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
47 views

Probability question cards

This is a probability question. There are 6 cards with letters a, c, e, i, m, n in a box. Somebody picks cards in a random order. What is the probability of getting the word “cinema”? Don't solve it ...
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1answer
73 views

Expectation/ independence of random variables

Let $X,Y$ be two correlated variables and $Z\sim N(0,1)$ independent of $X,Y$. Consider the expectation: $$E[f(X,Y)Z].$$ If $f(X,Y)$ and $Z$ are independent then clearly $E[f(X,Y)Z]=E[f(X,Y)]E[Z]=0$ ...
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2answers
913 views

Probablity a randomised four digit number does not have two specific consecutive numbers

I am trying to work out the probability a four digit number does not have two consecutive numbers, for example two consecutive 5's, not starting with a 0 is assumed. Now I could work out how many ...
1
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1answer
51 views

Sample Variance Part B)

The random variable $$ S^2=\sum_{i=1}^n \frac{(X_i- \overline X)^2}{(n-1)} $$ is called the sample variance. A) Show that $ (n-1)S^2=\left[\sum_{i-1}^n ((X_i-\mu)^2 \right]-n(\overline X-\mu)^2 $ ...
0
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0answers
58 views

Infimum of conditional probabilities

Let $f(\cdot,\cdot)$ be the joint density of random variables $X,Y$, and suppose that the joint density is symmetric, i.e., $f(w,w') = f(w',w)$ for all $w,w'$; denote the marginal density of $X$ by ...
1
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2answers
127 views

Family with two boys [duplicate]

You call randomly a family with two kids, and ask if there is a kid called Tom. The answer is yes. Then what is the probably that the family has two boys. So we want that $P(2 \text{ boys } | \text{ ...
1
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1answer
95 views

Sample Variance question

The random variable $$ S^2=\sum_{i=1}^n \frac{(X_i- \overline X)^2}{(n-1)} $$ is called the sample variance. A) Show that $ (n-1)S^2=\left[\sum_{i-1}^n ((X_i-\mu)^2 \right]-n(\overline X-\mu)^2 $ ...
1
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2answers
108 views

What will be expected value of smallest element of chosen set

We are given a set $X = \{x_1,x_2,\ldots,x_n\}$ where $x_i = 2^i$. A sample $S$ (which is a subset of $X$) drawn by selecting each $x_i$ independently with probability $p_i = \frac{1}{2^i}$. What will ...
1
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2answers
458 views

What combination of even and odd numbers would you prefer for $5$ digit lottery ticket?

I sometimes play lottery. The tickets are $5$ digit numbers, for example $34298$. I always buy a ticket which have $2$ even, $3$ odd numbers on it or vice versa, even though I know that it does not ...
3
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1answer
121 views

find the Law of probability Stopping time $T=inf\{n\ge 0: R_{n}\gt a\}$ for fixed number $a\gt 0$.

suppose $R_{n}=\sum_{i=1}^{n} X_{i}$ for $n\ge 1$ and $R_{0}=0$ , that $X_{i}\gt 0$ Random variables Are independent and distributed.find the Law of probability Stopping time $T=inf\{n\ge 0: ...
0
votes
1answer
49 views

Conditional Joint PDF given a value

Given $f(x,y) = (6-x-y)/8$ and $0<x <2$ and $2<y<4$ What is $P(2<Y<3|X = 1) $ How do i approach this problem? I got the marginal PDF for x $21/16 - 3/8x$
0
votes
1answer
51 views

MLE of Poisson - Is this correct and where do I go next?

Let's say I have a variable $X_i$ distributed as $Poisson(\lambda)$ for $i=0,1,2,...,m$ Now, we assume that we can write $\lambda = n_ip$ such that $X_i$ is distributed as $Poisson(n_ip)$. We assume ...
2
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0answers
43 views

Do these properties imply full rank?

Assume that we have a $M\times N$ matrix $P$ (where $M=N$): \begin{bmatrix} p_{1,1} & \ldots & p_{1,n} & p_{1,n+1} & \ldots & p_{1,N}\\ \vdots & & \vdots & \vdots ...
1
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2answers
271 views

Looking for a book on Probability and Statistics.

I am looking for a book or website on mathematical theory of probability and statistics for preparation of an examination. The syllabus written in the unit 4 of this document. Only multiple choice ...
1
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2answers
325 views

Chebyshev's Inequality

Consider $X_{1},...,X_{30}$ independent Poisson random variables with mean 1. I need to find a lower bound for $$ P(25 \le \sum_{i=1}^{30}X_{i} \le 35) $$ My first thought was that: $$ \bar{X}_{30} ...
1
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1answer
1k views

A pair of unbaiased dice are rolled together till a sum of either 5 or 7 is obtained. Then find the probability that 5 comes before 7..

