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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0answers
190 views

Probability, ball and bin with constraint

Assume that we have $n$ bins(indistinguishable and numbered from $1$ to $n$). We pick a bin and add a ball to it subject to the constraint that each bin can have no more than $n$ balls. If we know ...
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1answer
32 views

Show that $\mathbf{E} a\xi 1_{\xi>a} \leq \sup_{t>0} t^2 \mathbf{P}(\xi>t)$ for $a>0$ and a positive random variable $\xi$.

Show that $\mathbf{E} a\xi 1_{\xi>a} \leq \sup_{t>0} t^2 \mathbf{P}(\xi>t)$ for a positive random variable $\xi$. Here a>0 is arbitrary.
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1answer
934 views

Probability that a 3-digit randomly chosen number is divisible by 5

$$\text{Set}\; = \{0,1,2,3,4,5,6\}$$ Find the probability that a three digit number which does not have the digit $0$ chosen at the far left and is chosen at random from the set, is NOT a multiple ...
2
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2answers
702 views

Two cards are drawn without replacement. Find the probability the second card is a jack given the first is not a jack.

My calculations: I got $\frac{4}{51}$ because there are $4$ jacks in a deck and if we didn't have a jack then there are still $4$ left out of $51$ because we already chose one card. Is this right?
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2answers
544 views

if one three digit number (0 cannot be left digit) is chosen at random from all those that can be made from the set bellow.

SET = {0,1,2,3,4,5,6} find the probability that the three digit number chosen is NOT a multiple of 5.. how would i do this question?
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2answers
876 views

What is the probability of getting the sum of 5 or at least one 4 when you roll a dice?

I just want to know if my method is right: P(Sum of 5 or At least one 4) = 2+3, 3+2, 4+1, 1+4 [+] (4+1,4+2,4+3,4+4,4+5,4+6)*2 So that will be 4+12/36 Ans: 16/36 am i right here?
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2answers
90 views

Need help with derivation of conditional expectation

The following is taken from the book "Mathematical Statistics for Economics and Business": \begin{align*} E\left.\left( \left[ Y-h(x) \right]^2\ \right\vert\ x\right) =& E\left.\left( ...
3
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0answers
62 views

What is the probability that the store is closed?

Customers arrive at a store according to a Poisson process with a fixed rate $\lambda$ per hour. Now we only know that the store have served $m$ people during $T$ hours. What is the probability ...
4
votes
1answer
111 views

variance of number of divisors

Let $d(i)$ be the number of divisors of $i$. I know that $\frac{1}{n}\sum_{1\le i \le n} d(i)= \ln n+\Theta(1)$ as $n$ grows, this can be seen by asking, for each $j$, how many $i$ are there such that ...
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2answers
335 views

Conditional probability with Bayes' Rule

On a practice exam from statistics I encountered a very difficult exercise I couldn't manage to solve: In the tent next to you there is a family with two children. Early in the morning you see a ...
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2answers
289 views

Intuition about whether to switch in box problem

I ran across an apparent paradox which I then located in the paper The Box Problem: To Switch or Not to Switch as such: Imagine that you are shown two identical boxes. You know that one of them ...
1
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2answers
697 views

Linearity of conditional expectation (proof for n joint random variables)

Linearity of conditional expectation: I want to prove $$E\left(\sum_{i=1}^n a_i X_i|Y=y\right)=\sum_{i=1}^n a_i~ E(X_i|Y=y)$$ where $X_i, Y$ are random variables and $a_i \in \mathbb{R}$. I tried ...
0
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3answers
70 views

Probability of getting $70 \%$ or better

What is the probability of getting a $70\%$ or better on a $10$ question true/false test having not studied for the test?
0
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1answer
48 views

How can iid be from a conditional distribution?

Could someone explain to me how the following two things can hold at the same time: (a) $y$ is conditionally distributed to depend on $x$, that is $f(y|x)$ (b) $y$ is also i.i.d. (independent of ...
2
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2answers
79 views

What can be said about $E[1_A\mid\mathcal F]$?

