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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8
votes
6answers
565 views

Is $\frac{1}{\infty}$ equal zero?

After reading this paragraph: A simpler version of this distinction might be more palatable: flip a coin infinitely many times. The probability that you flip heads every time is zero, but it ...
3
votes
2answers
736 views

Examples of Student's T-distribution in real world empirical data?

I have recently stumbled onto some empirical (forecasting error) data that should be normally distributed. However, the normal distribution fits relatively poorly due to the abundance of data points ...
2
votes
2answers
167 views

a question on general conditional probability

Let $(\Omega ,\mathcal F ,P) := \bigl((0,1],\mathcal B((0,1]),u \bigr)$, where $u$ is the Lebesgue measure restricted to $\mathcal B((0 ,1])$. Let $X\colon\Omega\to\mathbb R$ be defined by $X(\omega) ...
13
votes
7answers
2k views

Good books on “advanced” probabilities

what are some good books on probabilities and measure theory? I already know basic probabalities, but I'm interested in sigma-algrebas, filtrations, stopping times etc, with possibly examples of ...
4
votes
1answer
62 views

Probability; can't understand the maths

For a random variable $x$, define a probability distribution $p[x=n]=c (3^n/n!)$ when $x=0, 1, 2, \dots$ and $p(x)=0$ otherwise. Find the value of $c$. My professor provided the solution $$ ...
0
votes
1answer
3k views

expectation of product of independent random variable

it is a problem from my exam praperation sheet Let $U$ , $Y$ be independent random variables. Here $U$ is uniformly distributed on $(0 , 1)$ . whereas $Y$ is $\frac{1}{4} \delta(0) + \frac{3}{4} ...
0
votes
1answer
510 views

How to use joint characteristic function to calculate characteristic function for single variables? [duplicate]

Possible Duplicate: probability question on characteristic function It is a problem in my practice exam. Defined on some common probability space, two random variables $X$, $Y$ have the ...
1
vote
2answers
49 views

Conditional Probability Independency Table Thinking

A have found an alternative definition of independency for a given conditional probability $P(A|B)$, they are independent, iff all columns of the probability table are equal. What does equal mean in ...
1
vote
1answer
107 views

A calculus/probability problem

Let $p_1,\ldots,p_n$ be numbers such that $0 < p_i < 1$ and $\sum p_i = 1$. Let also $x_1,\ldots,x_n$ be numbers such that $x_i \ge -1$. Define the function (for $0<y<1$) $$f(y) = ...
1
vote
1answer
353 views

probability question on characteristic function

I got a big problem with my exam practice question on characteristic function. Here are two. Let $X$, $Y$ be two independent random variables with the following characteristic functions: ...
1
vote
1answer
331 views

Difference between sample mean and true mean of a gaussian

Assume I have a gaussian distribution $\mathcal{N}(\mu, C)$ with mean $\mu$ and covariance $C$. I'm drawing $n$ random numbers from this distribution. Let $m$ be the mean of these numbers. Is there ...
3
votes
0answers
177 views

Size of an intersection with a randomly chosen subset

I'm hoping for some help with this excericse in probability. Let $V$ be a set and let $V'$ be a randomly chosen subset of $V$ such that each element belongs to $V'$ with probability $p$. Now, ...
9
votes
2answers
866 views

Most Probable Sum [duplicate]

Possible Duplicate: Probability of dice sum just greater than 100 A fair dice is rolled and the outcome of the face is summed up each time. We stop rolling when the sum becomes greater than ...
2
votes
1answer
103 views

Convergency of probability measures

While skipping through the class notes I noticed one exercice that I couldn't solve: Suppose we have $\mu$ - probability distribution in $\mathbb{R}$. Recalling that $\mu_k \rightarrow \mu$ iff ...
2
votes
3answers
76 views

Combinatorics: endless series

I have the following problem: In an urn, you have 1 blue and 9 white balls. You pull out one ball a time; if it is the blue one, you win. If it it is white, you throw it back in and pull again. ...
0
votes
2answers
352 views

Chebyshev's & Markov's Inequality

From a book, I found these 2 questions which I have not understand. (1) Suppose X is a discrete random variable with probability function ...
1
vote
3answers
1k views

Coin Toss Probability Question (Feller)

I'm working out of Feller's "Introduction to Probability and its Application (Vol I.)" textbook and I'm stuck on a coin toss problem. I'll list the full problem and show where I'm having trouble. ...
1
vote
1answer
526 views

