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 ...

learn more… | top users | synonyms (1)

0
votes
2answers
25 views

Formal proof that for distinct $A_i$ and $B \subset \bigcup A_i$, $\sum P(B\cup A_i) = P(B)$

What is a formal proof that for distinct $A_i$ and $\displaystyle B \subset \bigcup_{i\in I} A_i,$ $\sum_{i\in I} P(B\cap A_i) = P(B)?$ Is it just obvious?
0
votes
1answer
25 views

Independence of variables implies no functional representation?

My question's pretty simple, I just thought the title phrases it pretty well.. Anyway, the Doob-Dynkin lemma says that $X$ is $\sigma (Y)$-measurable iff there's a measurable $f$ such that $X=f(Y)$ ...
0
votes
0answers
21 views

Finding conditions for joint probability density larger than the product of marginals

I was wondering if you could help me out. I have a joint probability distribution with density $f(x,y)$ and marginals $g(x)$ and $h(y)$ defined over the real line. Now, I would like to find a class of ...
5
votes
1answer
112 views

Probability of random walk returning to 0

Given a symmetric 1-dimensional random walk starting at 0 -- what is the probability of the walk returning $k$ times to 0 after $2N$ steps? I know that the total number of paths it can take is ...
0
votes
1answer
16 views

Need help with these two probability questions!

For the first question doesn't it depend on if a and b are disjoint or not so if they are disjoint the we have P(A) and if they are not disjoint then we have P(A only) so how do we answer this when ...
1
vote
0answers
53 views

LimSup of Random Variable

I have a seemingly trivial question. Why does $$\forall a\in\mathbb{R},\mathbb{P}(\limsup X_n>a)>0\Rightarrow \mathbb{P}(\limsup X_n=\infty)=1$$ Clearly, we don't have (at least trivially), ...
0
votes
1answer
53 views

Prerequisites for studying Introduction to probability theory by William Feller,vol 1 & vol 2?

I know calculus, real analysis, discrete mathematics and applied probability, and want to know what else do I need to know to self-study both the probability books by Feller?
1
vote
1answer
19 views

Probability related puzzle

A student has to answer 9 out of 12 questions. How many choices has he if he must answer at least four of the first five questions? My logic: $^5 C_4$$*$$^7 C_5$+$^5 C_5$$*$$^7 C_4$=$105+35$=$140$ ...
1
vote
3answers
100 views

Probability of number of sets in tennis game

Player A and player B are playing tennis. Determine the probability of the game ending with 3, 4 and 5 sets played respectively, given that the probability of player A winning a set is 0.6 and that ...
3
votes
2answers
88 views

How to find nth moment?

I'm quite new to the field so please bare with me. Problem: Let ξ be a random variable distributed according to a log-normal distribution with parameters μ and $σ^2$, i.e. log(ξ) is normally ...
1
vote
1answer
23 views

Bayes with conditional independence

I have a problem that I can't work out I've two conditional independent A,B such as $P(A,B|C) = P(A|C)P(B|C)$ Now I've to find posterior formula for: $P(C | A,B)$, now what I got was pretty ...
2
votes
5answers
252 views

Find the distribution of $X_1^2 + X_2^2$? [duplicate]

Let $X_1$ and $X_2$ are independent $N(0, \sigma^2)$ which means (mean = 0, variance = $\sigma^2$) random variables. What is the distribution of $X_1^2 + X_2^2$? My approach is that $X_1\sim N(0, ...
0
votes
1answer
19 views

Is the statement: $p\left(\left.y\right|h^{-1}\left(\varphi\right)\right)=p\left(\left.y\right|\varphi\right)$ correct?

Say I have a likelihood function $p\left(\left.y\right|\theta\right)$ and I make the reparameterization $\varphi=h\left(\theta\right)$ using the bijective function $h$ with inverse $h^{-1}$. Then it ...
1
vote
2answers
168 views

How do I get the probability? (Balls picked from 2 urns)

An urn contains 10 white balls and 3 black balls. Another urn contains 3 white balls and 5 black balls. Two balls are drawn at random from the first urn and placed in the second urn. A ball is ...
1
vote
2answers
39 views

Poisson process counting process

Two individuals, A and B, both require kidney transplants. If she does not receive a new kidney, then A will die after an exponential time with rate $\mu_A$, and B after an exponential time with rate ...
2
votes
1answer
36 views

Probabilities with quadratic equations!

