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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0
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2answers
24 views

Conditional probability about card picking.

A card is picked at random from N cards labeled 1,2,3,,,,,N and the number that appears is X. A second card is picked at random from cards numbered 1,2,3,,,X and its number is Y. I am asked to ...
0
votes
1answer
7 views

Random Variable Modeling

I am trying to understand how to model a random variable. So using a biased coin with $P(Head) = q$. If I am to generate a random variable $Y$ that is equally likely to be either a or b depending on ...
0
votes
0answers
19 views

Struggling with the notation of conditional expectation

Here's the question. I know the fomula of $E(X|Y=y)$ where y is a paticular number. But for this question, I really don't understand what's $E(X|Z)$ meaning. Help!
-3
votes
0answers
16 views

Probability 2 balls in a bag [on hold]

consider an experience which consisit of drawing 2 balls with replacement from an urn containing 7 balls of which 3 are blue and 4 are yellow (i)what is the sample space (ii)define the events as a ...
1
vote
1answer
34 views

Find the value of $a$ and $b$ in $ F(x) = a + b \arcsin x $

Given $X$ is a continuous random variable and its probability distribution function is $$F(x)= \begin{cases} 0, & x < -1, \\ a+b\arcsin x, & -1 \le x < 1, \\ 1, & x \ge 1 ...
3
votes
2answers
51 views

Show that $P(X > \lambda) \geq \frac{(EX - \lambda)^2}{EX^2}$

Question: Let X be a nonnegative random variable and $0 < \lambda \leq EX$. Show that $P(X > \lambda) \geq \frac{(EX - \lambda)^2}{EX^2}$ At first glance I thought I could use some ...
0
votes
0answers
10 views

MATLAB code based on a binomial random matrix [on hold]

I have a matrix, A, with $40000$ binomial random elements. I am trying to complete the following code and would appreciate help: I need to create $40000$ arrays $X_n$, where $X_n$ represents the ...
1
vote
1answer
31 views

Let n>=2, k>=2. The set of all k-element subsets partitioned into 4 classes: (i) class of subsets containing both 1 & 2, how many k-element subsets?

Sorry for the long title, I'm new here & not sure of the appropriate way to post long questions. The full question is: Let n>=2,k>=2. The set of all k-element subsets of [n] may be partitioned ...
1
vote
1answer
15 views

Semimartingale jumps question

I am reading a statement which contains $\Delta X \cdot Y$ where $X$ is a semimartingale and $Y$ is a finite variation process and the notation means the lebesgue stieltjes integral. My problem is ...
0
votes
1answer
19 views

Poisson Process problem, transform the possibility notation

Question: Suppose that a store opens at 0 pm and customers arrive according to a non-homogeneous poisson process ${N(t),t\ge0}$ with the intensity function $\lambda(t)=2t+1$ per hour. Let $S_3$ denote ...
1
vote
2answers
40 views

Multnomial coefficient combinatorics problem

The following problem: Ten diplomatic delegates are seated in a row. There are two specific seating requirements: 1) France and Britain are sat next to each other, and 2) the U.S. and Russia are ...
2
votes
0answers
27 views

How to calculate probability of users generating distributed events reaching n events per 15 minutes?

We have games & apps that connect to services such as Facebook and Twitter to fetch information. These services have various rate-limit caps that you cannot exceed - typically based on a 15 minute ...
1
vote
1answer
38 views

What is the probability that a random K-bit odd-number is prime?

Is it $e/K$? In an experiment that created 1000 random RSA-2048 key-pairs, 2000 random 1024-bit primes were created. It turned out that $727,709$ random candidates were generated, to create 2000 ...
4
votes
0answers
22 views

An integral with respect to the Haar measure on a unitary group

Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate $$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$ where $dH$ is the unit invariant Haar measure on the group of unitary ...
0
votes
1answer
24 views

Challenging Problem of Linear Permutation by H.C. Rajpoot

How many numbers are lying between 20045757087 & 87050752074 when all the 11-digit significant numbers, formed by permuting the digits 0, 0, 0, 2, 4, 5, 5, 7, 7, 7, 8 together, are arranged in ...
0
votes
0answers
32 views

Expectation of matrix product

Suppose we have a random matrix $M \in \mathbb{R}^{n\times m}$ such that $\text{E}[M] = 0$ and $\text{E}[M M^\top] = \Sigma$. How does one compute $\text{E}[M^\top M]$?
-1
votes
1answer
38 views

Proving $P(A_i)=P(B_i)$ [on hold]

