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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1answer
98 views

How big is a large sample?

Short version According to the law of large numbers, how many samples do I need to take to reach the half-life of the convergence towards the mean? In other words: How big is a large sample? Long ...
0
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0answers
46 views

Apparent contradiction when manipulating conditional expectations

For a random vector $(T,S,\theta)$ with joint distribution $f(\cdot)$ define the following function (slightly abusing notation for the distributions ): ...
0
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1answer
846 views

Exotic 6-horse race betting probabilities

I'm gearing up for horse racing season, and I'm trying to teach some fellow engineering friends how to bet "exotic" bets by using colored dice to simulate horses. So, the odds for each horse winning ...
0
votes
1answer
24 views

Rewriting $p(A\cup B)$

We can write $p(A\cup B) = p((A\cup B)\cap S) = p((A\cup B)\cap (B\cup B^{c})) = p((A\cap B^{c})\cup B)$, how has the final step occurred?
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0answers
251 views

Product of standard normal and uniform random variable

I'm trying to find the PDF of the product of two random variables by first finding the CDF. I don't know where I'm going wrong. Let $X\sim N(0,1)$ and $Y\sim Uniform\{-1,1\}$ and let $Z = XY$, then: ...
2
votes
1answer
154 views

Probability of choosing 6 from 49 where no numbers are next to each other

What is a probability that when we choose 6 numbers form 1 to 49, we won't get adjacent numbers. In another we can't get a pair from those choosen 6 numbers whose difference equals 1. What i have ...
0
votes
2answers
91 views

joint density function e^(-x-y-z)

Suppose that X,Y and Z have a joint density function given by $$f(x,y,z) = \begin{cases}e^{-x-y-z}&\text{if }x,y,z>0\\ 0&\text{otherwise}\end{cases}$$ Compute $P(X<Y<Z)$ I think ...
1
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0answers
43 views

Showing an inequality relating two Poisson tail-probabilities

In my research, I've discovered that a property that I am interested in is equivalent to an inequality involving two tail-probabilities of the Poisson distribution. I belive this inequality to be ...
3
votes
4answers
514 views

Why does maximum likelihood estimation for uniform distribution give maximum of data?

I am looking at parameters estimation for the uniform distribution in the context of MLEs. Now, I know the likelihood function of the Uniform distribution $U(0,\theta)$ which is $1/\theta^n$ cannot ...
2
votes
0answers
71 views

Is it sensible to always assume that the “usual conditions” always hold?

I've read in several places that it is reasonable to assume that the usual conditions (that the filtered space is complete, and that the filtration is right-continuous) hold since one can always ...
0
votes
1answer
53 views

Probability that the $9$th extraction is the first one for which $3$ white balls are extracted

We are given an urn with $9$ balls: $5$ black and $4$ white. We extract $3$ balls from this urn at each step, check their colors, then put them back in and repeat. What is the probability that the ...
0
votes
1answer
34 views

Intersection of two unions

More specifically what is $$p((A\cup B) \cap (B\cup B^{c}))$$ where $B^{c}$ is the complement of $B$. Do we take the intersection of each set separately, ie. $$p((A\cap B)\cup (A\cap B^{c}) \cup ...
0
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0answers
19 views

The space of all normal covariances matrices

Let $\cal C$ be the space of all $k-$variate normal covariance matrices and $\cal M$ be the set of all $k\times k$ symmetric positive semi-definite matrices. As we know that if $k=1$ then ${\cal ...
6
votes
3answers
135 views

What is the probability to win? Die game [closed]

You have a die. If you get one pip at any point in the game you lose. If you get two,..., six pips you start adding the number of pips to a sum. To win the sum must get greater or equal to 100. What ...
1
vote
2answers
47 views

Probability of pairwise independent event, find its upper bound

So we were told that the events $A_1, A_2, A_3$ are pairwise independent such that $P(A_1)=0.1$, $P(A_2)=0.02$, and $P(A_3)=0.01$. What is the upper bound for the term $P(A_1 \cap A_2 \cap A_3)$? ...
0
votes
1answer
45 views

Moment Matrix Positive Semidefinite

Let $\phi(x)$ be a probability distribution on$[0,1]$, and consider the moment matrix $M$ where the $(i,j)^{th}$ entry is $$ M_{ij} := \int_0^1 x^{i+j}\phi(x)dx, $$ or in other words, the expectation ...
0
votes
1answer
199 views

Validity of cumulative distribution function?

