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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2answers
54 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 ...
1
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1answer
20 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 ...
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0answers
13 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 ...
1
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2answers
60 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. ...
2
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2answers
51 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 ...
4
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1answer
107 views
+50

Unbiased asymptotic variance

Problem: Let $X_1,...,X_n$ be indep. r.v.'s that satisfy, for $i = 1,...,n$, $E(X_i) = \mu_i(\theta)$ & $\mathrm{Var}(X_i)= \sigma_i^2(\theta)$. $\theta$ is the parameter of interest and the ...
0
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0answers
16 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: ...
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2answers
21 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
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2answers
35 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
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1answer
253 views

joint probability distribution of one discrete, one continuous random variable

This is a problem on the joint distribution of a discrete and a continuous random variable. Kitty Oil Co. has decided to drill for oil in 10 different locations; the cost of drilling at each ...
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0answers
24 views

Constraints KKT and LM problem

Hello everybody!! Days ago I solved an optimization problem with Lagrange Multipliers. I'll try to explain the general framework, (a copy-paste of something that was already checked to be correct): ...
0
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1answer
31 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
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1answer
35 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 ...
0
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3answers
47 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 ...
0
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1answer
219 views

How do I write the multinomial Naive Bayes Classifier Decision rule as a linear rule?

I am attempting to write the multinomial Naive Bayes Classifier Decision rule as a linear rule. A document is viewed as a sequence $d = (w_1, w_2,\ldots,w_l)$ of $l$ words and is classified as $$h(d) ...
2
votes
1answer
554 views

Mana Maximization (Hearthstone)

I recently started playing Hearthstone and a statistic / probability question came up my mind. Here's a quick breakdown: The game is a turn-based card game which involves "points" that you can used ...
0
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0answers
14 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?!
4
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2answers
125 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
1answer
22 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 ...
2
votes
0answers
75 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. $$P[X>\lambda E[X]] \geq (1-\lambda)^2 \frac{E[X]^2}{E[X^2]}$$ ...
-2
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1answer
12 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 ...
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2answers
32 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 ...
1
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1answer
40 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 ...
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0answers
10 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: $\sum_{L_i} L_ilog(p) + log(1-p)$, where L_i is a particular length. I take the derivatives and end up with ...
1
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2answers
19 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
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2answers
46 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
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1answer
32 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
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2answers
61 views

Is the space of probability distributions an infinite dimensional space?

Is the space of probability distributions an infinite dimensional space? If so, would you explain how? This is a follow up question to an answer to this question.
0
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1answer
67 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 ...
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0answers
9 views

Does $\sum_B p(L|B)p(B|G) = \sum_B p(L,B |G) = p(L|G)?$ [on hold]

Does $\sum_B p(L|B)p(B|G) = \sum_B p(L,B |G) = p(L|G)?$ By chain rule, does $p(L,B |G) = p(L|B,G)p(B|G)$? Does $\sum_B p(L|B)p(B|G) = \sum_B p(L|B,G)p(B|G)$
1
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2answers
65 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 ...
2
votes
1answer
66 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
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3answers
452 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
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1answer
20 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
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1answer
48 views
7
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1answer
118 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
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1answer
236 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
50 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 ...
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 $Y$?
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3answers
67 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
44 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
21 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
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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 ...
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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
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1answer
33 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
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0answers
9 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
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0answers
17 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 ...
0
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2answers
28 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 ...
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votes
1answer
63 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 ...
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1answer
22 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 ...