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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2
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2answers
116 views

Discrete probability problem

Problem: Assume the number of cars passing a road crossing during an hour satisfies a Poisson distribution with parameter $\mu$, and that the number of passengers in each car satisfies a binomial ...
1
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1answer
101 views

Central Limit Theorem problem.

There are 36 white and 64 black balls in the bin. After we have randomly picked some ball we put it back to the bin. How many times we must pick balls from the bin to be sure, that probability of ...
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2answers
104 views

Game with losing and winning a dollar

I found an interesting problem in my book: There is a game where player starts with $k\$$. In each step he wins or loses $1\$$ (both with probability $p=\frac{1}{2}$). The game ends when player ...
1
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2answers
145 views

Are waiting times always going to be exponentially distributed?

I'm studying for CAS/SOA Exam P/1 and a question I have here is: We have a portfolio of $20$ insurance policies. The number of claims per policy in a $3$-month period has a Poisson distribution ...
8
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5answers
548 views

Successful approaches to the modelization of ''randomness''

If you pick a number $x$ randomly from $[0,100]$, we would naturally say that the probability of $x>50$ is $1/2$, right? This is because we assumed that randomly meant that the experiment was to ...
0
votes
1answer
153 views

Is Lottery probability really the same for all combos?

http://justwebware.com/uklotto/uklotto.html Test run quickpick Test run 1,2,3,4,5,6 Test run (single digit,teens,twenties,twenties,thirties,forties) 1000 times or more each cycle for as many ...
1
vote
1answer
292 views

A sample of size n = 20 is drawn from a population with population proportion, p = 0.40. Find the mean

A sample of size $ n = 20 $ is drawn from a population with population proportion $ p = 0.40 $. Assume that the sample size is less than or equal to $ 5 \% $ of the population. Let $ \hat{P} $ be the ...
0
votes
1answer
207 views

Find the probability that 8 or more will feel that the system is adequate. Find the probability that exactly 8 will feel that the system is adequate.

Only 20% of the people in a large city feel that its mass transit system is adequate. If 20 persons are selected at random, find the probability that 8 or more will feel that the system is adequate. ...
1
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1answer
223 views

What is the probability that the 2002 mean salary of a random sample of 50 baseball players was within $20,000 of the population mean, μ .

Let μ be the mean annual salary of Major League Baseball players for 2002. Assume that the standard deviation of the salaries of these players is $107,000. What is the probability that the 2002 mean ...
0
votes
1answer
260 views

Calculating pdf for sum of two random variables

I have a random process Z(t).This function is defined as equal to A(t)+B. Well, It's clear that to obtain the pdf of Z(t), I need to take the convolution but I am not getting it how in the solution ...
0
votes
1answer
788 views

How many different combinations are there for the number of the letters of the alphabet chosen in a 500 type-up?

Suppose you want to make 500 draws, where you have 26 letters of the English alphabet in a bag. They come in equal quantities and their supply is unlimited, so that you could, for example, draw 500 ...
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0answers
48 views

How to model a system with multiple probability distributions, each for a part of the system?

I need to build a complex probability model to describe some "real world" scenarios. The system consists of several types of objects, and the contraints upon these objects and their interactions are ...
1
vote
1answer
67 views

Probability involving dependent normal variables

I have two independent, identically distributed normal random variables $X \sim N(0,\sigma^2)$, $Y \sim N(0,\sigma^2)$. I want to know $$Pr[X-Y>4\sigma \text{ and } X<3\sigma \text { and } ...
3
votes
2answers
133 views

Probability of a square landing within squares on a grid

A 2 x 2 square is tossed randomly on a grid of 3 x 3 squares. What is the probability that the 2 x 2 square falls completely within one of the 3 x 3 squares?
5
votes
1answer
188 views

Are these numbers $h_{r,s}$ irrational?

I came across these numbers in my work some time ago. This type of expressions do not exist in closed form (not to confuse with Vandermonde convolution), I already know that. To simplify I denote ...
2
votes
0answers
126 views

Independence of Brownian Motion with respect to a stopping time

Let $B_t$ be a brownian motion, $B_0=0$, and $\gamma \in \mathbb{R}$. Now, let's build the following stopping time: \begin{equation} T = \inf \{ t \geq 0 : |B_t + \gamma t| = 1 \}. \end{equation} If ...
1
vote
2answers
244 views

Probability Book Help

I have Ross a First Course on Probability and Bertsekas Introduction to probability book. However these two books do not exactly give me what I am looking for. The problem is Bertsekas book is ...
9
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1answer
198 views

