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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10 views

Drawing n intervals uniformly randomly, probability that one interval overlaps with all others

Randomly draw $n$ intervals from $[0,1]$, where each end point are selected from from the uniform distribution between $[0,1]$. What's the probability that at least one interval overlaps with all ...
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0answers
11 views

Product of (multivariate) Gaussian densities

One can frequently read that the product of two multivariate Gaussian pdf f1(x) and f2(x) is itself a Gaussian, with parameters as defined for example in: ...
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1answer
32 views

Two players toss a coin.

Two players $A$ and $B$ toss a coin. A has a coin $C_A$, B has a coin $C_B$. Probability of tail for $C_A = 1-a$, of head: $C_A = a$ Similary for $C_B$. Now, they are tossing on turn. The A starts. ...
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0answers
21 views

Does this sequence converge? If yes, what is the limit?

Assume $\{k_n\}_{n\geq 0}$ a sequence of natural numbers such that $k_0=0$, $k_n\leq k_{n+1}\leq k_n+1$, and $\lim_{n\rightarrow\infty} \frac{k_n}{n}=\alpha\in(0,1)$. So $\{k_n\}$ is an ...
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2answers
20 views

Find probability of event

Task is: Find probability of 4 aces laying in row in a deck of 36 cards. All possible shufflings of 36 deck is $36!$ I can place 4 cards in a row with $33$ different ways. And each way can be $4!$ ...
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1answer
21 views

Variance of a random variable between 0 and c.

My professor says we need to know how to solve a problem like this for our upcoming exam and I can't find anything in my textbook or notes related to this at all. Can anybody help make this ...
2
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1answer
11 views

Linear transforms of Normal dist

If $X_t = \sqrt{t} Z$ where $Z \sim \text{N}(0,1)$ Then show the distribution of $X_t - X_s$ for $s<t$ Just wanted to check, would this be $\sim \text{N}(0,t-s)$ or $\sim \text{N}(0,(t-s)^2)$ ?
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0answers
25 views

Guessing a password from a list of N passwords

I want to know if my solution to the problem is correct: I am given a list of n passwords to enter an account and only one will grant me access to it. I pick one and I test it. If it's incorrect, I ...
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1answer
16 views

Assume that P(B) > 0. Prove that if P(A1|B) < P(A1) then P(Ai | B) > P(Ai) for some i = 2, … , k.

Suppose k events from a partition of the sample space Ω, i.e., they are disjoint and ∪ i=1 to k over Ai = Ω. Assume that P(B) > 0. Prove that if P(A1|B) < P(A1) then P(Ai | B) > P(Ai) for some i = ...
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0answers
22 views

probability of collision approximation,matlab,maple

I'm working in the field of communication network. I have this equation p=((1-(1-(((2.*(1-2.p))./(1-p-p.(2.p).^m)).(1./Wi))).^(n-1))) I want to solve for p, but I failed(maple and matlab), can ...
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vote
1answer
12 views

Asymptotic stopping time for a ball-drawing problem

Take two different boxes, one with $N$ red balls and one with $N$ blue balls. Remove balls one at a time from either box with equal probability. When only one color is left, the (expected value of ...
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2answers
14 views

Two processes. Expected value.

On a computer running two processes $ X_1, X_2 $ at the same moment. $ X_1, X_2$ mean time work processes, respectively. $ X_1, X_2 $ have exponential distribution. $$ E (X_1) = E (X_2) = 60s.$$ ...
-2
votes
2answers
43 views

Clever way of finding $\int_0^\infty x\Phi(x)\phi(x)dx$

Suppose that $\Phi$ and $\phi$ are the Standard Normal c.d.f and p.d.f. respectively. Then, evaluate $$\int_0^\infty x\Phi(x)\phi(x)dx$$ There is no use of my trying to show my approach because ...
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1answer
8 views

Construction of Probability Generating Function in Branching Process?

