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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0
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1answer
11 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 ...
1
vote
0answers
8 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
votes
1answer
18 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 ...
0
votes
0answers
8 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) ...
0
votes
0answers
6 views

Absolute value of a sun 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 ${p}{}_{i}$, ...
0
votes
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
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
0answers
21 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
votes
1answer
19 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
votes
1answer
23 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 ...
0
votes
0answers
13 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 ...
-9
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0answers
32 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.
1
vote
1answer
19 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}) := ...
1
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0answers
20 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 ...
0
votes
1answer
57 views

Baye's Theorem Conditional Probability with multiple conditions

Lets assume I have a supermarket and I track the purchase history of my customers with 2 attributes of each customer - Gender (M/F) & Smiling (Y/N). Assume this is historical data of purchases: ...
1
vote
1answer
25 views

Generating Finite Groups By Random Premultiplication With Generators

Let $G$ be a finite group with identity $e$ and $S$ be a set which generates $G$. Is it always possible to define a procedure of the form: Start with $x=e$. With probability $p_1$, replace $x$ with ...
0
votes
2answers
31 views

Probability clarification

Consider the reading habits of the X class. If we know that 30% of the class students read USA Today daily, 40% read Salt Lake Tribune daily and 10% read both of them daily, what is the percentage of ...
0
votes
0answers
7 views

Copula vs Exprimental Copula

I have read some texts about finding/approximating copulas for a given sample based on known (famous) copulas. My question is: when we have the experimental CDF of (X, Y), why we should try to find a ...
0
votes
0answers
19 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 ...
-1
votes
0answers
37 views

Select the correct probability distribution for the given problem [on hold]

An airline company annually gives $5000$ vouchers to its clients. In the previous year, $67 $% of the clients that received a voucher redeemed it. (a) Let X be number of clients that will redeem the ...
0
votes
1answer
23 views

Compound Poisson Process property:$\mathbb{P}(\sum^{N_{t_4}-N_{t_3}}_{i=1}J_i \leq n)=\mathbb{P}(\sum^{N_{t_4}}_{i=N_{t_3}+1}J_i \leq n)$

I am trying to demostrate that the Compound Poisson Process has independent increments, and I have a problen because I have to use that: :$$\mathbb{P}(\sum^{N_{t_4}-N_{t_3}}_{i=1}J_i \leq ...
1
vote
0answers
20 views

Is the Martingale property still true for $\xi$ not necessarily $C^1$?

Denote $$M(t) = f(t, \alpha(t))\exp \bigg\{-\int_0^t g(u, \alpha (u)) \, du - \int_0^t h(u, \alpha(u)) \, d\xi(u)\bigg\}$$ Here $\xi: [0,\infty) \times \Omega \to \Bbb{R}$. If for each $\omega$ the ...
1
vote
1answer
41 views

What is the sum of all $k$ values?

In an urn there are a certain number (at least two) of black marbles and a certain number of white marbles. Steven blindfolds himself and chooses two marbles from the urn at random. Suppose the ...
0
votes
1answer
18 views

Prove that $\sigma$-algebras $A_1,\ldots,A_n$ are independent if and only if $A_i$ is independent of each $A_1,\ldots,A_{i-1}$, for all $i=2,\ldots,n$

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space and $\mathcal{A}_1,\ldots,\mathcal{A}_n\subseteq 2^\Omega$ be $\sigma$-algebras. How can we show, that ...
1
vote
1answer
31 views

Determining Probability Generating Function from Probability Mass Function and Convergence

I am trying to solve the following: Suppose $X_{nk}, k=1,2,\ldots,n, n≥ 2$ are i.i.d. random variables $$P(X_{nk}=0)=1-\frac{1}{n}-\frac{1}{n^2}\\P(X_{nk}=1)=\frac{1}{n}\\P(X_{nk}=2)=\frac{1}{n^2}$$ ...
1
vote
3answers
25 views

Product of Uniform Distribution and $\Gamma(2,1)$ Distribution

I ran into an old exercise but I seem to have messed up somehow. Can you tell me what went wrong? Let $U \sim \mathrm{Unif}(0,1)$ and $V \sim \Gamma(2,1)$ with $U,V$ independent. Show that $UV$ has ...
-4
votes
0answers
21 views

probability questions bolt [on hold]

Find the probability that at most four heads will occur when a coin is tossed ten times
0
votes
2answers
16 views

let x and y be uniformly distributed independent random variables on [0 ,1].the probability that the distance between x and y is less than 1/2 is?

I have a question about probability: let x and y be uniformly distributed independent random variables on [0 ,1].the probability that the distance between x and y is less than 1/2 is? can someone ...
0
votes
2answers
17 views

Odds of $X$ number of items getting picked twice or three times in a row

There is a list of $63$ items, from which $20$ items are picked in month $1$. What is the chance of $1$ item from the $20$, being picked again in the month $2$?(month $2$ also picks 20 items) Then ...
0
votes
1answer
25 views

Expected value of a sample

I am unsure of how to solve this question. I know from examples questions that expected value of a sample is usually very close to the population mean. However, it says to compute the expected value ...
3
votes
1answer
56 views

German Tank Problem, Confidence Level

Suppose you're in a city with n cabs. Each cab has a distinctive number from 0 to n. You take a cab 10 times, the choice of the cabs is independent and equiprobable. The cab with the biggest number ...
0
votes
0answers
24 views

This random variable converges in distribution?

