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
votes
1answer
54 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 $B\cup C$ ...
3
votes
3answers
450 views

Probability of having at least $K$ consecutive zeros in a sequence of $0$s and $1$s

I have a sequence of length $N$ consisting of $M$ ones and $N-M$ zeros. I am trying to find the number of possible arrangements that produce a sequence in which there exist at least K consecutive ...
1
vote
2answers
47 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
16 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 ...
6
votes
1answer
113 views

The Day Camp Stacking Game

My friend works at a day camp as a counselor and he told me about an interesting game he plays with his group of kids. You have a perfectly shuffled, regular $52$-card deck and a group of $2 \leq n ...
14
votes
1answer
235 views

The problem of the most visited point.

Represents the set $R_{n\times n}=\{1,2,\ldots, n\}\times\{1,2,\ldots, n\} $ as a rectangle of $n$ by $n$ as points in the figures below for exemple. How to calculate the number of circuits that visit ...
0
votes
4answers
46 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 ...
4
votes
2answers
108 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 ...
2
votes
2answers
375 views

What is the expectation of a random variable raised to the $n$th power?

If $Y=X^n$, with $n$ and the expectation and variance of $X$ known, what is the expectation and variance of $Y$?
1
vote
3answers
63 views

Time to $n$ heads when probability is a random variable

I have the following problem. I toss coins until I get a $n$ heads and then stop. The complication is that the probability of getting a head is itself a uniform random variable in the range $[0,1]$. ...
-1
votes
1answer
36 views

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

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 ...
0
votes
2answers
17 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 ...
0
votes
1answer
30 views

Simple conditional probability inequality

I'm reading on some branching process theory in Harris' Theory of Branching Processes and encountered an inequality which looks simple but is eluding me. The full version is a bit complicated to ...
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?
2
votes
1answer
29 views

Stochastic integration by parts formula to prove identity between iterated integrals

if $(M_t)_{t \geq 0}$ is a continuous local martingale, one can define the iterated integrals $I_0=1$, $I_1(t)=M_t$ and for $n \geq 2$ $$I_{n}(t) = \int_0^t I_{n-1} (s) \mathrm{d} M_s.$$ By noting ...
1
vote
0answers
8 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 ...
0
votes
0answers
15 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
2answers
34 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 ...
0
votes
2answers
25 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 ...
2
votes
1answer
53 views

proving probability inequality (how to return to chebychev?)

Supposing X is a random variable, $X>0$, $E[X^2]<+\infty$, $\lambda \in (0,1)$, I have to prove the following inequality. $$P[X>\lambda E[X]] \geq (1-\lambda)^2 \frac{E[X]^2}{E[X^2]}$$ once ...
0
votes
0answers
13 views

Probabilistic Graphical Model Diagram Notation, what does the box mean?

I'm just learning about probabilistic graphical models, I know the circles represent random variables, shaded being observed and unshaded being latent variables. But what does the box mean?!
-1
votes
1answer
61 views

Transforming distributions

There is an economy, populated by a large number of agents. A first order condition common to all agents, is the following: $$E[\exp^{(1-\theta)\eta_i}(r-R+\eta_i)]=0$$ the index $i$ indicates the ...
0
votes
1answer
21 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 ...
1
vote
0answers
23 views

How can I get the mean value of the variables with different time steps?

I am trying to get the mean value of the variables with different time steps. For example, I am trying to get the mean value of x at time t+dt which is E[x(t+dt)] as: ...
0
votes
1answer
24 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 ...
1
vote
1answer
17 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
30 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 ...
5
votes
0answers
281 views

An application of the Optional Sampling Theorem

let $S(k), k\geq 0$ a discrete random process. Suppose $S(N)$ is with probability one either 100 or 0 and that $S(0)=50$. Suppose further there is at least a sixty percent probability that the price ...
3
votes
1answer
40 views

Uniform sampling with replacement item frequency

Suppose we are sampling from $N$ distinct items uniformly with replacement $M$ times. What can be said about the distribution of frequencies of items drawn? For example, if I sort all the frequencies ...
0
votes
2answers
29 views

Computing problem in probability theory [on hold]

Josh takes a twenty-question multiple-choice exam where each question has five possible answers. Some of the answers he knows, while others he gets right just by making lucky guesses. Suppose that the ...
0
votes
1answer
14 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
17 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
0answers
19 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
285 views

Resistors to be used in a circuit have average resistance 200 ohms and standard deviation 10 ohms…

Resistors to be used in a circuit have average resistance 200 ohms and standard deviation 10 ohms. Suppose 25 of these resistors are randomly selected to be used in a circuit. a) What is the ...
2
votes
3answers
10k views

Four fair coins are tossed,what is the probability of at least getting two heads?