So I'm trying to construct a probability generating function for the following scenario: 1/5 of a rabbit population does not reproduce. 4/5 have 3 offspring each, and the probability of male or ...
0
votes
1answer
30 views

Counting and Probability String Length

Consider strings that can be made up from the set $\{a, b, c, d, e, f, \cdots, z, 0, 1, 2, \cdots, 9\}$ 1) How many strings of length 8 contain either the letter '$x$' or '$1$'? 2) What is the ...
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0answers
14 views

Probability of profit

I came across the following question in a book:- $Q.$ Cards are drawn one by one at random from a well shuffled pack of $52$ cards. $(a)$Find the probability that exactly $n$ cards are drawn before ...
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1answer
16 views

Proving the sequence $f_{x_n}(x)= (n+1)x^n$ converges in distribution

I am preparing for a final exam and just working on sample problems. Let $X_1,X_2,\dots$ be an infinite sequence of continuous random variables such that $f_{x_n}(x)= (n+1)x^n$ for $0<x<1$ ...
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0answers
8 views

Probabilities for the repetition of the same experiment $N$ times

Sometimes one experiment we want to discuss in terms of probabilities is in truth the same as performing another experiment $N$ times. I have a doubt on how to relate the probabilities for the ...
4
votes
2answers
40 views

What is the expected number of people who are shorter than both of their immediate neighbors?

A total of n people randomly take their seats around a circular table with n chairs. No two people have the same height. What is the expected number of people who are shorter than both of their ...
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1answer
17 views

Decision-making with random term

Consider the following situation. There are multiple options to choose from based on an attribute related to those options. For example: ...
2
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0answers
29 views

What is the probability of recovering from 0 − 40

A game in tennis consists of a sequence of points played with the same player serving. A game is won by the first player to have won at least four points in total and at least two points more than the ...
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vote
1answer
24 views

Show that $\Omega\setminus A_1, Ω\setminus A_2,\ldots, \Omega\setminus A_n$ are independent

Let $(\Omega, \Sigma, P)$ be a probability space and let $A_1, A_2, \ldots , A_n$ be independent events in this probability space. Show that $\Omega\setminus A_1, \Omega \setminus A_2, \ldots , \Omega ...
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2answers
11 views

Probability density function of random variable $X-Y$

Suppose $X$ and $Y$ are independent random variables. $X$ and $Y$ are continuous and given by exponential and uniform density functions. Find the probability density function of the random variable $X ...
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2answers
20 views

Probability and limit - proof of equality

Could anyone explain why this equality is true? Is there some intermediate step that could be used to prove it? If I were to guess, I'd guess it's certainly equal, but guessing is not enough I'm ...
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0answers
12 views

Conditional Distribution of Ordered Statistics [on hold]

Let $X_1,...,X_n$ be the order statistics of a set of $n$ independent uniform (0,1) random variables. Find the conditional distribution of $X_n$ given that $X_1=s_1,...,X_{n-1}=s_{n-1}$. I honestly ...
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2answers
44 views

Find the distribution - coin is tossed three times

A fair coin is tossed three times. Let $X$ be the number of heads that turn up on the first two tosses and $Y$ the number of heads that turn up on the third toss. Give the distribution of $X$, $Y$, $X ...
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1answer
8 views

Normal Distribution: Statistics

I'm having a lot of trouble trying to remember the formulas on how to calculate these questions. Any help would be great. An automobile insurer has found that repair claims are Normally distributed ...
0
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0answers
23 views

Probability for large nose and/or large ears

John has got a large nose and Mary big ears. Mary gives birth to their 5 children. Each one of them inherits the big ears with a probability of 0.5 and the large nose with 0.5 as well.Each child ...
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votes
1answer
30 views

Find distribution and the expected value of final grade [on hold]

A performance is graded independently by three experts (the possible grades are as follows: 1, 2, 3, 4, 5), and then the highest and the lowest mark are crossed out. The remaininggrade is the final ...
5
votes
1answer
35 views

At time n, randomly choose a natural number ≤n. How long is it until a single number is chosen three times?

To clarify, the number ≤n is chosen uniformly at random at each step. I wish to determine the expected value of $n$ at which a natural number is chosen three times (for the first time). (I would also ...
2
votes
2answers
31 views

Showing that supremum function is integrable

Let $g_1(\omega),g_2(\omega),...$ be integrable functions defined on $\Omega$ with $g_n\rightarrow g$ and $g$ is integrable and also $\lim \int g_n=\int g$ . Define $h(\omega)= \sup_n g_n(\omega)$. ...
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0answers
10 views

Normally distributed random variable

Could you please answer this question with an explanation, I am new in this subject.
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2answers
41 views

Find the probability that $X$ is even when following a geometric distribution [on hold]

