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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1answer
18 views

$12$ Rolls of a dice: Probability of obtaining each face exactly 2 times.

I was wondering about the situation in the title... would it be $(1/6)^{12}12!\approx 0.22? $ It seems a large number for an event that doesn't seem very probable.
0
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2answers
17 views

Finding standard deviation from proportions.

There are 365 people. $1/6$ of the people eat $0$ cookies, $1/2$ eat $1$ cookie and $1/3$ eat $2$ cookies. What is the probability that between 400 and 450 cookies are eaten? What I've done so far: ...
0
votes
1answer
12 views

Bounds on the Neighborhood Size of a Random Vertex in $G(n,p)$

Let $G(n,p)$ be the Erdős-Rényi random graph model. I am interested in the regime $p = c/n$, where $c > 0$. Further, let $B_G(d)$ denote the neighborhood of depth $d$ of a randomly chosen vertex ...
-2
votes
1answer
20 views

Simple question in probability [on hold]

Given n students and n courses, is also well known that every student has successfully passed k courses. For a particular course i, what is the probability of a student to pass it successfully? $ ...
1
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1answer
12 views

question about brownian hit time and reflexion principle

I have a Brownian motion W(t) I consider 2 events, where T is fixed : A : W(T) is above a, a > 0 B : W(t) hit the level b, b < 0, at least once between 0 and T I am trying to compute P(A and B) ...
0
votes
1answer
8 views

Beta-binomial random number generator

Could someone help me find a random number generator from a Beta-Binomial distribution in MATLAB, R or SAS? Thank you!
0
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0answers
25 views

Upper bound and Chernoff bound

I would be happy to get help for the following question: Given n students and n courses, is also well known that every student has successfully passed k ($ 1< k <n $) courses (in randomly). In ...
0
votes
2answers
16 views

Roll 3 dice. Calculate the size of the event where the largest value of d1, d2, d3 is 6.

The way I approached this is: Any set of values $ (6,5,x) $ are allowed. So we get $ 6 \times{3} = 18 $ Can anyone correct me if I'm wrong. Cheers.
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0answers
20 views

Combining normal distributions?

60% of the population are men and 40% of the population are children. The mean height of the men is 72 inches with a standard deviation of 10 inches. The mean height of the children is 48 inches with ...
0
votes
1answer
16 views

How many possible ways can I choose a 5 card hand of 3 spades from a deck of 52 cards

I have the answer to this question. But I'm not 100% sure how it works. My original way of approaching this was to get the sample space, 52 choose 5, and use bernoulli trials to get the probability of ...
0
votes
1answer
22 views

density joint function

I got a question and I was stuck for more than 15 minutes... Here is the question, And the question was: Find F(1/2,2). I tried to reason but the answer was different from what I got, here is the ...
0
votes
0answers
6 views

Trying to find article by Tukey

I am trying to find a commonly cited paper by John Tukey published in 1960 called "A survey of sampling from contaminated distributions", from a monograph(?) called "Contributions in Probability and ...
0
votes
0answers
27 views

Calculating the probability in $m$ steps of a Homogeneous Markov Chain

I have the next problem: Consider a homogeneous Markov chain $\{X_n: n = 0,1,2, ... \} $ with state space $E = \{0,1,2, ... \} $, with the following transition probabilities where $ 0 <\theta ...
2
votes
1answer
12 views

Series of independent gaussian variables and brownian motion

I am checking the proof of the construction of a brownian motion in $[0,\pi]$. We show that \begin{gather*} t \mapsto B^m_t = \frac{t}{\sqrt{\pi}}X_0 + \sqrt{\frac{2}{\pi}}\sum_{n=1}^{2^m-1}X_n ...
3
votes
2answers
24 views

Probability that someone will pick a red ball first?

A father and son take turns picking red and green balls from a bag. There are 2 red balls and 3 green balls. The first person to pick a red ball wins. There is no replacement. What is the probability ...
5
votes
2answers
38 views

Prove the probability to even number of “Heads” is $\frac{1}{2}$.

Let $n$ coins, where at least one of them is a fair coin. Each one of the $n$ coins is tossed - Prove the probability to get even number of "Heads" is $\frac{1}{2}$. I'd be glad for a direction. ...
0
votes
1answer
22 views

How to prove the bound on the probability?

I am trying to prove, given a random sequence of $2k$ bits, the probability that the sequence contains exactly $k$ ones is less than or equal to $1/2$. I tried ${2k \choose ...
3
votes
2answers
24 views

Inverse Probability and conditional probability.