It is known that $E[X\mid 1_A]$ is of particularly nice form. What can be said about the form of $E[1_A\mid\mathcal{F}]$ for "general" $\mathcal{F}$? Is it true that $E[1_A\mid\mathcal{F}]=1_B$ for ...
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4answers
81 views

Elementary set proof

On a statistics trial exam I encountered the following proof I was supposed to give but I have no idea how to start with this proof and solve it: $P(A\cap B)$ $\geq$ $1 - P(A') - P(B')$ where $A'$ is ...
1
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1answer
98 views

Gebelein's Inequality and convergence of distribution

We know that for a bivariate standard normal vector $Z=(Z_1,Z_2)$ it holds that \begin{align*} \operatorname{Cov}(1\{Z_1\leq u),1\{Z_2\leq u))\leq \operatorname{Cov}(Z_1,Z_2). \end{align*} This ...
0
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1answer
286 views

Lottery, probability of losing

Let's assume a imaginary lottery where each draw, we are allowed to buy the following tickets: ...
1
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1answer
57 views

Simple probability problem

A bag contains $(2m+1)\,$ coins. It is known that $m$ of these coins have a head on both sides and the remaining coins are fair. A coin is picked up at random from the bag and tossed. If the ...
2
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1answer
68 views

what does it mean when $\operatorname{E}[X^2]$ diverges?

is it possible for a random variable $X$, such that the expected value of $X^2$, $\operatorname{E}[X^2]$ is a divergent integral? If it is impossible, does that mean the probability density function ...
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1answer
60 views

Computing mean and standard deviation of binomial variate $X$

Let $X$ be binomially distributed with $n = 60$ and $p = 0.4$. (1) When i randomly generate $60$ numbers, 15 to 30 occur most frequently. Why? that is, what does ...
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1answer
60 views

Writing the definition of the expectation $E(X+Y \mid Z=z)$

If I want to write out the "definition" of the conditional expectation $E[X+Y \mid Z=z]$, would it be (for the continuous case): $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}(x+y)\,f_{X,Y\mid ...
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votes
3answers
4k views

Notation of random variables

I am really confused about capitalization of variable names in statistics. When should a random variable be presented by uppercase letter, and when lower case? For a probability $P(X \leq x)$, what ...
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2answers
134 views

Change in expected value by repeating an experiment

I made this question myself which I think is a question on the conceptual level of expected values which can be solved only by logic. However being a beginner at expectancy, I am unable to solve this. ...
2
votes
1answer
567 views

Affine transform of multivariate gaussian

If $X_1, \ldots, X_n$ are iid $N(0,1)$ or in other words $\mathbf{X}=(X_1, \ldots, X_n)$ is distributed $N(\mathbf{0}, \mathbf{I})$, then $A\mathbf{X}+\mu$ is distributed $N(\mu, AA^t)$. Showing that ...
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2answers
995 views

A die is rolled until a 6 comes up. Should the sample space of this experiment contain the set of all infinite sequences which do not contain a 6?

Is there a standard way to view this? The problem is, In an experiment, die is rolled continually until a 6 appears, at which point the experiment stops. What is the sample space of this ...
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2answers
33 views

How can i find the following probabilities?

Let $X$ be binomially distributed with $n = 60$ and $p = 0.4$. Now i have to compute (a)$P(20\leq X$ or $X\geq40)$ (b)$P(20\leq X$ and $X\geq10)$ i know $P(x\leq ...
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votes
1answer
196 views

Poker, probability of players true raising frequency

The game is texas holdem, you get dealt two cards out of a 52 card deck. Hence there are 1326 possible combinations of two cards you can get. Let's say our opponent has oppened the pot with a raise 5 ...
3
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1answer
85 views

$k$ cards summing to $n$

This was a problem posted about probability involving Fibonacci numbers that I thought was really interesting so I decided to repost a portion of it regarding a general closed formula. The problem ...
4
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1answer
1k views

Convergence of binomial to normal

Problem: Let $X_n \sim \operatorname{Bin}(n,p_n) $ where $p_n \xrightarrow{} 0$ and $np_n \xrightarrow{} \infty$. What I need to show is that $$\frac{X_n - np_n}{\sqrt{np_n}} \xrightarrow{d} N(0,1) ...
3
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1answer
99 views

Shannon's entropy in a set of probabilities

Let $P = p_1, \ldots, p_N$ be a set of probabilities (i.e., $0 \leq p_i \leq 1$). I can compute the Shannon's entropy as follows: $$ H(P) = -\sum_{i=1}^N p_i \log_2 p_i $$ Now, suppose I perform the ...
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0answers
33 views

Probability of a sum being above a threshold in a n dimensional vector

I am having a problem measuring a certain probability. Suppose a n-dimensional vector a1,a2,...,an. What are P(a1+a2+...+an < 1) , P(a1+a2+...+an = 1) and P(a1+a2+...+an > 1) equal to? ais are all ...
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2answers
45 views

if $X\sim U(0,1)$ show that $(b-a)X+a \sim U(a,b)$

I'm using the MGF method, this is what I get: $$ \begin{align} Y&=(b-a)X+a\\ M_Y(t)&=E[e^{(b-a)X}e^a] \\ &=E[e^{(b-a)X]}e^a &\text{I think this is my error} \\ &= M_x((b-a)t)e^a\\ ...
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2answers
328 views

Probability of randomly generated quadratic equation having equal roots

Could any one help me to solve this problem? Given that the coefficients of the equation $ax^2+bx+c=0$ are selected by throwing an unbiased die, we need to find what is the probability of the ...
0
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2answers
47 views

Common distribution with sharper edges?