Suppose that $p \in (0, 1)$ and that $X$ is a discrete random variable with a geometric distribution with parameter $p$. Find the probability that $X$ is even.
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0answers
12 views

conditional expectation uniqueness

Conditional expectation is unique up to a set of probability measure zero, but if $Z=E[X|Y]$ and $Z_2$ almost surely equals $Z$, then is $Z_2=E[X|Y]$ still the case? I think this is false but can't ...
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1answer
13 views

Prove that a symmetric distribution has zero skewness

Prove that a symmetric distribution has zero skewness. Okay so the question states : First prove that a distribution symmetric about a point a, has mean a. I found an answer on how to prove this ...
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0answers
34 views

Another fun card game involving probability

Two people, (call them C and D), decide to play a card game for fun. They use an ordinary fair deck of $52$ cards, well shuffled, and randomly draw cards from it one a time without replacement, both ...
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0answers
34 views

Help with two probability questions. Classic definition of probability.

The first can be done using condition probability, but was wondering how to do it just with the classic definition of probability? Both questions are in the same part of the book, and therefore i ...
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3answers
30 views

Problem on Baye's formula

I was reading A First Course in Probability by Sheldon Ross. I think I quite understood the below problem but I still feel fuzzy. Problem: In answering on a multiple choice test, a student either ...
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0answers
16 views

Is this an algebra of events?

An algebra of events is a non-empty $F$ family of subsets of the sample space $\Omega$ closed under the union and the complement. That's to say $F \subset P(\Omega)$(power set) that verifies: 1st) $F ...
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1answer
32 views

Jee Main 2015 Question. Probabilty

If $12$ identical balls are to be placed in $3$ identical boxes, then the probability that one of the boxes contains exactly $3$ balls is: (1) $22 \times(\frac{1}{3})^{11}$ (2) $\frac{55}{3} \times ...
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0answers
18 views

Interchanging infinite double sum and expectation

Let $\xi_i$ be a sequence of independent and identically distributed standard normal random variables and consider sequences $\{b_i\}$ and $\{c_j\}$ such that $\sum_i b_i<\infty$ and $\sum_j ...
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0answers
15 views

Random Variables and Statistic

I'm studying Statistical Inference by Casella and I'm confused with the definitions of random variable & statistic. So let we have the probability space $(\Omega, F, P)$ where $\Omega$ is the ...
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0answers
15 views

Find upper bound of probability value using Chebyshev's inequality [on hold]

Given density function of random variabel X is f(x) = 1/(2√x), for -√3 < x < √3. Use Chebyshev's inequality to find upper bound of probability value P(IxI≥3/2).
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9 views

Find lower bound of probability value using Chebyshev's inequality

Given density function of random variabel $X$ is $f(x) = 3x^2$, for $0 \lt x \lt 1$. Use Chebyshev's inequality to find lower bound of probability value : $P(5/8 \lt x \lt 7/8)$ $P(1/2 \lt x \lt 1)$ ...
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1answer
11 views

Find the probability P(x is even) of given cumulative distributive function [on hold]

Given cumulative distributive function (CDF) $F(x) = 1 - (1/2)^{(x+1)}$ for $x = 0, 1, 2, ...$ Find the probability value P(x is even).
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0answers
13 views

The probability of two dependent events occurring

If you wish to calculate the probability that both of 2 dependent events A and B will occur and you draw a tree diagram with A and B as the first two branches and then A and B again as two branches ...
0
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1answer
16 views

matrix sampling and its rank preservation

Assuming matrix $X\in R^{m\times n}$ is row orthogonal of rank $m$. Then, if I construct a new matrix $Y\in R^{m\times t}$, whose columns are directly sampled from $X$ with or without replacement ...
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1answer
18 views

Derivation of negative binomial distribution

Let $X, Y$ be geometric distribution where $ \mathbf P(X=k) = \mathbf P(Y=k) = (1-p)p^{k-1}$ for $k = 1, 2, 3...$ Using the convolution formula: $$\mathbf P(Z=z)=\sum_{n=1}^{z} \mathbf P(X=z) ...
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0answers
16 views

expectation calculation problem small problem

a Continuous, positive random variable X, whose PDF is proportional to $(1+x)^{-4}$, where $0<x<\infty$, determine $E(X)$ i tried to solve it directly by integrating from 0 to infinity to get ...
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0answers
8 views

Bayesian update multivariate normal based on one-dimensional signal: simple rule

Is there a simple rule to update the linear combination of normal distributions based on a one-dimensional signal? The unconditional joint density of $(\eta,\theta)$ is multivariate normal ...