An unbalanced die (with 6 faces, numbered from 1 to 6) is thrown. The probability that the face value is odd is 90% of the probability that the face value is even. The probability of getting any even ...
3
votes
0answers
22 views

Generalized Binomial Model independent in the limit

Start with a generalized binomial model $$P(X_{n+1}=1\mid \mathcal{F}_n)=\theta_n+ n^{-1} d_n \sum_{i=1}^n X_i$$ $$P(X_{n+1}=1)=p_{n+1}=\theta_n + n^{-1}d_n \sum_{i=1}^n p_i$$ Then if we impose ...
-2
votes
1answer
31 views

How central limit theorem invoke two random variables jointly Gaussian or independent? [on hold]

$X$=$\sum_{i=1}^n d_icos\theta_i$ and $Y$=$\sum_{i=1}^n d_isin\theta_i$ If n is large enough, and $d_i$ and $\theta_i$ are independent, then X and Y are jointly Gaussian? Is it ture? and if is please ...
0
votes
2answers
34 views

Express the CDF of $Y=X^2$

Let $X$ be a random variable with CDF $F$. Express the CDF of $Y=X^2$ in terms of $F$.
1
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0answers
14 views

NIMO 16.8 Expected Value + Probability

Let $p=2^{16}+1$ be a prime. A sequence of $2^{16}$ positive integers $\{a_n\}$ is monotonically bounded if $1\leq a_i\leq i$ for all $1\leq i\leq 2^{16}$. We say that a term $a_k$ in the sequence ...
0
votes
1answer
27 views

Conditional Distribution: how to set up Limit of Integration of a joint density

I have a question in conditional probability. I'm asked to find the conditional distribution, however, I'm unsure about the answer given and would appreciate someone helping straighten out the theory ...
0
votes
0answers
27 views

Uniform distribution in a cube

I came across the following problem and got stuck. Problem: Let $X_1,X_2,...$ be independent Unif$(-1,1)$ and $S_n=X_1^2+...+X_n^2$. Let $$A_n=\{x\in ...
0
votes
0answers
22 views

How to find expected number of games between two players?

I couldn't understand an answer to this question, so I'm asking it again. Can someone explain the answer or solve it by another method? The one think I didn't understood in answer is why ...
-1
votes
2answers
71 views

What is the distribution of Z=min(X,Y) [on hold]

Let X and Y be independent geometric random variables. What is the distribution of Z=min(X,Y)?
0
votes
1answer
12 views

Show that the σ-algebras generated by the collection of all intervals are equivalent

Show that the σ-algebras generated by the collection of all intervals of the form [a,b]⊂R and by the collection of all the intervals of the form (−∞,b]⊂R are equivalent. i am having trouble with ...
0
votes
1answer
28 views

Martingale $X_n \to \infty$ a.s.

Construct a martingale $X_n$ such that $X_n \to \infty $ a.s. I have trouble coming up with such an example and prove it. Can someone provide an example?
1
vote
1answer
41 views

How find this the expected number of games played, if A won $n$ consecutive games?

Question 1: A and B two people play a game, where the odds of winning for one per game is $\dfrac{1}{2}$. If someone first win $n$ games consecutive,then the games end.Find this expect with the ...
3
votes
1answer
21 views

If the probability density on a random vector is symmetric, then each variable is identically distributed?

Let $X$ be a random vector with joint distribution $F$ and density $f$. If $f$ is symmetric, is this equivalent to each random variable being identically distributed? We say $f$ is symmetric if it is ...
0
votes
0answers
5 views

Use copula to find the joint distribution of two random variables

Assuming that there are two random variables $x$, $y$ both having an Arcsine marginal distribution $F$, i.e. $F(x)=\frac{2}{\pi}\arcsin{\sqrt{x}}$ The density of their Gaussian copula can thus be ...
1
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0answers
25 views

Probability and search Tree

I need some help with the following question. Given the random permutations of $ n > 2 $ numbers. Now, creating a binary search tree and puting it the organs one by one. Denote the input organs ...
6
votes
0answers
56 views

Shooting bullets

This is from http://domino.research.ibm.com/Comm/wwwr_ponder.nsf/challenges/May2014.html Every second, a gun shoots a bullet in the same direction at a random constant speed between 0 and 1. The ...
0
votes
3answers
45 views

Is it possible for a reality to exist where the law of large numbers does not apply? [on hold]

Being more specific, is the law of large numbers more empirical than it is rational? That is, is it more a feature of the observable universe that it is something that is true based on our definition ...
1
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2answers
14 views

Calculating the variance of speed measurements

Some speed measurements (km/h) outside Furutåskolan has been observed. They are supposed to be outcomes from a random variable with expectation . Result: $29, 31, 36, 34, 33$ (a) Construct a ...
0
votes
0answers
13 views

Show the sample mean $\mathfrak{T}_t$ converges to the population mean faster than $n^{1/3}$.