$\delta_x$ is a Borel probability that $\delta_x(x)=1$ and $\mu_n$ is a uniform distribution in this interval $(1, 1 + \frac{1}{n})$ The variable $X_n\sim \frac{1}{2}\delta_{\frac{1}{n}} + ...
0
votes
1answer
24 views

identifying sudden change in value given a list of values over time

I have a list of the average price of an item in a game over time. Things don't tend to move much. I am wondering how I can detect whether a new value inserted is a surprising movement in price. I ...
2
votes
1answer
36 views

Application Problem: Conditioning Poisson Process

I am trying to solve the following application problem: There are $n$ components with independent lifetimes which are such that component $i$ functions for an exponential time with rate $\lambda_i$. ...
1
vote
1answer
30 views

Application Problem: Expectation and Variance of Compound Poisson Process

I am solving the following: Let $Y1, Y2,…$ be a random sample from $\Gamma(p,a)$ distribution, where p and a are positive real numbers. $Y$ is damage in thousands of dollars caused to a car in an ...
0
votes
1answer
12 views

how do I parametrise a stochastic matrix

I have a matrix $\mathbf{t}$ that maps one $d$ dimensional probability distribution to another $\mathbf{t}^T x = q$, i.e. with $\sum\limits_i t_{ij} x_i = q_j$ and $\sum\limits_j t_{ij} = 1$ $\forall$ ...
2
votes
0answers
24 views

Probability that a birth--death process crosses level $n$ in $(0,T)$

This question is inspired by this question. Jobs arriving according to a Poisson process with rate $\lambda$. Jobs stay in the system for a fixed amount of time $d$ and depart thereafter. Let $X(t)$ ...
0
votes
0answers
14 views

Comparing Two Ways of Scoring Data [on hold]

I have used two different methods to give a score to the same data set. One is a discrete method and the other continuous. How do I show that the continuous method is more sensitive to changes in ...
0
votes
1answer
41 views

Variance of absolute value sums of random variables

Let $X=\left|\sum _{i=1}^n Z_{i} \right|$ and $Y=\sum _{i=1}^n |Z_{i} |$ where random variables $(\textit{Z${}_{i}$})$ are i.i.d, and $Z_{i} =0$,$+1$ or$-1$, with probability ${p}{}_{0}$, ...
0
votes
0answers
29 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 ...
1
vote
1answer
26 views

Borel-Cantelli question

If $X_1...X_n$ are i.i.d. and $\mathcal{N}(0,1)$ how can Borel-Cantelli lemma helps us to proof a.s. of: $$\max\{X_{n^2+1},X_{n^2+2},\dots,X_{n^2+2n}\}\ge5 \text{, }\forall n>N$$ Thank's for your ...
2
votes
0answers
20 views

Queue theory - M/D/k - Probability of never having a queue before a time T

This is probably a known result, but I couldn't find any resource pointing directly to the issue I'm trying to solve. Suppose you start a logistic mission that needs that during its time $T_m$ a ...
1
vote
0answers
20 views

construction of a path of quadratic variation

Consider a path $x: [0,1] \to \mathbb R$. it's $p$-variation on an interval is defined by $$V_{p}(x, [a, b]) = \lim_{|\Pi| \to 0} \sum_{i=1}^{n}|x(t_{i}) - x(t_{i-1})|^{p}$$ where $\Pi = \{a= ...
0
votes
0answers
26 views

Distribution of the minimum of two exponential random variables

$X$ and $Y$ are two exponential random variables with rate 1 and 2. lets define random variable $Z$ such that: $z_i = min(x_i,y_i)$, where $i =1,2,3,...N$. Let $V$ be another random variable and ...
0
votes
1answer
51 views

Find the probability of opening all the boxes

Suppose there are $20$ boxes which $1-20$ are printed on each box. There is a key in each box which are also marked with $1-20$. So only the key with the same number with the box can open it. For ...
0
votes
1answer
14 views

When do I use Law of total variance?

For example, at the beginning of doing this problem (http://math.illinoisstate.edu/krzysio/3-6-10-KO-Exercise.pdf), I was thinking of using $\text{Var}(\text{Total loss}) = \text{Var}(N \cdot L)$, ...
0
votes
1answer
18 views

how to find the cumulative density function

Consider $$f(x)=3x^{-4} \qquad \mbox{on} \qquad x\geq 1.$$ Let $X$ be a continuous random variable on $x\geq 1$. Find the cumulative distribution $F(x)$ for $X$. I know that CDF for a continuous ...
47
votes
9answers
8k 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, ...
0
votes
1answer
33 views

Assume a die is rolled repeatedly. Find the markov matrix $P$ for the random variable of the time until the next $6$.

Assume a die is rolled repeatedly. Find the markov/transition matrix $P$ for the random variable $X_r$ = the time until the next six at time $r$. My solution was: For $i,j \geq 0$, $P$ is given ...
2
votes
3answers
32 views

The non-uniform probability of sums from the throw of multiple dice

I'm reading The Drunkards Walk by Leonard Mlodinow. In the book, the author writes: From a throw of three dice, a sum of 9 and 10 can be constructed in an equal combinations. However, the outcome ...