Suppose that $P$ is a probability on a field $F$. Consider three events $A_1,A_2,A_3 \in F$ so that $P(A_i \cap A_j) = 0$ for all $i \neq j$. Let $B_1 = A_1, B_2 = A_2 \cap A^c_1$ and $B_3 = A_3 \cap ...
1
vote
1answer
23 views

combinatrix & probabilities

probabilities have always been something tough to comprehend for me, may be someone can help me on this. So here's the problem: Bob tosses a coin but can't see the result, his friend John can see it, ...
1
vote
2answers
52 views

Deriving a joint cdf from a joint pdf

I see that a similar question was asked last year, but I am still confused. I have $f(x,y) = 2e^{-x-y}$, $ 0 < x < y < \infty $ and need to find the joint CDF. I have a solution that ...
1
vote
1answer
17 views

Does the statistical frequency of patterns manipulate the probability of a given event? [on hold]

This is a question I've encountered when I first read about the Gambler’s Fallacy, I'm really wondering why it's considered fallacious? Taking statistics into consideration, If you studied the results ...
2
votes
0answers
29 views

How to combine two conditional exponential CDF's?

Suppose one has two machines (machine A and machine B) in sequence with time to machine break down exponentially distributed with rate parameters $\lambda_A$ and $\lambda_B$. Machine A and B have a ...
1
vote
1answer
35 views

Showing That Two Normal-Based Random Variables Have the Same Distribution

Above is my question. $\overline X$ has distribution $N(0,1/n)$ - that's fine to work out. Similarly, $X_n / \sqrt{n}$ has distribution $N(0,1/n)$. These follow from the general relation $$ ...
0
votes
1answer
37 views

Conditional expectation of $X$ given $Z$, where $Z = 1$ if $X > Y$ and $-1$, otherwise

Let $X\sim\operatorname{Exp}(1)$ and $Y\sim\operatorname{Exp}(2)$ be independent random variables. Define $Z$ by $$ Z = \begin{cases} 1,& X>Y\\ -1,& X\leqslant Y. \end{cases} $$ I want to ...
0
votes
2answers
49 views

Expected value problem on dice reroll

The question is here: Roll N* 3-sided dice(0,0,1), roll them twice and choose a better result, what is the expected value? If possible I would also like an answer for dice {0,1,2} or {1,2,3} if ...
0
votes
1answer
51 views

Lottery winning

This is a ratter simple probabilistic problem but i have not seen any similar. My local lottery works like this: There are 48 numbers in total (numbered from 1 to 48) You have to pick 5 numbers from ...
0
votes
2answers
27 views

How many 5-element subsets of [10] contain at least one of the members of [3]?

Here [10] denotes the set {1,2,3,4,5,6,7,8,9,10} & in the same manner [3] denotes {1,2,3}. I'm attempting to solve this for my combinatorics course. My method would be to solve 10 permutation 5, ...
3
votes
1answer
30 views

Monty Hall problem with pre-specified probabilites

Suppose that a player is given the probabilities for a prize behind each of the three doors. $p_1$, the probability of the prize being behind door 1, is $p_1=\frac{1}{2}$, the other probabilities are ...
1
vote
0answers
8 views

Distribution of the sample mean of correlated exponential random variables

My question is how to determine the PDF of $X = \frac{1}{N}\displaystyle\sum_{k=1}^N \frac{X_k}{(X_k + a)^2}$ where $X_k$ are dependently, identically exponential random variables with mean $\lambda = ...
1
vote
1answer
19 views

Find the probability generating function of $2X$.

If $X$ follows a poisson distribution with parameter $\lambda$ (mean). Then find the probability generating function of $2X$. I'm getting stuck with forming the expression, as I'm getting confused ...
0
votes
2answers
14 views

Let $X$ be a Random Variable. Define $2X$.

I would like to know what exactly the changes are in the values the random variable($2X$) can take, if for example $X$ follows a Poisson or Binomial Distribution. If suppose $X$ follows a Poisson ...
0
votes
0answers
12 views

Probability of lead Between two Candidates

Suppose in an election cadidate A receives n votes and cadidate B receives m votes $m<n$.If all orderings are equally likely what is the probability that A throughout leads B?I think the number of ...
0
votes
0answers
16 views

Probability of rest of votes, when some votes are already counted

Say, for example, that we had $n$ people voting YES or NO and we have already counted some amount $d$ of the votes and of those $r$ have been YES's. How does this effect (or does it) the distribution ...
1
vote
2answers
43 views

How do you calculate P(A/B), when event B occurred after event A?