Let $\mathfrak{T}_{t}$ be an iid random variable with support $\mathfrak{T}_{t} \in [0,1]$. Prove $n^{1/3}\frac{1}{n} \sum\limits_{t=1}^{n} (\mathfrak{T}_{t} - \mathbb{E}[\mathfrak{T}_{t}] ) ...
1
vote
1answer
14 views

Combined Distribution of Random variable

How to compute $P[T1 \le T2 \le t]$ for T1, T2 is independent random variable with exponential distribution in terms of cmf, pdf of T1 and T2? Similarly for $P[T1 \le T2 \le T3.. \le t]$ ? I tried ...
5
votes
2answers
61 views

Find probability that random triangle covers centre of circumscribed circle

We are given the equilateral triangle A. On each edge of the triangle we pick a point: randomly (probability distribution is Gaussian) independently of others We construct new triangle B ...
4
votes
2answers
27 views

Showing that $p^n(1-p) \leq \frac{1}{en}$

I am reading a paper and found the following Lemma without a proof. Let $X_1, \ldots, X_{n+1}$ be independent Bernoulli random variables, where $\Pr[X_i = 1] = p$. Let $E$ be the event that the first ...
0
votes
2answers
17 views

Covariance of uniform distribution and it's square

I have $X$ ~ $U(-1,1)$ and $Y = X^2$ random variables, I need to calculate their covariance. My calculations are: $$ Cov(X,Y) = Cov(X,X^2) = E((X-E(X))(X^2-E(X^2))) = E(X X^2) = E(X^3) = 0 $$ because ...
0
votes
2answers
23 views

Find the probability density function of $Y=X^2$

Consider the random variable X with probability density function $$f(x)=3x^2$$ if $0<x<1$, and $$f(x)=0$$ otherwise. Find the probability density function of $Y=X^2$. This is the first question ...
0
votes
0answers
9 views

Reference requests for an opt-cited result in Jennrich (1969)

Lemma 2 on page 637 of Jennrich (1967) states that: Let $Q$ be a real-valued function on $\Theta\times Y$ where $\Theta$ is a compact subset of a Euclidean space and $Y$ is a measurable space. ...
5
votes
3answers
200 views

Probability that a cow is black given that I've observed at least one side is black

I'm on a farm with six cows; three are white, two are black and one is completely black on one side and completely white on the other. I see one cow from the side, who appears to be black (that is, ...
2
votes
2answers
28 views

Probability a truck full of stones weighs more than 1800kg?

Stones of a particular kind weigh in mean 10 kg and have standard deviation 1 kg. Assume the weight of a truck is normal distributed with mean 1000 kg and standard deviation 100 kg. 50 stones are put ...
0
votes
1answer
17 views

expected value of three uncorrelated random variables

Random variables ξ, η and ζ are pairwise uncorrelated. It means that E(ξ*ζ) = E(ξ)*E(ζ), etc. Is it true that in this case E(ξηζ) = EξEηEζ ? How it can be proven? Note: we don't know if they are ...
10
votes
8answers
4k views

% of % - Please Help Me Prove My Friend Wrong

Here is the situation: My friend and I are at an impasse. I believe I'm correct, but he's so damn stubborn he won't believe me. Also, I'm not the most articulate at explaining things. Hopefully ...
0
votes
1answer
36 views

Conditional expectation of second moment given sum of iid variables.

We have $\xi_i \geq 0$, $\forall i = \overline{1,n}$ (i.i.d. variables). Assume that $S_n = \xi_1 +...+ \xi_n$. It is easy to show that $\mathrm{E} (\xi_1\vert S_n = 1) = \frac{1}{n}$. Now we want ...
1
vote
0answers
15 views

CDF of maximum of iid rvs [duplicate]

I am having a small doubt regarding maximum of random variables. I have $$Z= \max\{ X_1, X_2,\dots X_p, \dots X_N\}$$ where all $X_i$ are independent, identically distributed. Now, If for sure, I know ...
1
vote
0answers
36 views

A fair dice is thrown six times and the list of numbers showing up is noted. The probability that among the numbers 1 to 6 only 4 nu…

Question : A fair dice is thrown six times and the list of numbers showing up is noted. Now how to find the probability that among the numbers 1 to 6 only 4 numbers appear in the list Please ...
1
vote
1answer
21 views

create a Gaussian distribution with a customize covariance in Matlab

the Matlab function 'randn' randomize a Gaussian distribution with $\mu= \begin {pmatrix} 0\\0\end{pmatrix}$ and $cov= \begin {pmatrix} 1&0\\0&1\end{pmatrix}$ Ineed to randomize a Gaussian ...