Basically i have never seen probabilities linked with quadratic equations like these and so have no idea how to interpret this question, would appreciate any help!
1
vote
3answers
56 views

Probability of even number of successes in a series of independent trials

Consider a series of independent trials at each of which there is a success of a failure with probabilities $p$ and $1-p$ respectively. I am finding it difficult to derive the probability of even ...
1
vote
2answers
35 views

Probability of obtaining equal number of each outcome of a fair die at the nth trial

Suppose a fair die is tossed repeatedly. I am concerned in deriving the probability of the occurrence of obtaining equal number of each possible outcome at the nth trial. Clearly this is only possible ...
1
vote
1answer
28 views

Speed Networking for Small Group (10 people)

I'm hosting a speed networking session where there are 10 people at 5 tables (2 people per table.) Each table is hosted by a different person. There will be four rotations. It does not matter if ...
0
votes
0answers
27 views

How do I convert a Gamma-distributed random variable, $ \Gamma(2,1)$ to Erlang distribution?

I know that a Gamma-distributed random variable with $\alpha$ as an integer can be converted to Erlang distribution but how? and how do I write it's new probability density function?
3
votes
0answers
63 views

determine type of probability distribution

let us consider following model $$y(t)=A_1 \sin(\omega_1 t+\phi_1) + A_2 \sin(\omega_2 t+\phi_2) + A_3 \sin(\omega_3 t+\phi_3)+ \ldots +A_p \sin(\omega_p t+\phi_p)+z(t)$$ we have three parameter ...
0
votes
1answer
26 views

Probabilistic solution Poisson problem

Let us consider the Poisson problem \begin{cases} \frac{1}{2}u''=-f\qquad\text{in}\,\,(a,b)\\u(a)=u(b)=0 \end{cases} where $f:(a,b)\to\mathbb{R}$ is continuous and bounded. We have obtained ...
1
vote
1answer
53 views

What is the distribution of the service-starting time lag w.r.t. two concurrent customers from two parallel $M/M/1/1$ queues?

Consider two parallel, independent $M/M/1/1$ queues (denoted $Q_i, Q_j$) with identical arrival rate $\lambda$ and service rate $\mu$, using FCFS (First Come First Served) discipline. Note that the ...
0
votes
3answers
29 views

Limit of some random variable sequence: $\lim_{n\rightarrow\infty}\dfrac{\max_{1\le i\le n} Y_i}{n}$

Suppose that we have some infinite sequence of random variables $\{Y_i\}$. They may or may not be independent. I wonder whether there is some existing theory or method which would deal with ...
2
votes
2answers
46 views

Given an urn of $k$ distinct balls. Do $n$ drawings with replacement. What is the probability that every ball is drawn at least once?

The space of outcomes is $k^n$ since for each of the drawings, there are $k$ outcomes. Deciding the numerator seems harder. Originally the problem was "What is the probability that every ball is not ...
2
votes
1answer
21 views

Discrete distribution with the minimum variance

Consider a discrete random variable $X \in \{x_1, x_2, \ldots, x_n\}$, where $n < +\infty$ and $x_1 < x_2 < \ldots < x_n$. Let pose $p_i = \text{Pr}(X = x_i)$, with $\sum_{i=1}^N p_i = ...
1
vote
2answers
45 views

Find vector of expected values ​​and covariance matrix

For vector (X,Y) with density $f(x,y)=C exp \{ -4x^2-6xy-9y^2 \}$ find constans C, vector of expected values ​​and covariance matrix. How to do this kind of exercises?
1
vote
2answers
62 views

Maximum Likelihood Estimator for Uniform Distribution

Can somebody please explain this example to me. I am struggling to see why the likelihood is $\frac{1}{\theta^n}$ only if theta is greater than the maximum x. Furthermore, why is it the case that ...
0
votes
1answer
29 views

Distribution of distance from 0 of gaussian point

Suppose $X_1,...,X_d\sim\mathcal{N}(0,1)$ are i.i.d.'s, each distributed normally around 0 with variation 1. It looks like $\mathbb{E}\left(\sum X_i^2\right)=d$. Why is that true? And how $Y=\sum ...
1
vote
1answer
173 views

mean and variance normalization of vectors

I have vectors $x \in \mathbb{R}^n$ and I expect some multivariate normal distribution. I want to normalize the vectors in such a way that $y = M (x - b)$ has mean zero ($\operatorname{E}[Y] = 0$) ...
1
vote
2answers
67 views

What is the probability of escaping from jail if…?