The function F ( x ) = 1+sin( x ) is not a valid cumulative distribution function. Why not? What properties make a ...
0
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1answer
32 views

initial probability , 2rd drawn is red ball

There are $2$ red and $4$ blue balls. One ball each time without replacement is drawn. What is the probability that the $2$nd drawn is red ball? The answer is treated as if it's the initial drawn. ...
0
votes
1answer
121 views

The probability distribution function of uniform random variables is as given

Given $U_1, U_2, \dots, U_n$ where each $U_i \sim U[0,1]$, then use uniqueness theorem to show probability distribution function of $X = U_1 + U_2 + \ldots +U_n$ (sum of independent uniform random ...
3
votes
3answers
74 views

Probability Dice Game

Paul, Dave and Sarah are rolling a fair six sided die. Paul will go first, always followed by Dave, who is always followed by Sarah, who is always followed by Paul, and so on... What is the ...
0
votes
0answers
31 views

Little O Bound, Combinatorics

I am reading a book on combinatorics. I tried deriving the result in the following sentence, but could not get it. Can someone show me the algebra? Theorem 1.2.1: If $\dbinom{n}{k} {(1- ...
2
votes
1answer
37 views

A monotonicity property for ratios of power means

Let $Z$ be any non-degenerate positive random variable with pdf $% g(z)$. Let $a>0$ and $r\neq 0$ denote arbitrary real numbers. Define the "$r$ -mean" of $Z$ shifted by the constant $a$ as ...
0
votes
1answer
470 views

Probability four people pick their own names out of a hat

I have a problem where 5 people put their names in a hat. Each person takes a name out, and does not replace it. I have these two questions: what is the probability that a person, call him "Joe" ...
0
votes
1answer
24 views

(possibly?) equivalent defintiioons of indepdnt events

Let $A_1$ and $A_2$ denote two events. Now I'm sure from school we've read that $A_1$ and $A_2$ are called independent when $P(A_1)P(A_2) = P(A \cup A_2)$ Now I've read another definition (albeit ...
1
vote
3answers
394 views

Distribution of the first passage time of a Gaussian random walk

Does anyone know the distribution for the first passage time of a Gaussian random walk i.e. $$ S_n = \sum_{i=1}^n X_i $$ where $X_i$ are iid normally distributed random variables. The first passage ...
0
votes
2answers
58 views

how to compute this expectation value

A random variable $X \sim N(0,1)$, compute $\Bbb E(X^n)$ . I manage to do this by characteristic function. Now I try to compute this by moment generating function or do it directly. So I have 2 ...
1
vote
3answers
54 views

Permutations of numbers

Given the five digits $1,3,4,6,$ and $7$. In the following question, it should be understood that repition of a digit is not allowed. (i) How many three-digit numbers can be formed from the ...
1
vote
2answers
32 views

cumulative distr. $X,Y$ independent

what is wrong here? ($X,Y$ are independent): $1 - P(X > u, Y >u) = 1 - (1 - P(X \leq u, Y \leq u)) =1 - (1 - F_X(u)F_Y(u)) $. it should say, in the last equality:$ = 1 - (1 - F_X(u)) (1 - ...
1
vote
2answers
137 views

Salesmen in a supermarket Poisson

We have a supermarket in which customer enter at Poisson rate 2. There are two salesmen near the door who offer passing customers samples of a new product. Each customer takes an exponential time time ...
0
votes
0answers
42 views

Are there “necessary” conditions for a solution to the multivariate, truncated Hausdorff moment problem?

I am looking for NECESSARY conditions for a solution to the multivariate, truncated Hausdorff moment problem (i.e., conditions under which a given finite sequence of numbers is the sequence of first ...
0
votes
1answer
89 views

Probability of there being at least one defective component with three independent events + 1 more event.

The question is stated as follows. "There are three separate independent components in a machine, each with a defection probability of [p = 0.001]. Further more there is another independent glitch in ...
3
votes
1answer
93 views

Random walk where increment depend on current position

Consider the following stochastic process, $$b(i+1) = b(i) + \xi_i (b_i),$$ where $\xi_i(b_i) \in \{-1, k \}$ are the independent increments having the following distribution: $$\begin{align} P (\xi ...
2
votes
1answer
85 views

How much space probability should have in Statistics learning

Of late I have started self-learning. I have bought a few well-advised Statistics books such as Statistics for Management by Levin and First Course on Probability by Ross. I observe that Ross' book ...
0
votes
2answers
98 views

Sample Space & Random Experiment (Hat Check)

The hatcheck experiment with n = 4 hats. (In this experiment, n people check their hats. When someone comes to claim his/her hat, they are given one of the unclaimed hats, not necessarily their own. ...
2
votes
3answers
157 views

