
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1answer
17 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 ...
1
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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 ...
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2answers
41 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. ...
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0answers
31 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 ...
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2answers
23 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 ...
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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 ...
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0answers
13 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 ...
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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 ...
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1answer
37 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 ...
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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 ...
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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 ...
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2answers
32 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?
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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 $f(b)>0$? ...
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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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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 ...
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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 $4$ ...
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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 ...
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0answers
18 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 $p = q = \frac{1}{2}$. ...
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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 ...
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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 ...
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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 ...
0
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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 ...
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0answers
12 views

Definition of expected value of a continuous random variable [duplicate]

Let $X$ be a random variable with the probability desntiy function $f$. Then, according to the book "Intro to probability and statistics" by Rohatgi, the expected value of $X$ is defined as: $$E(X) ...
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0answers
21 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 $1-a_i$, ...
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2answers
14 views

Two different samples from different time periods.

I have a sample of grades from 1000 students. The average mark was 60 with a standard deviation of 3. A year later I collected a sample of grades from 50 students sitting the same test. The average ...
2
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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 ...
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0answers
30 views

Find the expected value .

I am having a trouble in in the following questions. I would like if some one could help me. Thank you so much for your time. Simplify $$K=\int_0^T\left[\int_0^t e^{-\alpha(t-s)}dJ(s)\right] dt $$ ...
0
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1answer
21 views

Probability Of Birthday Months

In a class of 20 students, what is the probability that at least one will be born in november or december? Using the complement we can look at $P(\overline A)=(\frac{10}{12})^{20}$ so ...
0
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1answer
28 views

Geometric sum of geometric random variables

I am looking to find the probability mass function of $Y=\sum_{i=1}^NX_i$ where $X_i\sim\textrm{Geometric}(a)$ and $N\sim\textrm{Geometric}(b)$. I attempted to do this by finding the probability ...
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0answers
14 views

Optimal decision for sampling a distribution.

I was wondering which probability distribution is best sampled with $\pm\alpha^n, n\in\{1,2,\cdots\}$ for various values of alpha. Sampling means to pick the one which is closest, store the sign and ...
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35 views

Plz help me I need the answer urgently [on hold]

So if 15 people join a raffle, and they are giving out 4 prizes. What is the profanity of me winning a prize. Each person only put in one slip. They also took out the winners name after each round.
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1answer
22 views

How to show that $\mathbb{E}(\lim_{n \to \infty} X_n) = 0$ when $X_n(x) := n \cdot 1_{[0,\frac{1}{n}]}(x) \qquad (x \in [0,1])$

from the answer of Exchanging limit and expectation for $L^2$ random variables: Consider for example the probability space $(\Omega,\mathcal{A},\mathbb{P}) := ...
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0answers
21 views

Closeness in distribution implies closeness in statistics?

I am aware that convergence in distribution does not necessarily imply convergence in the mean. I browsed through some examples of statistical distances here ...