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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1
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1answer
27 views

Mean of $k$ highest out of $n$ i.i.d. draws from unit uniform distribution

I am stuck with the following question: There is a random variable $x\sim U[0,1] $. There are $n$ different draws (i.i.d.) being made. I am trying to compute the expected value/arithmetic mean of the ...
1
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0answers
12 views

First and second moments of two correlated functions

I am trying to find the first and second moments of the following: $k=mk^{-b}$ where $m \sim U[a,b]$ (discrete) and $k^{-b}$ is a power law distribution I know how to find the first and second ...
3
votes
1answer
36 views

Probability to pick couples of numbers from a set of the first $n$ natural numbers

I am stuck on the following problem: I have a set $A$ of the first $n$ natural numbers. I define a new set $B$ picking randomly $m$ numbers from $A$. What is the probability to have at least $k$ ...
1
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0answers
15 views

Probability of a noisy model

Suppose I have a model $y = x + n$. The law of $x$ and $n$(noise) are known. $y$ is the observation. Thus I write: $ p(y|x) = p_n(y-x)$. Using the Bayes's rule I have: $p(x|y) \propto p(y|x) p(x) $. ...
0
votes
1answer
19 views

Factorization of trivariate joint distribution under independence

I know that if $X$ and $Y$ are independent, then we have $P(X,Y)=P(X)P(Y)$. Does the independence statement also imply any type of factorization of the trivariate joint $P(X,Y,Z)$?
0
votes
2answers
24 views

Negative binomial distribution?

We throw a coin with success probabilty $p$ and $Y$ is the amount of coin tosses we need untill we have $n$ successes. Now I want to show that $P(Y=n+i)=\begin{pmatrix}n+i-1\\ i ...
0
votes
0answers
18 views

Probability matrix from a adjancency matrix

i have this adjacency matrix of 3 nodes: |0 1 0| |0 0 1| |1 1 0| Now i need to find the associated probability matrix. Naturally i would say it would look ...
1
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0answers
18 views

Find probability dense function of multiple random variable

Suppose we have random variable such that $S=g(X_{1},X_{2},...,X_{n})$ and $X_{i}, 0\leq i\leq n$ are all independent and uniformly distributed. I have done my best to find the cumulative distribution ...
0
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0answers
18 views

Show there exist $N $ such that $p_{s,s}(n) > 0$ for all $n \geq N$ for an aperiodic state $s$ of a Markov chain

Show there exist $N \geq 1$ such that $p_{s,s}(n) > 0$ for all $n \geq N$ for an aperiodic state $s$ of a Markov chain. Where $p_{s,s}(n)$ is a transition probability. My approach: because $s$ is ...
0
votes
0answers
9 views

Relation between Poisson representation of extremes and GPD representation of extremes

I want to derive the theoretical relation between the parameters in a point process model for extremes and the parameters in the GPD model for extremes. I'm following Coles - An introduction to ...
2
votes
0answers
35 views

How to generate correlated random numbers with specific distributions?

After read the answers of some similar questions on this site, e.g., Generate Correlated Normal Random Variables Generate correlated random numbers precisely I wonder whether such approaches can ...
1
vote
1answer
26 views

Statistical significance and sample size

I have a device that is said to succeed at doing some task at least 99% of attempts, and fails no more than 1% of attempts. If I want to be 98% sure that it achieves that success rate, how many ...
1
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0answers
18 views

Machine Learning: are there other functions similar to the softmax?

Recall in probability and machine learning softmax is defined as: $\sigma(\mathbf{z})_j = \dfrac{e^{z_j}}{\sum_{k=1}^K e^{z_k}}$ for $j = 1, ..., K.$ where $\sigma: \mathbb{R}^k \to (0,1)$ ...
0
votes
0answers
13 views

Density function of a gaussian vector

Let $X=(X^{(1)}, ..., X^{(n)})$ be a random gaussian vector with mean $m\in\mathbb{R}^{n}$ and covariance matrix $\Sigma$ with $\text{det}(\Sigma)\neq 0$. Prove that the distribution of $X$ is ...
1
vote
2answers
38 views

A stick is broken into two pieces, at a uniformly random chosen break point. Find the CDF.

I'm having trouble understanding how the CDF is found in the solution below: We can assume the units are chosen so that the stick has length $1$. Let $L$ be the length of the longer piece, and let ...
1
vote
1answer
34 views

is Cov(X) and Var(X) same? when X is random vector

i'm studying with hogg. introduction to mathematical statistics. and i learned about random vector but i wonder whether Cov(X) and Var(X) is same or not. as intuitive thinking , if X is a random ...
3
votes
1answer
22 views

Total expected time

The first task has a probability of success of $p$, $0 < p < 1$. Assume each attempt are independent. After the first task is completed, the second task (trials also independent) has a ...
1
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0answers
34 views

Bayes' Theorem and Law of total propability for CDF

The calculation of conditional probability is the same for conditional PDF and CDF(according to a number of questionable sources: first, second) (I will use rough notation, with just $x$ and $y$): ...
1
vote
1answer
21 views

Finding the marginal distribution for problem with n balls.

