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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3
votes
2answers
81 views

How to give rigorous proofs of these two limit statements?

Let $X$ be a random variable with cumulative distribution function $F(x)$. Then how to rigorously prove the following two limit statements? $\lim_{x \to - \infty} F(x) = 0$. $\lim_{x \to + \infty} ...
2
votes
2answers
74 views

Brainteaser: Player A has £1, Player B £99. They flip a coin. The loser pays the other £1. Expected number of games before one is bankrupt?

Player A has £1, Player B £99. They flip a coin. The loser pays the other £1. What is the expected number of games they play before one is bankrupt? I have struggled at this for hours now with little ...
9
votes
2answers
358 views

What is the expected value of the number of circles formed?

There are $99$ identical square tiles, each with a quarter-circle drawn on it. When the tiles are randomly arranged in a $9$ by $11$ rectangle, what is the expected value of the number of full circles ...
0
votes
1answer
45 views

How is this paper using probability notation?

I am trying to understand this paper about documents and sentences. At the end of page three, they say: Let g(wi, wj ) be the distance between two events (1 if in the same sentence, 2 in neighboring, ...
6
votes
2answers
45 views

What is the expected value of the number of anchors of $S$?

For any subset $S\subseteq\{1,2,\ldots,15\}$, call a number $n$ an anchor for $S$ if $n$ and $n+ |S|$ are both elements of $S$. For example, $4$ is an anchor of the set $S=\{4,7,14\}$, since $4\in S$ ...
5
votes
2answers
56 views

What is the expected value of A?

The Happy Animals Kennel has 18 cages in a row. They allocate these cages at random to 6 dogs, 6 cats, and 6 pot-bellied pigs (with one animal per cage). All arrangements are equally likely. Let A ...
0
votes
0answers
24 views

Expectation of a logarithmic/trigonometric function

I am trying to find a closed form solution of the following expectation: $$\mathbb{E}[\log(a+b\cos(\phi))]$$ where $a$ and $b$ are real constants, and the expectation is with respect to $\phi$. If ...
1
vote
1answer
28 views

Fudge Dice: Reroll vs. Bonus

A "fudge die" is a die with equal probability to result in -1, 0, or +1. The commercially produced fudge dice are generally 6-sided dice with two "–", two "+", and two blank sides. In the ...
0
votes
1answer
25 views

CDF of random variables

due to my lack of knowledge in probability theory, I have first to apologize if the following question is not formulated in a proper language. I was wondering if there is any formal expression of the ...
-2
votes
1answer
33 views

Mean of max vs max of mean

If I have say an $n$ collection of 10 random variables $X_1, \ldots, X_{10}$ (so an $n \times 10$ matrix of values) from some underlying distribution whether Gaussian or uniform, and I calculate ...
8
votes
10answers
2k views

Should I throw the dice again if I have rolled 4?

My math skills are very basic so it might be a stupid question, I had a discussion with my brother in law and now we have a 'math problem'. We were playing a game with dices and he threw 4. The ...
1
vote
1answer
44 views

Find one-dimensional distribution function $F(y\mid t)$ of random process $Y(t)$

$ Y(t)=tZ^2;\quad Z\sim U(-2;2); \quad t\ge0. \quad$ I need to 1) find one-dimensional distribution function $F(y|t)$ of random process $Y(t)$. 2) calculate probability that trajectory of the ...
2
votes
2answers
204 views

10 little dwarves

A dwarf-killing giant lines up 10 dwarfs from shortest to tallest. Each dwarf can see all the shortest dwarfs in front of him, but cannot see the dwarfs behind himself. The giant randomly puts a ...
0
votes
0answers
32 views

Probability of infinite intersections

While I was studying Probability and random processes I came across the following question. Say I have $A_1,A_2, \ldots, A_n$ events such that $A_i$ is in $E$ but not equal to $E$. What is: ...
1
vote
1answer
34 views

A Bayesian estimate of the bias of a coin

Consider a coin with unknown probability $p$ of landing on head. I will toss the coin and stop as soon as I get a head. Say this is after $n$ tosses. If my prior belief for $p$ was uniform on ...
1
vote
3answers
26 views

Probability of getting a certain group of students when choosing three at random out of 25

A teacher randomly chooses a group of three students from her class of 25 students. Find: a) Probability that friends Suri, Lily and Violeta are chosen for the group? b) If he ...
3
votes
2answers
49 views

Parity of the sum of consecutive Bernoulli random variables

$\newcommand{\Var}{\operatorname{Var}}$I have $X_1,X_2,\ldots,X_{n+1}$ i.i.d. rv, each $X_i$ is a Bernoulli rv with parameter $p$, i.e. $X_i \in \{0,1\}$, $P(X_i=0)=1-p$ and $P(X_i=1)=p$ with $0 \leq ...
1
vote
2answers
87 views

How to obtain probability distribution from the generating function $G(s) = e^{a(s-1)^2}$?