Recommendations for probability books

i do IT work, and the "it" thing these days is to throw the occasional probability question out there. The last time i stumbled on this, i'd just sat the GMAT and had probability somewhat down... ...
2
votes
1answer
126 views

Laplace Transform of a Truncated Random Variable

Let $B$ be a random variable, and $S=\mbox{min}(1,B)$. Can you help me see why the laplace stieltjes transform of $S$ is given by $$ E[e^{-\alpha S}]=1-\alpha\int_{0}^{1}e^{-\alpha y}P(B\geq y)dy$$
2
votes
5answers
1k views

Elementary Probability Question

So the problem is in Tanis & Hogg, Probability and Statistical inference section 1.5 Independence of Events. An Urn contains five balls, one marked WIN and four marked LOSE. You and another ...
2
votes
1answer
97 views

Joint distribution gives two marginal

In the following exercise I got two different distributions for $Z.$ I want to know where my mistake is. Every hint or comment is appreciated. The exercise goes as follows: Let $(X,Y)$ be a ...
2
votes
1answer
129 views

Weak a.s. convergence VS a.s.weak convergence

Let's consider a sequence $(\mu_n)_n$ of random probability measures on $\mathbb R$, and let $C_b$ be the Banach space of bounded continuous functions on $\mathbb R$. I am considering the following ...
1
vote
3answers
189 views

Probability of sharing cards drawn from different decks?

I have been struggling with the following problem, which I have been trying to solve combinatorially, but without much success. Suppose n players each have a deck of cards. Each player randomly draws ...
55
votes
6answers
4k views

Is it generally accepted that if you throw a dart at a number line you will NEVER hit a rational number?

In the book "Zero: The Biography of a Dangerous Idea", author Charles Seife claims that a dart thrown at the real number line would never hit a rational number. He doesn't say that it's only ...
0
votes
1answer
51 views

Invariant markov chains and stopping time question

I have two Markov chains $X_n$, $Y_n$ with the same transision matrix P, which is non-periodic and non separable. The initial distribution is $\pi_x = \frac{1}{3}[1,1,1]$ and $\pi_y$ is unknown. ...
2
votes
1answer
48 views

Computing probabilities involving committees

A committee consisting of 6 members is randomly selected from 25 students, 5 teachers, and 10 parents. I wish to find the following: (i) the probability of having no teacher on the committee (ii) ...
2
votes
1answer
67 views

Figuring out probability in a sequence of coin tosses

I have a question I have worked out and I would like to check my solution. I am told that 4 fair coins are tossed in succession. I am to find the probability of getting 4 tails given that the first 2 ...
0
votes
1answer
206 views

Probability space for stochastic processes

In Sinai's book on stochastic processes, the definition for discrete time stochastic processes is "a sequence of random variables $\{X_{n}\}_{n\in{}T}$ defined on a common probability space ...
0
votes
1answer
370 views

Binomial probability distribution

(1) Three items are selected at random from a manufacturing process & classified defective or non-defective. A defective item is designated a success. Assume $25\%$ of the production is defective. ...
0
votes
1answer
77 views

Comparison of implicit functions of random variables

Assume that we have a continuous random variable $X$ with bounded support $[a,b]$ such that $0<a<b$ and two implicit functions of $X$, $g:R^+ \rightarrow R^+$ and $g:R^+ \rightarrow R^+$. We ...
1
vote
0answers
86 views

What are the applications of normalization in math

I know what a normalization of numbers mean in probability. As far as I know; If a=3, b=2,c=1: Then nomalized value of a is 3/(1+2+3), b is 2/(1+2+3) & c is 1/(1+2+3). But I dont know why do we ...
2
votes
2answers
662 views

Definition of Conditional Probability by Measure Theory

I was reading a book on information theory and entropy by Robert Gray, when I saw the following definition of conditional probability: Given a probability space $(\Omega,\mathcal{B}, P)$ and a ...
1
vote
1answer
202 views

lower bound for probability of no 2 balls per bin.

There are $n$ balls and $m$ bins and every ball is placed independently and uniformly at random into a bin. I'm trying to show that there exists a constant $c$ such that, if $m=c\sqrt{n}$ then with ...
2
votes
6answers
444 views

Variation of the Monty Hall Problem.