Questions about Bayesian inference

From Wikipedia The prior distribution is the distribution of the parameter(s) before any data is observed, i.e. $p(\theta \mid \alpha )$. ... The sampling distribution is the distribution of ...
2
votes
1answer
160 views

Catching fish in a pond

Suppose the amount of fish in a pond follows a Poisson distribution. A fisherman catches each fish (independently) with probability $\frac{1}{2}$. If $N$ is the total number of fish he catches, ...
-1
votes
1answer
352 views

Expected utility and St. Petersburg paradox

Can someone explain to me how they get the $10.94$ at the Expected utility theory section of the solutions to the St. Petersburg paradox? My problem is that they use a formula to calculate the ...
0
votes
1answer
50 views

Empirical formula of coverage probability

Can someone explain me how this formula calculates the coverage probability. Suppose I have a time series of size $n$. Then I can fit a model to this series and get its one-step ahead forecast and ...
2
votes
2answers
133 views

Conditional expectation as Borel function

Let $X,Y$ be random variables with $E|X|< \infty$. Prove that there is a Borel function $h:\mathbb{R}\rightarrow \mathbb{R}$ such that $E[X|\sigma(Y)]=h(Y)$ almost surely. (Here $\sigma(Y)$ is ...
1
vote
2answers
345 views

Probability of picking out coloured balls of different colours

I am stuck on a question. I know how to attempt it but I was wondering if there is an easier way to do it. Question Suppose you have one red ball, one green ball, one blue ball, and three white ...
1
vote
1answer
48 views

Lower bound functional binomial r.v.

I am trying to find a bound of the type $\mathbb{E}(|B-\frac{N}{2}|) \geq C \sqrt{N}$ Where $B$ is a binomial variable with parameters $(N,\frac{1}{2})$. The bound doesn't need to be very tight in ...
1
vote
1answer
40 views

ordinary question: choosing balls from urn, confuse about the “ordering”

I am confuse about the ordering for the probability question. Say we have $5$ red, $4$ blue balls, and we pick $3$ balls, what is the probability that $2$ of them are red? My solution: ...
0
votes
1answer
68 views

If there is $\epsilon >0$ s.t. $\forall n, m(A_n)\ge\epsilon$, then there is at least one point that belongs to infinitely many sets $A_n$.

I want to prove the following: If $A_n,n\ge 1$ are Borel sets on Lebesgue space $([0,1],B(0,1),m)$, and there is $\epsilon >0$ s.t. $\forall n, m(A_n)\ge\epsilon$, then there is at least one point ...
0
votes
1answer
49 views

$x_1,x_2,\dots,x_n$ are i.i.d RVs uniformly distributed on $\{1,2,\dots,N\}$.

Let $x_1,x_2,\dots,x_n$ be independent identically distributed random variables uniform on $\{1,2,\dots,N\}$, and let: $Y_n:=\text{the number of different elements in } \{x_1,x_2,\dots,x_n\}$. Let ...
0
votes
4answers
444 views

Expected Value of a Probability Density Function with Absolute Value

I'm given a probability density function $f(x) = c |(x^2 - 1)|$ for $-2 \leq x \leq 3.$ I found $c$ by by integrating from -2 to 3 and setting this equal to 1 and got $c = \frac{3}{28}$, so $f(x) = ...
3
votes
0answers
105 views

The distribution of the null space of a random Gaussian matrix

Each element of a `fat' matrix is i.i.d standard normal distribution, is the distribution of the element in its null space still normal? For example, $A$ is a $2\times 3$ matrix, each element of ...
0
votes
1answer
140 views

Poisson Distribution?? finding the probability of randomly distributed trees.

Assume that the aggressive invasive tree known as European Buckthorn is randomly distributed in a degraded forest preserve with λ =40 trees/seedlings per 100m^2. If two 100 m^2 plots are randomly ...
0
votes
1answer
89 views

the characteristic function of this distribution is equal to 0 everywhere except at the origin, mistake?

I wanted to compute the characteristic function of the distribution in question here: How to multiply a standard normal RV times a uniform{-1.1} RV? Let $X$ be standard $N(0,1)$, $Y$ be Uniform ...
1
vote
1answer
240 views

Cloudy and Sunny Days

In Freedonia, every day is either cloudy or sunny (not both). If it's sunny on any given day, then the probability that the next day will be sunny is $\frac 34$. If it's cloudy on any given day, then ...
0
votes
1answer
114 views

Probability that a person has a disease

Assume that the rate of a disease in the general population is 4 per 10,000. If a test has been developed with a sensitivity of 0.97 and a specificity of 0.96, then what is the probability that a ...