There's really only one question I can't begin to handle when it comes to probability, literally. It's not the only type of question I struggle with, though it's the type of question where I can't ...
0
votes
0answers
10 views

Drift of Brownian motion conditioned on Hitting Time

Suppose we have a Brownian motion started from height b>0, with constant negative drift $\lambda$. We can 'calculate' the drift in the following seemingly ridiculous way. We condition on the first ...
0
votes
0answers
38 views

Multiplication rule and regular conditional probability

I've been studying the conditions of existence of the regular conditional probability and have a question about it. Let's $(\Omega, \mathcal{B}, P)$ be a product probability space, and let's say the ...
0
votes
0answers
31 views

$E(X_T; T < \infty) \leq E(X_0)$ with $T$ stopping time

I'm doing this exercise: $(X_n)$ is a non-negative supermartingale and $T$ a stopping time, then $$E(X_T; T < \infty) \leq E(X_0)$$ My attempt: $(X_n)$ is a negative supermartingale, and so ...
1
vote
0answers
24 views

Probability of collecting all the sticker types

This question is in the context of tuning a training procedure, whereby the learner may receive random stickers for good performance. I am trying to figure out the probability of any given learner ...
1
vote
4answers
69 views

Difference between $E[X^2]$ and $E[X^3]$

Hope to ask a dumb question. $Y = aX$,with $a \in N_+$. Here, we know the correlation coefficient is 1. Now, suppose $X \sim N(0,1)$. Here, we know $X, Y$ are not independent. Cov($X,Y$) = ...
0
votes
0answers
42 views

Let $X_1,X_2\sim N(0,1)$. How to find joint pdf of $\,Y_1=X_1^2+X_2^2\,$ and$\,\,Y_2=\frac{\displaystyle X_1}{\displaystyle \sqrt{X_1^2+X_2^2}}$?

Let $X_1,X_2\sim N(0,1)$. How to find joint pdf of $\,Y_1=X_1^2+X_2^2\,$ and$\,\,Y_2=\frac{\displaystyle X_1}{\displaystyle \sqrt{X_1^2+X_2^2}}$? $$$$ I have tried to use Jacobian matrix to do ...
1
vote
0answers
21 views

How to compute the covariance matrix of a random variable uniformly distributed in an ellipsoid

Suppose that x is a random variable uniformly distributed in an ellipsoid \begin{equation} x^{T}Mx\leq\delta, \end{equation} where $x\in \mathbb{R}^{n}$. Clearly, the mean of $x$ is zero. The ...
0
votes
0answers
31 views

Question with the value at risk (VaR) criterion

Let $X$ and $Y$ be the random payoffs from two different investment strategies. Recall that the Value at Risk (VaR) criterion with parameter $\gamma \in (0,1)$ decides $X \succ Y$ if and only if ...
0
votes
1answer
30 views

standard deck and the probability of at least one card,exactly one void and two voids

The question is this: if 13 cards are dealt from a standard deck of 52, what is the probability that these 13 cards include a)at least 1 card from each suit b) exactly 1 void(e.g no clubs)? ...
0
votes
1answer
13 views

mutually exclusive events where one event occurs before the other

This question has been asked before. Here is the link: Mutually exclusive events Here is the description to the problem: Let E and F be mutually exclusive events in the sample space of an ...
1
vote
0answers
16 views

Conditional expectation with disjoint $\sigma$-algebras

Let $(B^1,B^2)$ be independent Brownian motions with corresponding filtration $\mathcal{F}_t$. Let $\mathcal{F}^2_t$ be the filtration generated by $B^2$. How does one prove that for any $s<t$ and ...
1
vote
1answer
66 views

Probability of Random Event and Conditionality

A company has been running a television advertisement for one of its new products. A survey was conducted. Based on its results, it was concluded that an individual buys the product with probability ...
0
votes
1answer
24 views

bernoulli trials and job interviews

I was trying to think of a way to give a hopeful spin to my friend's unsuccessful job interview outcome and I remembered Bernoulli trials which apply to anything with 2 outcomes like "heads or tails" ...
1
vote
0answers
29 views

Expected size of largest weakly connected component?

Given an undirected graph of n vertices and n randomly assigned edges, one edge from each vertex, what is the expected size of the largest connected component? For example, with four vertices, there ...
4
votes
3answers
178 views

Probability of one stock price rising, given probabilities of several prices rising/falling

So this is the problem: An investor is monitoring stocks from Company A and Company B, which each either increase or decrease each day. On a given day, suppose that there is a probability of ...
0
votes
1answer
11 views

Gamma distribution - closed towards multiplication

First observe how the gamma distribution function can be written in terms of the incomplete gamma function. $\boldsymbol{(1)} \qquad G(y) = \int_{0}^{y} \dfrac{c^{\gamma}}{\Gamma(\gamma)} x^{\gamma - ...
-2
votes
0answers
18 views

Probability in 3 multiple choice exams [on hold]

a student is taking 3 multiple choice exams in which each question has 4 choices. there are 16 multiple choice questions on each exam and the minimum passing grade is 10 correct questions. What is the ...