Problem : A pair of unbaiased dice are rolled together till a sum of either 5 or 7 is obtained. Then find the probability that 5 comes before 7.. My approach : Probability of getting 5 ( let it ...
1
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1answer
139 views

Probability of a 5 card hand from a standard 52 deck containing all 4 suits

The answer to this is $\dfrac{4 {13 \choose 2} {13 \choose 1} {13 \choose 1} {13 \choose 1}}{ {52 \choose 5}}$, but what I'm trying to figure out is why $\dfrac{{13\choose 1}{13\choose 1}{13\choose ...
0
votes
1answer
172 views

Marginal density of bivariate density that is a circle

I have the following density function: $f(x,y) = \frac{2}{\pi}$ for $x^2 + y^2 \leq 1$ and $y > x$. I figured out that this represents half of the unit circle (the upper half when cut along the ...
1
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1answer
57 views

Conditional pdf

$f(x) = 2(1-x)$ for $0 < x < 1$ Given that $X$ exceeds $0.5$ what is the probability that $X$ is less than $0.75$? How do I go on about thia problem? I can calculate the probability of ...
1
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2answers
28 views

How to calculate a series with binomial terms invovled

I'm studying probability and having trouble in understanding the following calculation How to get from left to the right on the first line, with the condition that m could only be even numbers? Any ...
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1answer
128 views

Balls and Bins problem

Below is the problem I wanted to solve : ...
1
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1answer
52 views

Probability: PDF and CDF of a disc

A point is chosen at random from a disc of radius 1. We use the uniform distribution on the disc meaning that the probability of a subset of the disc is equal to the area of the subset divided by π, ...
0
votes
1answer
74 views

log partition function of exponential family

In an exponential family $$p_{\theta}(x)=\exp \left(h(x)+\sum\limits_{i=1}^s \theta_iT_i(x) - \phi(\theta) \right) $$ is the log partition function $$ \phi(\theta)=\log \int \exp ...
0
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1answer
193 views

Prove that $f(X)$ and $g(Y)$ are independent if $X$ and $Y$ are independent

Let $X$ and $Y$ be independent random variables. Prove that $f(X)$ and $g(Y)$ are independent for any choice of $f$ and $g$. This sounds very obvious, but I have no idea how to approach it. ...
0
votes
1answer
102 views

Expected running time

Suppose A = A[1] . . .A[n] of length n (where A[i] is either 0 or 1). We want to determine if at least half the elements in A are 1’s. ...
4
votes
2answers
151 views

Intuitive explanation for $\mathbb{E}X= \int_0^\infty 1-F(x) \, dx$

I can see by manipulating the expression why $\mathbb{E}X$ works out to be $\int_0^\infty 1-F(x)\,dx$, where $F$ is the distribution function of $X$, but what is an intuitive explanation for why that ...
0
votes
4answers
225 views

laws of probability

Suppose that there is a $60\%$ probability that the product will be a success on the market (that means, the probability of failure is $40\%$). If the product is a success, you will get a profit of ...
0
votes
1answer
18 views

Probability question for an employee raffle

We are holding an employee raffle and there's one big prize that everyone wants. There are 145 people in the pool. We want to give the big prize at the end to build suspense but an employee said we ...
4
votes
1answer
89 views

Conditional expectation: Is $X/E[X \mid \mathcal{G}] \in L^p$?

Let $(\Omega, \mathcal{F}, P)$ be a probability space, and let $X$ be a strictly positive random variable with finite moments of all orders (i.e. $E[X^q] < \infty$ for all $1 \le q < \infty$). ...
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3answers
94 views

Probability and combinations question

Find the probability that a poker hand of 5 cards from a standard deck will contain exactly 2 face cards (i.e. J,Q,K) (event A), given that it contains exactly 1 cards smaller than 8 (i.e. ...
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5answers
2k views

Example of two dependent random variables that satisfy $E[f(X)f(Y)]=Ef(X)Ef(Y)$ for every $f$

Does anyone have an example of two dependent random variables, that satisfy this relation? $E[f(X)f(Y)]=E[f(X)]E[f(Y)]$ for every function $f(t)$. Thanks. *edit: I still couldn't find an example. I ...
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0answers
76 views

Markov Chain Problem

I have been stuck on this question for days and really need some help. There are two methods, A and B, to finish a work. Method A succeeds with probability 1/3, but if it fails one tries method B ...
1
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2answers
96 views

probability regarding three people throwing a die

There are 3 players, A, B, C, taking turns to roll a die in the order ABCABC.... What's the probability of A is the first to throw a 6, B is the second, and C is the third? The answer said it's ...
0
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1answer
42 views

How to solve this probability exercise?

