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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3
votes
1answer
544 views

Classic $2n$ people around a table problem

A total of $2n$ people, consisting of $n$ married couples, are randomly seated (all possible orderings being equally likely) at a round table. Let $C_i$ denote the event that the members of couple $i$ ...
0
votes
1answer
19 views

Number of necessary stickers to complete a sticker album

I have the following problem, and I was hoping you guys could help me solve it: Consider a set of $t$ unique, collectable stickers (that accounts for the universe of collectable stickers, i.e., ...
3
votes
5answers
84 views

$\sum\limits_{n=1}^\infty n(\frac{1}{2})^{n}$

I am trying to find the expected value of the number of even numbers rolled before the first odd number when rolling a fair die until an odd number comes up. I arrived at $\sum\limits_{n=1}^\infty ...
0
votes
0answers
14 views

How to find the discrete probability vector given a transition probability matrix?

I have a transition probability matrix for the above Discrete Time Markov Chain I want to find the 'discrete probability vector' of this state space. My understanding is that the discrete ...
5
votes
2answers
208 views

How minimize $\sum p_b \ln{p_b}$?

I have a multiset $A = \{a_1,\dots,a_n\}$ of integers. Let $q = P(a_i = a_j)$ when $i$ and $j$ are chosen independently and uniformly from $\{1,\dots, n\}$. Let $B$ be the set of integers in $A$. We ...
0
votes
2answers
43 views

Discrete math: probability of picking certain hands with a preset condition

In 5-card draw poker, a player receives an initial hand of 5 cards, and is then allowed to replace up to three of her cards with the remaining cards in the deck. (b) Suppose that, among the initial 5 ...
6
votes
1answer
56 views

I pull $17$ balls out of a bag, and there are $13$ distinct colors in the sample. About how many colors are in the bag?

I have a bag filled with different colors of balls. My goal is to determine the number of distinct colors that in the bag, but I am limited to taking a small sample. From a sample of $N$ balls, I ...
0
votes
2answers
24 views

MAP Estimator with Laplacian Noise

I need to calculate the MAP estimator of $ x $ in the following case: $$ \left [ \begin{matrix} {y}_{1}\\ {y}_{2} \end{matrix} \right ] = \left [ \begin{matrix} x\\ x \end{matrix} \right ] + ...
1
vote
2answers
36 views

Forth Moment of Sum of Normal with Equal Correlation

I have $X_1,\dots,X_n$ identically normal distributed $N(0,\sigma^2)$ and $\operatorname{corr}(X_i,X_j)=\rho $ for all $i\neq j$. I'd like to compute \begin{equation} E\left(\sum_{i=1}^nX_i\right)^4. ...
2
votes
1answer
9 views

Bound on expectation of function of standard normal, $\mathbb{E}[\exp(Z^a)]$

I'm trying to find the maximum (or sup) of the value of $a$ such that $$\mathbb{E}[\exp(Z^a)]<+\infty$$ where $Z\sim \mathcal{N}(0,1)$. Obviously for $a=1$ the expectation is finite since it is the ...
0
votes
1answer
13 views

Functional Choice for p in a Bernoulli Distribution

Why is the functional choice $p = \exp(x)/(1+\exp(x))$ to model $p$ a good one in a Bernoulli distribution? Is it because it is limited at $0$ as $x$ approaches $0$ and $1$ as $x$ approaches ...
0
votes
0answers
19 views

Bounding second moment of entropy

We know that entropy is defined as the $E(-\log(P(x))$, and we know it's bounded by $log(r)$ (when $r$ is the size of alphabet). Defining the second moment of this, meaning $E(\log^2(P(x))$. How can ...
4
votes
1answer
30 views

Probability of finding a Hamilton circuit in a graph

In short, I would like to know either/both the probability that there exists a Hamiltonian circuit within a graph, or the number of circuits expected to exist. (Without actually finding all the ...
0
votes
1answer
19 views

probability and applied statistics 3 [on hold]

given two urns, suppose urn 1 contains four black and seven white balls. urn 2 contains three black , one white , and four yellow balls . we select an urn at random and then draw a ball . what is the ...
0
votes
1answer
18 views

Average of IID Cauchy RVs

Suppose that $X_i$'s are iid Cauchy RV's with pdf $f_u (x) = \frac{1}{\pi} \frac{u}{u^2+x^2}$. I am aware that the RV $Y:=\frac{1}{N}\sum_{k=1}^N X_k$ has the same density as the $X_i$'s. I am trying ...
-1
votes
0answers
12 views

Given two players competing, what is the probability of one getting ahead x wins vs the other getting ahead y

We have 2 players, A and B, competing. The probability that A wins a match is p, making the probability that B wins a match (1-p) = q. The game is won by player A as soon as he gets one more win than ...
0
votes
1answer
22 views

measurable function and composition of function

Show that if $f$ is a measurable function and $g$ is a continuous function on $\Bbb R$ then $g\circ f$ is measurable. please tell me how to prove it !
2
votes
1answer
26 views

what is the distributions of the random variable?