$P(x,n) = \frac{x(n-1)!}{n^{x}(n-x)! }$ — What is the name of this probability distribution?

$$ P(x,n) = \frac{x(n-1)!}{n^{x}(n-x)! } $$ I'm having a really tough time describing what this distribution does, but it's simple in code. So if you know code, then read on: ...
1
vote
1answer
55 views

Calculating $P(X-E(X)\geq \mathrm{SD})$ for a normal random variable $X$

If you'd have to calculate the following probability of an $X\sim \mathrm{Normal}$: $$P(X-E(X)\geq \mathrm{SD}),$$ how would you calculate it? And if the calculation will be for $4\cdot\mathrm{SD}$, ...
5
votes
1answer
156 views

Calculate $\mathbb{E}[F(Y)]$

I try to resolve this problem, but I have some difficulties to get a clear result. The problem : Let X be a normal random variable with mean 0 and variance 1 (ie. $X\sim \mathcal{N}(0,1)$). Let Y ...
1
vote
1answer
72 views

Flip another coin each time, whats the mean times required?

Start with one coin, flip all coins, if all land on tails then stop, otherwise add another coin and repeat. What is the mean average number of flips? This is not a homework question.
4
votes
1answer
311 views

Compute $P(X>40\; |\; X>10)$ where $X$ has an exponential distribution

Please could someone advise if I have interpreted this problem correctly Let $X$ have an exponential distribution with a mean of $i = 20$ (1) Compute $P(X>40 \;| \;X>10)$ I believe the ...
0
votes
1answer
77 views

Expected value where benefit and probability depends on stochastic variable

I am trying to calculate the expected benefit of an action, where both the benefit and probability of carrying out the action depends on a parameter c, which is distributed according to f, F'=f. The ...
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1answer
204 views

the upper and lower bound of the max of two iid variables

If $x$ and $y$ are independent standardized random variables with zero mean and unit variance, what are the upper and lower bounds of $$E[\max(x,y)]$$ Can I get a useful reference to this?
2
votes
1answer
46 views

Derivation of equalities from odds ratio

Could someone give me the consecutive steps to derive the following equations: $$P(Y=1) = \frac{\mathrm{OR}}{1+\mathrm{OR}}$$ and $$P(Y=0) = \frac{1}{1+\mathrm{OR}}$$ from ...
1
vote
2answers
6k views

Multiplication of a random variable with constant

Suppose $X$ is a random variable which follows standard normal distribution then how is $KX$ ($K$ is constant) defined. Why does it follow a normal distribution with mean $0$ and variance $K^2$. ...
0
votes
1answer
290 views

Exercise 2.8 in Mackay's Information Theory, Inference and Learning Algorithms

[Editing question per Leon's suggestions - thanks for these!] Could someone walk me through a solution to Ex 2.8? 2.7: Bill tosses a bent coin $N$ times, obtaining a sequence of heads and tails. ...
0
votes
1answer
37 views

How the pdf of this equation calculated?

I am not getting it ,how this pdf is obtained.I was given a cdf and they took its derivative to obtain the pdf but how this cos(tk) remain inside as well as outside.
1
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1answer
123 views

A conceptual doubt on steady state probability of a DTMC

Consider a DTMC with state space $\{1, 2,\ldots {}{} \}$. Now we want to calculate the probability that state 1 is followed by state 2 in the long run i.e, $P(X_n=1, X_{n+1}=2)$ as $n$ tends to ...
2
votes
1answer
78 views

Distribution of largest sample from normal distribution.

Given $n$ independent random variables $X_i$ with normal distribution, mean $\mu$, variance $\sigma^2$, what is the distribution of $\max\limits_{i=1}^n(X_i)$ ? In particular I am interested in ...
3
votes
1answer
64 views

Is Uncorrelatedness sufficient for the CLT?