We have a box and we have on it 6 balls with numbers from 0 to 5. We push out 3 balls in the way that after pushing out a ball we turn it back again in the box. What is the probability that the sum of ...
0
votes
1answer
31 views

correlated random variables

Suppose that X~Normal(m1, var1) and Y~Normal(m2,var2) and X and Y are correlated. Is this true to conclude that Z=[X|X>0]+Y is a normal variable with the following parameters: ...
2
votes
1answer
44 views

Probabilities with three events

I have a probability problem where I have to calculate the total probability and a Bayes probability from two events. The chances as given are: $P(A) = 0.1, P(A^c) = 0.9$ $P(B|A) = 0.9, P(B|A^c) = ...
0
votes
1answer
69 views

Probability: Joint PMFs

The probability that a particular student passes a test is 0.75. The number of tests required until the student receives a passing score is thus distributed geometrically, so if $X$ is the number of ...
0
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0answers
38 views

markov definition by bounded measurable function

I need to prove: $$P(X_t|\mathcal{F}_s) = P(X_t|X_s) \Leftrightarrow E[f(X_t)|\mathcal{F}_s] = E[f(X_t)|X_s]$$ where $$\mathcal{F}_s= \sigma(X_u,u\leq s)$$ (I have to prove that this two ...
0
votes
1answer
45 views

Probability of having lots of unique elements

If you sample $n$ integers from the range $1$ to $n$ inclusive it seems intuitive that you are likely to get a lot of numbers exactly once. Call $X_n$ the number of integers you get that occur ...
0
votes
2answers
34 views

Variance problem with probability

Three cards are drawn sequentially from a deck that contains 16 cards numbered 1 to 16 in an arbitrary order. Suppose the first card drawn is a 6. Define the event of interest, A, as the set of all ...
1
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0answers
137 views

A spaceship travelling to infinity while avoiding star collisions

Consider placing countably infinitely many points labeled $S_i$ randomly over $\mathbb{R}^2$, with asymptotic density points/area $µ$. Then, what is the largest $r$ such that we can find a a ...
1
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0answers
108 views

Mean value theorem and random variables

Let $(Z_i)_{i\in\mathbb{N}}$ be a (stationary) sequence of real valued random variables with finite variance where $Z_i:\Omega\rightarrow\mathbb{R}$ for all $i\in\mathbb{N}$. Further let ...
0
votes
2answers
74 views

Probability theory - Dice

Two guys are playing dice with each wagering $50. Player 1 chooses 2 as his lucky number, and Player 2 chooses 6. Every time their lucky number appears as a result, the player gets one point. The ...
3
votes
2answers
139 views

How can it be meaningful to add a discrete random variable to a continuous random variable while they are functions over different sample spaces?

It seems that one usually use the set of all possible values of $X$ as the sample space of random variable $X$. (Therefore discrete random variables have countable sample space, continuous random ...
0
votes
1answer
235 views

Autocorrelation function, cumulants and probability distribution

I have a doubt. Is it possible to get the cumulants of a probability distribution from the autocorrelation function? or the probability distribution?. For example, the variance (the second cumulant) ...
0
votes
1answer
37 views

Chance $U_1$is bigger than all other random variables

Please, help with the following would be highly appreciated. Again, I have an idea and a solution, but would like to see what other people think. Let $X_1, X_2,\dots, X_N$ be iid random variables ...
0
votes
1answer
69 views

Conditional Distribution of a random variable in a Multinomial Distribution

Given $X_1,X_2,X_3$ ~ $Multinomial(n, \theta_1, \theta_2, \theta_3)$ what is the conditional distribution of $X_2$ given that $X_1=x_1$? My thoughts are: P($X_2=x_2$ | $X_1=x_1$) = $n-x_1 \choose ...
0
votes
0answers
27 views

Limit of $n(1-U_{n:n})$.

Let $U_{n:n}$ be n-th order statistic of uniform distribution on segment [0,1]. Let's think about $n(1-U_{n:n})$. I'm supposed to find its limit for n approaching $\infty$. And another question is: ...
2
votes
1answer
103 views

Intuition behind product rule of probability

I have an intuitive understanding of the definition of conditional probability: $$P(x \vert y) = \frac{P(x, y)}{P(y)}$$ based on a Venn diagram. I imagine that given $y$, we zoom in on this region to ...
2
votes
2answers
124 views

Show Independence of two random variables $X$ and$Y$

Suppose that $X$ and $Y$ are independent random variables and $g$ is a real-valued function on $R$. Show that $g(X)$and $Y$ are independent. Okay, so i dont really know where to begin with this to ...