If moment generating function is $m(t)=[(1/3)e^{t}+(2/3)]^{5}$, then what is the distributions of the random variable?
0
votes
1answer
18 views

Conditioning twice?

I know that $P(X, Y)=P(X|Y)P(Y)$. How can we apply this to $P(X,Y|Z)$? We have already conditioned on $Z$, so can we condition it again on $Y$? Thanks!
0
votes
4answers
131 views

Probability Urn Question(Recurrence Model)

From an urn containing a white and b black balls, balls are drawn one by one at random according to the following rules: (i) at any drawing, if the ball drawn is white, then it is returned to the ...
0
votes
0answers
27 views

Show that if E is measurable set and f is continuous on E, then f(E) is measurable set

Please tell me how to prove or disprove it ! Show that if E is measurable set and f is continuous on E, then f(E) is measurable set
0
votes
1answer
32 views

Hammersley–Clifford theorem

I'm reading this paper http://image.diku.dk/igel/paper/AItRBM-proof.pdf and I got stuck in page 4 with equation (1) that's based on Hammersley–Clifford theorem. I'm not good in reading set theory ...
-1
votes
1answer
27 views

A basic question on uniform distribution [on hold]

I want to know under what condition on random variable $X$, $\{\log_{10}X\}$ is uniformly distributed. Here $\{x\}$ is the fractional part of $x$.
7
votes
2answers
288 views

What is the most unfair set of three nontransitive dice?

In a set nontransitive dice, each die is superior to another die, but is inferior to a third. It is similar to the game of rock-paper-scissors. Here is one example: ...
0
votes
2answers
182 views

Is every symmetric positive semi-definite matrix a covariance of some multivariate distribution?

One can easily prove that every covariance matrix is positive semi-definite. I come across many claims that the converse is also true; that is, Every symmetric positive semi-definite matrix is a ...
4
votes
1answer
27 views

If $X$ ~$Uni(-1,1)$ show that $X$ and $X^2$ are not independent

I provide my approach in solving this but I amd not entirely sure whether I am correct. Since X~uni(-1,1) $f_X(x)=1/2$ and $F_X(x)=(x+1)/2$. $F_{X^2}(x)$=$Pr[X^2≤x]$=$2F_X(√x)$=$(√x+1)/2$. Hence ...
1
vote
0answers
45 views

Proof of the Surfer Model Pagerank formula

How do you prove this formula for the Surfer Pagerank algorithm mathematically? ...
2
votes
3answers
62 views

Probability, random line up

Five distinct families arrive to a party. Each family consists of 3 people. The 15 participants of the party are arranged randomly in a line. Let X be the number of families that their members sit ...
0
votes
0answers
14 views

confidence coefficient z value

I'm having a bit of difficulty understanding a concept in my notes and was wondering if someone could help me. This is probably a really simple concept that I've just completely overcomplicated but ...
1
vote
2answers
40 views

Probability of number formed from dice rolls being multiple of 8

A fair 6-sided die is tossed 8 times. The sequence of 8 results is recorded to form an 8-digit number. For example if the tosses give {3, 5, 4, 2, 1, 1, 6, 5}, the resultant number is $35421165$. ...
1
vote
1answer
200 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 ...
1
vote
1answer
23 views

probability, need help on the marginal densities

I need help on the marginal densities. In particular, I know you just integrate the joint pdf f(x,y) from y=-infinity to +infinity, but in the context of the below question, I have trouble to define ...
0
votes
0answers
27 views

Erdos-Renyi Model Intuition

I was reading the Wikipedia article on the Erdos-Renyi model and I was wondering how they came up with the probability for each connection. I see that there are $n \choose 2 $ nodes to check out, but ...
2
votes
0answers
22 views

Basic probability questions about dice rolls

This is from the Probability for Dummies text: Suppose you roll two dice, one red die and one green die. Let event $A$ represent getting an odd number when you roll a red die and let event $B$ ...
4
votes
0answers
128 views