I am looking for a "counterexample" to a central limit type setup. Here is my question: Is there an example of a sequence of identically distributed random variables $(X_n)_{n\in\mathbb{N}}$, with ...
0
votes
1answer
62 views

Hash Functions and Probabilty

We are considering bit strings of length 160. Let there be some input x, and hash function $H(x) \rightarrow \left \{ 0,1 \right \}^{160}$. How many turns at least it takes to make collision: ...
2
votes
2answers
153 views

Approximation of discrete distribution

I have a discrete random variable $X$ which takes the values $+1$ and $-1$ with equal probability $\frac{1}{2}$. Can I approximate this with a normal distribution ?
0
votes
1answer
151 views

Probability of at least 2 adjacent events happening out of $n$ events

I've been having some trouble with what I thought at first should be quite a simple problem. I have n events in total and they can only be 0 or 1 (so it's a binomial). Lets say the probability of a ...
1
vote
1answer
94 views

A problem in Probability

I have a doubt in a sum of probability. The sum states : "There are 2 girls. One is born in the year 1989 and the other in 1990. What is the probability that they both have the same birthdays?" Can ...
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4answers
77 views

Probablity of a random variable

I really feel ashamed to ask this question however I don't have time for revision. Also not a native English speaker, so excuse my lack of math vocabulary. I am writing a program that requires ...
2
votes
1answer
645 views

Passing a limit into expectation

While reading about random walks, I started thinking about this and got a headache: Given a random process $\{X_n\}_{n\in \mathbb{Z}^+}$ with a real state space (i.e., $X_n$ takes on real numbers), ...
6
votes
1answer
368 views

A problem about strong law of large numbers of Shiryaev's Probability

This is a problem after the section "Strong Law of Large Numbers" of Shiryaev's Probability: Let $\xi_1,\xi_2,...$ denote independent and identically distributed random variables such thatt ...
2
votes
2answers
133 views

(yet another) length of random segment

Take a random point $Z$, i.d. in [0,1], which defines a stick. Break the stick in two, (random i.d.). Take the left part of the broken stick and break it again in two (i.d.) You thus obtain three ...
0
votes
1answer
102 views

Computation of Conditional Expectation

Let the stochastic process $\{X_n\}$ be constructed inductively as follows: $X_0=0$, and for $n\ge 1$, and conditionally on $\mathcal{F}_{n-1}=\sigma(X_0,\ldots,X_{n-1})$, we set $\;\,\,\, ...
0
votes
1answer
64 views

Which distributions should be used to model the winning & 2nd bids in second price auctions?

With second price auction which distributions should I use to model the winning bids and 2nd bids (separately)? I'm thinking of using Gaussian. However for the winning bids r.v, it has to satisfy: $$ ...
1
vote
1answer
58 views

I think R(X) ={1,2,3,4} but how to find $f_x(k)$

$9$ tires from different brands are ranked from $1$ to $9$ (best to worst). Let $X$ be the rank of the best tire among $6$ randomly chosen tires. I think $R(X)$ = $\{1,2,3,4\}$ but how do you find ...
1
vote
0answers
104 views

lazy counting - failing to check equivalence classes are distinct

Declaring that a set has 10 elements requires two things: find some elements show they are all distinct showing you have all of them Sometimes these are easier said than done. E.g. count all ...
0
votes
1answer
47 views

We have eight balls. Find E[X], E[Y]

We have 8 balls numbered 1,2,...8. We draw randomly one ball. Let $X$ be smallest number in the box after removal, $Y$ the greatest number in the box after removal. Find $E[X]$ and $E[Y]$? How to ...
1
vote
1answer
144 views

Large Deviations Result for non iid variables/ Conditional Large Deviations?

Let $X_n, n\in\mathbb{Z}$ be a sequence of independent random variables (finite, let's say the size of the alphabet is $2$ to simplify things) with mean zero and variance less than $1$. Is there a ...
2
votes
2answers
134 views

length of a random segment

Take three random points, iid in [0,1]. The first two points devide the interval in three. What is the expected length of the segment where the third point is lying? Generalise for n.
6
votes
4answers
2k views

If you toss $1000$ fair coins $10$ times each, what is the probability the *some* coin will get $10$ heads?

The answer to this is supposedly close to $0.63$. However, I get approximately $0.9765625$ for the following reason: The probability of a fair coin flipped $N$ times resulting in all heads is ...
1
vote
1answer
107 views

Supermartingale with vanishing drift

Is a continuous supermartingale with vanishing drift already a martingale? In my concrete problem, I have a continuous nonnegative local martingale $ (X_t) $ on $ \left[0, T\right] $ which is bounded ...
3
votes
1answer
130 views

Brownian motion interesting question

I found this interesting question on the internet, but unfortunately I could not solve it. What is probability that Brownian motion (starting at origin) has value 1 before having value -2?
3
votes
1answer
164 views

Brownian motion and hitting frequency

Suppose we have a Brownian motion $B_t$ with $B_0 = 0$ and $B_t - B_s \sim N(0,t-s)$. Every time $B_t$ hits $\pm h$, where $h$ is some "barrier" $>0$, I pay someone £1 and the brownian motion ...