Suppose instead of the normal Monty Hall scenario in which we have two empty doors and a car residing behind the third, we instead have three types of objects. One is a car, one is a hard drive, and ...
1
vote
1answer
914 views

Integrating a probability density function

Let the pdf defined as: $P(x, \bar{x}, \sigma) = \exp\left(\frac{-(x-\bar{x})^2}{2\sigma^2}\right)$. How can we integrate this probability density for some values of $x$ that are higher than a given ...
2
votes
1answer
172 views

Densities of Levy Processes Continuous

I've been reading the literature and I am not sure whether this is a necessary condition or not... Suppose the probability density function $f$ is infinitely divisible and generates a Levy process ...
2
votes
1answer
1k views

Relationship between binomial and negative binomial distributions (how to extend the probability space?)

I wonder a technique to extend the discrete probability space. Here's an example from Concrete Mathematics EXERCISE 8.17: Let $X_{n,p}$ and $Y_{n,p}$ have the binomial and negative binomial ...
5
votes
1answer
498 views

Calculation of the moments using Hypergeometric distribution

Let vector $a\in 2n $ is such that first $l$ of its coordinates are $1$ and the rest are $0$ ($a=(1,\ldots, 1,0, \ldots, 0)$). Let $\pi$ be $k$-th permutation of set $\{1, \ldots, 2n\}$. Define ...
0
votes
3answers
3k views

probability of a horse winning a race.

Lets suppose ten horses are participating in a race and each horse has equal chance of winning the race. I am required to find the following: (a) the probability that horse A wins the race ...
2
votes
1answer
54 views

combinations and probablity

A committee of 5 members is selected randomly from 10 parents, 16 students and 4 teachers. how can one find the probability that a teacher will be the chairman if the first person selected is to be ...
5
votes
5answers
459 views

What does it mean to do MLE with a continuous variable

I am struggling with the semantics of continuous random variables. For example, we do maximum likelihood estimation, in which we try to find the parameter $\theta$ which, for some observed data $D$, ...
2
votes
1answer
156 views

Is the expectation $E[\xi U'(\xi)]$ finite?

I encounter the following problem today. It seems a simple question. Let $U$ be a real function from $R^+\rightarrow \bar{R}$ satisfying the following conditions: (1) $U$ is concave, continuous, ...
4
votes
1answer
205 views

Bounding $\|(X'X)^{-1}X'\mathbf{1}\|_{\infty}$ in probability

Let $X \in \mathbb{R}^{n \times d}$ be a random matrix with independent elements $N(0, 1)$ and let $\mathbf{1}$ be an n-vector of ones. Assume that $d \asymp n^\alpha$ for some $\alpha > 0$. I ...
3
votes
1answer
114 views

given vector a, how to find vector b such that inifinitive norm of $a -b$ is smallest

I'd appreciate any hint for the following problem: Given a vector $a=(a_1, a_2,\ldots,a_n)$, how can I find a vector $b=(b_1, b_2,\ldots,b_n)$ such that $b_1 \leq b_2 \leq \cdots \leq b_n$ and the ...
0
votes
2answers
2k views

Ratio Distribution: Poisson Random Variables

Suppose two Poisson processes. For example, during the time interval, $\Delta t_{1} = t_{1} - t_{o} = 50\mu s$ , $x$ photons are incident on a detector with rate $\lambda_{1} = 10$x$10^4 s^{-1}$. At ...
0
votes
1answer
375 views

taylor expansion of exponential function

To prove CLT of binomial distribution, $$X \sim \mbox{bin}(n,p)$$ $M_X(t)=(p e^t+q)^n$ where $M$ is mgf. Let $Z=\frac{X-np}{ \sqrt{npq}}$, $\sigma =\sqrt{npq}$, then $$ \begin{align} ...
0
votes
1answer
281 views

discrete random variables transformation

I have two discrete random variables $X$ and $Y$, let $P_X$ and $P_Y$ be the PMF of the random variables, if $Y=X^2$ ,I want to know the PMF of Y in terms of PMF of X ? I know how to do it with ...
1
vote
1answer
252 views

Explicit examples of smooth entropy computation

Smooth classic entropies generalize the standard notions of entropy. This smoothing stands for a minimization/maximization over all events $\Omega$ such that $p(\Omega)\geq 1-\varepsilon$ for a given ...
3
votes
2answers
399 views

Generating function for Banach's matchbox problem

Here's the description for Banach's matchbox problem from Concrete Mathematics EXERCISE 8.46 (edited) Stefan Banach used to carry two boxes of matches, one containing $m$ matches and the other one ...
3
votes
2answers
311 views

Probability of getting two consecutive 7s without getting a 6 when two dice are rolled

Two dice are rolled at a time, for many time until either A or B wins. A wins if we get two consecutive 7s and B wins if we get one 6 at any time. what is the probability of A winning the game??