I was trying to get the probability distribution $p(n)$ from a generating function $G(s)$ like this: $G(s) = e^{a(s-1)^2}=\sum s^np(n)$ I need first to do Maclaurin expansion of the exponential and ...
0
votes
1answer
39 views

Integrability condition

Suppose that \begin{align} \mathbb{E}\int_{0}^{T}f^{2}(t)dt <K \end{align} Does it also hold that \begin{align} \int_{0}^{T}f^{2}(t)dt <K \end{align} ? Here, T, K>0 are fixed. I am a bit ...
0
votes
1answer
58 views

Maximum likelihood estimators

I have $X_1,X_2,\dots,X_n$ as random samples from a binomial distribution, with probability function: $$p_X(x) = Pr(X=x) = {m \choose{n}}\alpha^x(1-\alpha)^{m-x},x=0,1,2,\dots,m$$ where $m$ is given ...
1
vote
4answers
84 views

Estimate bias of a coin

Consider a coin with probability $p$ of landing on head. You can estimate the prob by tossing it lots of times and looking at the proportion of heads one gets. In my problem I just want to know if ...
-2
votes
1answer
15 views

Standard deviation of travel times

Suppose that travel times for Swinburne students are normally distributed with mean of $32.5$ minutes and a standard deviation of $5$ minutes. Complete the following sentence, giving figures correct ...
2
votes
2answers
42 views

Probability problem: n different balls in n different boxes

Problem Suppose $n$ different balls are distributed in $n$ different boxes. Calculate the probability that each box is not empty when distributed the balls. I'll define the sample space as ...
2
votes
1answer
34 views

Invariance Properties of Brownian Motion

I am trying to make sense of the Scaling-Invariance and Time-Inversion properties of Brownian motion by producing a sample path. For the record, I am using the following definitions. Let $B(t)$ be the ...
1
vote
3answers
92 views

Probability that the red fish are the first species to become extinct

I have a doubt in the solution of the next problem: A pond contains $3$ distinct species of fish, which we will call the Red, Blue, and Green fish. There are r Red, b Blue, and g Green fish. ...
1
vote
2answers
22 views

probability that the white balls are left in the urn

I don´t understand the solution of next problem: An urn contains n white balls and m black balls. The balls are withdrawn one at a time until only those of the same color are left. Show that with ...
1
vote
1answer
44 views

Probability of drawing the king of hearts and a red card

Two cards are drawn from a standard deck of cards at the same time. Find: a) Probability of drawing the King of hearts and a red card b) Probability of drawing the King of hearts and a black ...
0
votes
0answers
14 views

Check work for finding Max log-Likelihood of a geometric Distribution

Here is my geometric distribution: $P(L=n)=p^{n}*(1-p)$ To find the max likelihood, I do: $\sum_{L_i} L_i\log(p) + \log(1-p)$, where L_i is a particular length. I take the derivatives and end up ...
0
votes
1answer
81 views

Why is the expected average displacement of a random walk of N steps not $\sqrt N$?

Let $D_N$ be the expected average of the displacement of a random walk on $\mathbb Z$ from the origin, where $N$ is the number of steps, each of which is either $-1$ or $1$. We take the definition of ...
1
vote
2answers
69 views

what is the difference between average and expected value?

I have been going through the definition of expected value in Wikipedia (http://en.wikipedia.org/wiki/Expected_value) beneath all that jargon it seems that the expected value of a distribution is the ...
1
vote
1answer
30 views

Conditional probability with a normal distribution

Given that Y and L are normally distributed, the expectation of L given Y is $\mu (Y)$ and the variance of L given Y is $\sigma ^2 (Y)$, why is the conditional probability $P(L > x| Y) = \Phi ...
0
votes
1answer
52 views
0
votes
2answers
25 views

A question about moment-generating function

Suppose $X$ is a r.v. and $\phi(\theta)=\mathbb{E}(e^{\theta X})$ Let $\theta_+=\sup \{\theta:\phi(\theta)<\infty\}$ $\theta_-=\inf \{\theta:\phi(\theta)<\infty\}$ Why ...
-1
votes
1answer
44 views