I am trying to solve the following problem: A box contains N balls: $N_1\ white, N_2\ black,\ and\ N_3\ red\ (N = N_1 + N_2 + N_3).$ A random sample of n balls is selected from the box (without ...
0
votes
0answers
17 views

Ratio of Two Sample Mean of Gamma Random Variables.

Let $X_1,\ldots, X_n$ are iid $\mathrm{Gamma}(\alpha,\beta)$, $Y_1,\ldots, Y_n$ are iid $\mathrm{Gamma}(\alpha,\gamma)$ and independent of $X_i$. What will be the distribution of $\frac{\bar X}{\bar ...
1
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0answers
14 views

Is this an exponential family of distributions? from casella and berger 6.20

I am trying to do 6.20 in Casella and Berger part d. The solutions manual says that the order statistics are minimal sufficient and not complete. I understand their logic, but why doesn't this work? ...
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votes
2answers
25 views

probabilities when $n$ independent Bernoulli trials are carried out with probability of success $p$. [on hold]

I don't understand on how to calculate the following probabilities: Let the probability of success be $0.30$. Assume a sample size of $10$ a.The probability of no failures b.The probability of at ...
2
votes
0answers
16 views

Independent coordinates of a gaussian vector

Let $(X^{(1)}, \ldots, X^{(n)})$ be a gaussian random vector. For $i\neq j$, Prove that $X^{(i)}$ and $X^{(j)}$ are independent $\iff \text{Cov}(X^{(i)},X^{(j)})=0$. I'm trying to work with ...
21
votes
5answers
3k views

Probability of being poisoned

You are playing a game in which you have $100$ jellybeans, $10$ of them are poisonous (You eat one, you die). Now you have to pick $10$ at random to eat. Question: What is the probability of ...
2
votes
1answer
30 views

Finding $\mathbb{P}(A\cup B\cup C)$ under following assumptions

$$\mathbb{P}(A)=0.3$$ $$\mathbb{P}(B)=0.4$$ $$\mathbb{P}(C)=0.5$$ $$\mathbb{P}(B|C)=0.5$$ $A$ and $B$ are mutual exclusive $$(A\cap B=\emptyset )$$ and A and C are ...
2
votes
2answers
44 views

An elevator containing five people can stop at any of seven floors.

An elevator containing five people can stop at any of seven floors. What is the probability that no two people get off at the same floor? Assume that the occupants act independently and that all ...
2
votes
1answer
31 views

Many urns with colored balls

This is 1.7 from Grimmett's Probability book: There are n urns of which the r th contains r − 1 red balls and n − r magenta balls. You pick an urn at random and remove two balls at random without ...
0
votes
1answer
28 views

Chance of getting a car toy in a chocolate

A mom brings her child every day a chocolate with a toy inside, the toy is random. The boy is happy when he gets a car as a toy. His mom decided to look in which supermarket the probability of getting ...
0
votes
2answers
28 views

Finding the distribution of a n tossed fair coin

I am trying to solve the problem: Consider a sequence of n tosses of a fair coin. Let X denote the number of heads, and Y denote the number of isolated heads, that come up. (A head is an ...
0
votes
0answers
30 views

Question related to expected value I think (Probability) Need help

Hi Im stuck on this question im wondering if someone can help. You dont need to solve it, i think this is an expected value question, but not too sure, which is why im struggling lol. I'd just like ...
1
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2answers
33 views

A group of 60 second graders is to be randomly assigned to two classes of 30 each…

A group of 60 second graders is to be randomly assigned to two classes of 30 each. Five of the second graders, Marcelle, Sarah, Michelle, Katy, and Camerin, are close friends. (a) What is the ...
1
vote
1answer
23 views

Balls in bin. Adding similar coloured ball at every turn.

The bin has 3 black balls and one white ball. In one turn, a ball is picked at random and returned to the bin with another of the same colour. After 60 such turns, what is the probability that the ...
3
votes
0answers
15 views

weak convergence and unbounded functions with bounded moment

I want to prove the following: Given a topological space (it is a Lusin space, but I think that does not matter) $\Omega$, a function $f \in C(\Omega,\mathbb{R})$ and a sequence of Radon measures ...
0
votes
1answer
35 views

2 classes in the same classroom each with 100 seats and the same 100 students, find the probability that no one has the same seat for both classes

The question is as follows: Harvard Law School courses often have assigned seating to facilitate the “Socratic method.” Suppose that there are $100$ first year Harvard Law students, and each ...
0
votes
0answers
16 views

Derivation of spacing distribution of independent events

A crude approximation of the spacing of energy levels $E_i$ of complex nuclei (like uranium) is that the energy levels appear independently, with known average spacing $D$. I'm trying to understand a ...
0
votes
0answers
13 views

ACVF: Is Steiner's THM being used here?