Normal probability distribution is common, but it has only mean and stdev parameters being well defined. There are no singular points which could serve as minimum and maximum (all suggestions like 3 ...
0
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2answers
30 views

Expectation with a “regular” function

I hope this is not a silly question. I know that the expectation of a constant is just a constant (i.e. $E[c]=c$ for $c\in \mathbb{R}$), and that for a function $g$ of a random variable X, ...
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2answers
41 views

Dice Probability question

The question is, When Six side dice rolled until the first time T that a four turns up. What is the Sample space for T and distribution function? Also, What is the probability T>3? I ...
1
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2answers
214 views

How many ways to draw consecutive fibonacci numbers from deck of cards

In a deck of cards there are 4 suits of 13 cards each. If the face value of the aces is defined as 1 and the jack, queen, and king are 11, 12, and 13 respectively, then: 1) What is the probability ...
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1answer
388 views

Examples of Metaphors in Statistics and Probability?

I have a couple of questions about teaching of Probability and Statistics for high school students: 1. Can I find metaphors for the teaching of basic concepts of Probability and Statistics? (Please, ...
2
votes
1answer
181 views

Given that the first child draws $10\$$ from his envelope, what is the probability that the second child has an envelope that contains a 20$ note?

Three children each receive an envelope from their grandparents. It is known that each envelope contains three banknotes and that in the three envelopes together there are two $5\$$ notes, four $10\$$ ...
3
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2answers
6k views

Roulette betting system probability

The Fibonacci is a popular Roulette betting system that is based on a naturally occurring mathematical sequence. The sequence itself is cumulative. In other words, the next number is equal to the sum ...
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0answers
34 views

Population size and accuracy of expected value

If I have a series of populations, and a set of outcomes for these populations, how can I be certain that the observed proportions are, in fact, credible? I have investigated certain sampling methods ...
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1answer
144 views

Efficiency of the Sieve of Eratosthenes.

It's well-known that the sieve of Eratosthenes, using the first $m$ primes {$p_1, p_2, ..., p_m$}, sifts out all composite numbers up to $(p_m+2)^2$, since every composite $n \lt (p_m+2)^2$ contains ...
0
votes
1answer
45 views

How to generate sequence like this?

Can you tell what algorithm can generate sequence $x_1, x_2, x_3, x_4, ...$ satisfying: $x_n$ is real, and always $0<x_n<1$. Every change between $x_n$ and $x_{n+1}$, such as increase or ...
1
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1answer
81 views

Absorbing state for a collection of random walks

Further to this question; having learned some stuff since I posed it. Consider a collection of random walks $X_i$ which take finite integer values. These evolve as time-inhomogeneous Markov Chains. ...
2
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2answers
62 views

Why do we care about specifying events in a probability space?

Why aren't probability spaces just defined as $(\Omega, p)$ pairs with $\Omega$ as the sample space, $\sum_{\omega \in \Omega}p(\omega) = 1$, and for a subset $A \subseteq \Omega$, $\Pr(A) := ...
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2answers
233 views

Probability (arranging around a circle)

What is the probability that when arranging n people around a circle, two people with the same birthday (assume no leap years) will be adjacent to each other?
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1answer
103 views

An upper bound related to Binomial distribution

In $m$ independent trials, each with probability at least $p$ of success, the probability that there are at most $k$ success, where $k<mp$, is at most ...
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1answer
101 views

Expected Value for a Sequential Poisson Random Variable

The set up: A mouse nest contains $n$ female mice. In a particular year, the number of female offspring that each female mouse produces has the following pmf: $$ f(x) = \begin{cases} ...
4
votes
1answer
500 views

Convergence in distribution of Gaussian processes

Assume given a sequence $(W_n)$ of Gaussian processes indexed by, say, $\mathbb{R}^p$, with mean zero and covariance function $R_n$. This means that for each $n$, the finite-dimensional distributions ...