Probability that a randomly marked multiple choice test is all correct/incorrect [closed]

A quiz is made up of five multiple choice questions each with 4 possible answers. Suppose you randomly select an answer for each question. Determine the following probabilities. Express your answer as ...
1
vote
3answers
28 views

probability of throwing three adjacent numbers

When throwing one dice 3 times in a row, what is the probability of getting adjacent numbers in right order, for example 2,3,4 or 4,3,2?
1
vote
1answer
35 views

probability of bingo

It is the first time I heard about bingo game and I would like to learn more on this game by mathematical analysis. To make it simple, I consider the American BINGO with 75 balls used and each game ...
1
vote
0answers
12 views

Is there a Burkholder-Davis-Gundy inequality for martingale increments?

is there a Burkholder-Davis-Gundy inequality for martingale increments? More specifically, I would like to find a finite bound of order $h^{p/2}$ for the expectation $$\operatorname{E} \left[ \sup_{t ...
1
vote
2answers
53 views

Problem solving: Counting and probability

i am a little bad at probability, i'm studying to overcome this lack. Since i'm not with a tutor i need some help on the correct way to approach a basic probability problem. I would appreciate your ...
2
votes
1answer
69 views

Proving that three events are mutually independent

Suppose that: the events $A$ and $B\cap C$ are independent. the events $B$ and $A\cap C$ are independent. the events $C$ and $A\cap B$ are independent. the events $A$ and $B\cup C$ ...
0
votes
2answers
30 views

Conditional probability for random variables with different distributions

Random variables $X$ and $Y$ are independent, where $X$ is exponentially distributed with parameter $1$ and $Y$ has uniform distribution on $[-1,1]$ interval. Find $\mathbb{P}(Y>0|X+Y>1)$. My ...
0
votes
1answer
26 views

Combining independent predictions into an overall probability

I am trying to understand the mathematical basis of combining independent probabilities, as described here: http://www.paulgraham.com/naivebayes.html Suppose that being over 7 feet tall indicates ...
4
votes
2answers
142 views

A counter example of Brownian Motion

Here is an example in my textbook to illustrate why we need the continuous sample path in the definition of Brownian motion. Let $(B_t)$ be a Brownian motion and $U$ be a uniform random variable on ...
1
vote
1answer
23 views

Branching process: Why does the population die or explode?

Consider a population such that each member, independently from other members, at a certain instant of time is replaced by its offspring. Lets denote with $X_n$ $({n\ge 1})$ the amount of the ...
0
votes
1answer
38 views

how to compute $E[e^{a^2/2}N^2]$, $N$ is $\mathcal{N}(0,1)$

I have to show that $E[e^{(a^2/2)N^2}]=E[e^{(aNN')}]$ and tell for which values of $a$ these quantities are finite. $N$ and $N'$ are independent $\mathcal{N}(0,1)$ random variables I computed the ...
0
votes
1answer
40 views

Finding the variance problem

I am working on the following problem and the explanation was not clear to me, so I am seeking for help. The following is the problem. A fire occurs with a probability of 0.01. The damage Y ...
1
vote
1answer
48 views

How to compute this conditional probability in Bayesian Networks?

I met a problem related to conditional probability from the article "Bayesian Networks without Tears"(download) on page 3. According to the Figure 2, the author says $$P(fo=yes|lo=true, ...
0
votes
1answer
15 views

expected value product dependent random variables

My question is strictly operative, if I have, for instance, two random variables $X$ and $Y$, $X$ is a $\mathcal{N}(m,\sigma^2)$ and $Y=e^{h(X-m)-1/2(h^2\sigma^2)}$. $E[Ye^X]$ is $\int y e^x p(x) ...
2
votes
3answers
22 views

What is the probability any one item in a set of 10 items is picked from a pool of 30?

Consider that a set contains 30 distinct items. User must pick 10 distinct items. What is the probability that any given item will appear in the set of items picked? The probability that an item is ...
0
votes
4answers
58 views

Probability of drawing a red ball

An urn has $2$ balls and each ball could be green, red or black. We draw a ball and it was green, then it was returned it to the urn. What is the probability that the next ball is red? My attempt: I ...
0
votes
0answers
19 views

Overflow and underflow of a probability value

I am evaluating the probability that the minimum of a process is a above a a barrier $\log(H)$. The probability is given by $$P_i=1-\exp\left(-2\frac{(\log(H)-x)(\log(H)-x_b)}{\tau\sigma^2}\right).$$ ...