I have that {$Z_t$} $\in IID(0,1) $ and if t is even then $X_t=Z_t$ and if t is odd then $X_t=\frac {Z_{t-1}^{2}-1} {\sqrt{2}}$. In the book they write that for even t the ACVF, with $h=0$, is $$ ...
0
votes
2answers
32 views

Showing that the multivariate normal density integrates to 1

This is NOT the same as How to show the normal density integrates to 1?. Let $\mathbf{x} \in \mathbb{R}^d$ be a multivariate normal random vector, with $\mathbb{E}[\mathbf{x}] = \boldsymbol\mu$ and ...
-7
votes
0answers
32 views

Probability of event will occur in next year [on hold]

Let's say an event occurs 2 times in 2014, 3 times in 2015, 0 times in 2016.How can I find the probability that the event will occur in 2017
1
vote
1answer
21 views

Probability that a random polynomial over a finite field can be factorized to linear terms.

Suppose that $f\in\mathbb{F}_p[x]$ is a degree $d$ random univariate polynomial with coefficients from a finite field $\mathbb{F}_p$. What is the probability that $f$ can be written as: ...
0
votes
1answer
29 views

Indicator function integral

Let $(\Omega, \mathcal A, \mathbb P)$ be a probability space. Let $A, B\in\mathcal A$. Assume that $\mathbb P(A) = 0.5$, $\mathbb P(B) = 0.4$ and $\mathbb P(A\cap B) = 0.1$. Find the integral over ...
2
votes
4answers
231 views

Probability of having a Girl

A and B are married. They have two kids. One of them is a girl. What is the probability that the other kid is also a girl? Someone says $\frac{1}{2}$, someone says $\frac{1}{3}$. Which is correct? ...
0
votes
1answer
27 views

Convolution of Gaussian and error function

I am trying to evaluate the following integral: $$ \int_{-\infty}^{\infty}e^{-\frac{x^2}{2}}\Phi(x-t)dx $$ where $$ \Phi(y) = \frac{1}{2} + \frac{1}{2}erf\left(\frac{y}{\sqrt{2}}\right) $$ I have ...
0
votes
1answer
57 views

Finding the distribution from the moment generating function

Let $X_1, X_2, · · · , X_n$ be a random sample of size n from a geometric distribution withpmf $f(x) = 0.75 · 0.25^{ x-1} , x = 1, 2, 3, ··· .$ (a) Find the mgf $M_{Y_n} (t)$ of $Y_n = X_1 + ...
-1
votes
2answers
74 views

Find an expression of the probability function when $f(x)=\frac{x}{10}$ is the PMF. [on hold]

Suppose that $$ f(x) := \begin{cases} \dfrac{x}{10}, & x \in \{ 1,2,3,4 \} \\ 0, & \text{otherwise} \end{cases} $$ is a probability mass function. Find an expression of the ...
0
votes
1answer
40 views

A binomial sum identity

Let \begin{align*} f(n, r, \pi, k) &= \sum_{z=0}^{n}\sum_{s=0}^{r}\binom{z}{s}\binom{n}{z}\binom{n-z}{r-s}(-1)^{r+s}\left(\frac{\pi}{1-\pi}\right)^{r/2-s}\pi^{z}(1-\pi)^{n-z}z^k \end{align*} I am ...
0
votes
1answer
18 views

Pearson product-moment correlation coefficient of a coin toss

A fair coin is tossed 3 times. Let $X$ be a random variable representing the number of $H$'s appeared in the first 2 tosses, $Y$ the number of $H$'s appeared in the last 2 tosses, and $Z$ the number ...
-4
votes
1answer
31 views

The $\%$ chance that I'm beat [on hold]

If I'm up against $2$ opponents, and each of them has $50\%$ chance of having me beat. What are the $\%$ chance that they got me beat?
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0answers
9 views

Multivariate transformation of random variables on Maple 17

I'm trying to workout a multivariate transformation of multiple random variables on Maple 17. I know how to work out for a single variable but couldn't get it to work for two variables. I'm trying to ...
0
votes
1answer
21 views

Goldberg compound probability problem- guess the correct colour ball from an urn

I am working my way through an example problem from Goldberg's "Probability: An Introduction". There are x red balls and x green balls in an urn. Total number of balls in the urn is 5. You must guess ...
0
votes
0answers
30 views

Computing average of a distribution.

Suppose on a day we toss an unfair coin $60$ times (each trial are independent) and we assume getting a head "H" is favorable event and the probability of getting a head is $0.04$. Then the case is ...