Four fair coins are tossed,what is the probability of at least getting two heads? How can I find the probability without drawing out all the results ?
2
votes
0answers
41 views

Dice: Expected highest value with a tricky condition

I know how to calculate the expected value "E" of a roll of n k-sided dice if we are supposed to keep the highest number rolled. If I am not wrong, the formula is: E = k − (1^n + 2^n + ... + ...
0
votes
0answers
15 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 $$P_i=1-\exp\left(-2\frac{(\log(H)-x)(\log(H)-x_b)}{\tau\sigma^2}\right).$$ ...
1
vote
1answer
18 views

Filtration from a Brownian Motion

The textbook I am reading defines the filtration induced from a Brownian Motion as follows. Let $\{B(t): t \geq 0\}$ be a Brownian Motion defined on some probability space, then we can define a ...
0
votes
0answers
20 views

Normal approximation with dependent variables

I have a sequence of $N$ dependent random variables $$y_i = \frac{x_i}{||\vec x||_2} \quad \mathrm{for} \quad \vec x \sim \mathcal N(0,\mathbb{1}_N),$$ where the $x_i$ are the iid elements of $\vec ...
-2
votes
1answer
336 views

geometric series word problem help [on hold]

Brennen has been playing a game where he can create towns and help his empire expand. each town he has allows him to create 1.15 times as many villagers. The game gave brennan five villagers to start ...
1
vote
2answers
36 views

What is p(x=1) of this moment generating function?

So for a MGF like so $M_x(s) = \frac14e^s + \frac34e^{5e^s-5} $ What is P(x=1)? How do I take into account of the 5e^s?
2
votes
1answer
52 views

Chance of exactly one birthday out of 336 to be January 1st.

What is the possibility that out of 336 birthdays, exactly one of them is January 1st? I'm assuming not a leap year. This is what I have so far. If we imagine the list birthdays to be a string, there ...
0
votes
4answers
160 views

How many combinations of four letters each can be made from the word PEPPER?

As the title says. I know that if there is no certain number of letters to choose you would have to just do 6!/3!2!. But what would you do if you have to only choose 4 letters?
0
votes
1answer
50 views

distribution of books among students

There are $p$ students and $q$ books where $q>p$ and all books are different, but each student will get a minimum of $1$ book and a maximum of $(p – 1)$ books. Find the total number of ways of ...
2
votes
3answers
72 views

“Practical” Claim about Hypothesis Testing of Bernoulli Distribution Parameter

First, let me state the original problem (in my own wording): Describe the decision procedure for testing the hypothesis about the parameter $p$ (success rate) of a Bernoulli distribution. The ...
0
votes
0answers
32 views

Can we show this?

Let us assume that we have $x\in \{0,...,x_0\}$. We have $p(x)\geq 0$ for all $x$ and $\sum_{x=0}^{x_0} p(x)=1$. Define $\mu:= \sum_x x p(x) = \omega(1)$. Let $P(x)=1$ iff $x p(x) = ...
2
votes
3answers
40 views

Finding the expected value of a function of random variables

I'm having troubles with finding marginal density functions and expected values in my probability theory class. I was hoping someone would be able to walk me through the solution to this question (I ...
21
votes
14answers
5k views

Simulate a 7-sided die with a 6-sided die

What is the most efficient way to simulate a 7-sided die with a 6-sided die? I've put some thought into it but I'm not sure I get somewhere specifically. To create a 7-sided die we can use a ...
0
votes
4answers
43 views

Fair dice throwing, expected value

We throw a standard fair dice until we threw $5$ and then $2,4$ or $6$ (not necessarily one after another). What is the expected value and variation of number of throws (let it be $X$)? I was ...