Disclaimer: The story given below is purely fictional and does not, in any way, relate to a prison break. :P Four roads lead away from a jail. A prisoner is trying to escape from the jail and ...
0
votes
1answer
56 views

Cauchy-Schwarz inequality with Expectations

Cauchy-Schwarz inequality has been applied to various subjects such as probability theory. I wonder how to prove the following version of the Cauchy-Schwarz inequality for random variables: ...
1
vote
1answer
33 views

Probability to remove complete buckets from histogram

I have some data and its distribution as a histogram. Let's say for example there are the following 20 data items: 3 times a A 5 times a B 4 times a C 4 times a D 3 times a E 1 times a F Now I ...
0
votes
0answers
20 views

Convergence of Probability Theorems

This does not make too much sense to me. So if I have the following conditions: $\sum_{i=1}^n h(X_i; \theta ) =0$; $E_\theta[h(X_i; \theta ) ] = 0$; Then, $\sum_{i=1}^n h(X_i; \theta ) =0 ...
0
votes
1answer
50 views

Standard deviation with multiple means and deviations

The amounts of a certain mineral that can be produced in a day from mines $1$, $2$, and $3$ are independent normal random variables with means equal to $80$, $90$, and $75$ pounds, respectively, ...
0
votes
1answer
57 views

Simplifying an Expected Value formula

Recently I asked a question related to Expected Value over here. The case is that interns have to pass to 2 training programs consecutively. Each program has a success rate of 40% ($p$) and each ...
4
votes
3answers
82 views

N balls and M boxes, probability that there is at least 1 box contains at least 2 balls

I have $N$ balls and $M$ boxes. Balls are thrown to the boxes at random. What is the probability that there is at least 1 box contains at least 2 balls? Thank you very much
0
votes
0answers
17 views

A doubt on markov decision process

Given that a policy is a function from a state action pair to probabilities, the set of policies for a MDP forms a POSET (the partial order is due to value function for a policy). Why there should be ...
1
vote
1answer
23 views

Exponential deviation with two $x$ values

I recently got interested in this topic of standard deviation. My TA did not have any time to go over this topic so I was trying to teach myself it recently. My TA said if he had more time he would ...
1
vote
1answer
47 views

Pictures for Expectation

Is there a good way to visualize the formula: $$ E(x) = \int_{0}^{\infty} 1 - F(X) \,\mathrm{d}x $$ ? for positive continuous random variables? I understand the formula as far as basic calculus ...
1
vote
1answer
38 views

Expected Value of a Carnival Game Paradox [duplicate]

I was in my Algebra 2 class today, and we're learning basic probability. Talking with my friend, he proposed this carnival game. The game begins with two dollars in the bowl, and every turn you flip a ...
5
votes
1answer
161 views

Continuous probability distribution with no first moment but the characteristic function is differentiable

I am looking for an example of a continuous distribution function where the first moment does not exist but the characteristic function is differentiable everywhere. Cauchy distributions do not ...
0
votes
1answer
21 views

Is a continuum of mixtures of stable distributions (e.g. normals) stable?

Take some random variable $X$ and indices $i \in [0,1]$. Let $X$ be a stable distribution, (i.e. for any copies $i,i'$, $a,b>0$, $a X_i + b X_{i'} \sim c X + d$ for some $c$ and $d$). This ...
0
votes
0answers
20 views

An online reference for proof of BIC approximation

I have been looking for an online note giving a proof of the BIC asymptotic approximation. It has been surprisingly difficult to find. In fact I only found the original paper by Schwarz 1978, which ...
3
votes
3answers
102 views

How does one 'correct' a table that doesn't add up to $100\%$?

I have a table consisting of a number of whole percentages $x_i$ between $0\%$ and $100\%$. However, they don't add up to $100\%$ (rather they add up to $101\%$). But they 'should'. Assuming that any ...
1
vote
2answers
48 views

What's wrong with this random variable proof?

Let $X$ be a Binomial random variable $\sim B(p, n)$. Show that for $\lambda > 0$ and $\epsilon > 0$, $P(X - np > n\epsilon) \le \mathbb{E}\{\displaystyle e^{\lambda(X - np - ...
1
vote
2answers
46 views

Dividing into 2 teams

In how many ways can $22$ people be divided into $ 2 $ cricket teams to play each other? Actual answer : $\large \dfrac{1}{2} \times \dbinom{22}{11}$ My approach : Each team consists of $11$ ...
0
votes
1answer
23 views

definition of distribution function of random variable

please help me to understand fully following definition : i am using this book http://www.math.harvard.edu/~knill/books/KnillProbability.pdf page 79,i can't understand some part,in spite of this ...
0
votes
0answers
28 views

discrete time Markov chain, difference between absorbing and recurrent classes.

In a discrete time Markov chain, are there any differences between an absorbing and a recurrent class? Recurrence is that we with probability 1 will reenter a state that we are in, this is a class ...
0
votes
0answers
14 views

MLE for mean of symmetric but otherwise unknown distribution

Given i.i.d. draws $x_1,...,x_n$ from $X$, where: $X$ has a finite mean $E[X]=\mu$, $X$ is symmetric about its mean, meaning $f_X(\mu+c)=f_X(\mu-c)$ for all $c$., The probability density function ...