Asymptotics of sum of binomial distributions

Definition 1: For any random variable $X$, we define $\mathrm{Bin}(p,X)$ as a variable with binomial distribution having parameters $p$ and $X$. Definition 2: For all $i \in \mathbb{N}$, define ...
-1
votes
2answers
24 views

Probability Help - Independent Events and Complements [on hold]

I can't seem to figure out this question in my test-prep textbook.... Help would be appreciated! The probability that an event $A$ occurs is $P(A)=0.3$. The event $B$ is independent of $A$ and ...
0
votes
1answer
29 views

Clique factorization

I'm reading about Clique factorization in wikipedia: http://en.wikipedia.org/wiki/Gibbs_random_field#Clique_factorization but I'm unable to understand this: What is $X_C$ here? Ok I understood ...
1
vote
1answer
43 views

Martingales problem how to

I am unsure how to approach the following question. Given $\{X_1,X_2,...\}$ let $\displaystyle S_n=\sum_{i}^n X_i$ and $F_n=\sigma(X_1,...X_n)$. Suppose that for all $n\geq 1$, $\mathbb ...
0
votes
2answers
30 views

Simple propability issue

I have a simple problem: I have 16 kitties. There is 1/2 chance to get a male and 1/2 chance to get a female. What is the probability to get at least a female? Then, if 10 kitties die, what is the ...
0
votes
2answers
30 views

Explanation for the uniformity of the distance between a Gaussian variable to its nearest integer?

earlier I asked the question Expected distance for a gaussian variable to its nearest integer. and got a good answer. The expected distance is highly close to $1/4$, which is very similar to the ...
0
votes
1answer
202 views

Normalize data with large spread in values.

I'm currently trying to rank a set of data. The issue is that my initial rank comes from a search on google and the returning result set. The spread in values is ranging from 33 all the way up ...
-1
votes
0answers
46 views

how to do this probability question ?

I have two six-sided dice, each with faces numbered from 1 to 6. One of the dice is fair, but the other is not – it will land on numbers 1 to 5 with equal probability, but lands on 6 with a different ...
0
votes
1answer
18 views

Convolution of uniform random variables [on hold]

Let $X$ and $Y$ be IID $U[0,1]$ random variables. Find $\text{Prob}(0 \leq X^2 < Y < X^{0.5} \leq 1)$.
1
vote
1answer
236 views

Unknown number of colours Bernoulli Urn

Okay, so, in the traditional Bernoulli Urn problem, we have an urn with a number N, possibly infinite, of coloured balls, and there are k possible colours. That one I grok. However, what if I don't ...
0
votes
1answer
40 views

How can I prove that $\: \operatorname{Pr}[Y=0] \leq (\operatorname{E}[Y^2] - (\operatorname{E}[Y])^2)/\operatorname{E}[Y^2] \:$?

How can I prove that $\: \operatorname{Pr}[Y=0] \leq (\operatorname{E}[Y^2] - (\operatorname{E}[Y])^2)/\operatorname{E}[Y^2] \:$? I know, $\: \operatorname{Pr}[Y=0] \leq (\operatorname{E}[Y^2] - ...
1
vote
0answers
11 views

Forest fire simulation; analytically constructing a function for tree residual after fire

Consider a Cellular Automaton with an $n \times n$ grid, where each cell corresponds either to a tree or dirt. We assign a tree to cell $(i,j)$ by probability $p$. Next, we initiate a fire in some ...
2
votes
1answer
34 views

Expected distance for a gaussian variable to its nearest integer.

Consider a Gaussian variable with arbitrary mean and variance, $N(\mu, \sigma^2)$. I am interested in its expected distance to its nearest integer : $$ E[|X - R(X)|] $$ where $R(X)$ is the round ...
-1
votes
0answers
8 views

how to find a series of r.v. which is not complete convergences but a.s. [on hold]

Plesase give a sequence of random variables which converges a.s. but not complete convergence
2
votes
1answer
36 views

Roll 2 dice, what's the probability that at least one will come up 6?

Let $P(E)$ be the probability that at least one roll out of the two will come up as 6. I thought of doing $P(E)=1-P(E^{c})$, which is basically $1-P($neither of the 2 rolls are 6$)$. So ...
0
votes
1answer
14 views

Probability of 2 of three independent events occuring

Three objects are thrown at a target. The probabilities the individual objects will connect with the target is .75, .85 and .90. Find the probability that at LEAST two of the objects hit the target? ...