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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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 ...
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1answer
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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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10 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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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 ...
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2answers
25 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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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: ...
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0answers
24 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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1answer
18 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
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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
19 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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11 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
41 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
7 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 ...
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0answers
22 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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1answer
28 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 ...
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0answers
9 views

Determining the expected number of Grim Patrons after a bouncing blade on a board with no minion cap [on hold]

The title of this question refers to the card game Hearthstone and a particular card interaction. See http://hearthstone.gamepedia.com/Grim_Patron http://hearthstone.gamepedia.com/Bouncing_Blade So, ...
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2answers
29 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
9 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
37 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
11 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
20 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
23 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
9 views

product of two multivariate normal densities - with one only specified for a subset of the random vector

I am wondering if the following problem can be solved as a special case of the product of two normal densities (for the same vector of random variables): A random vector x with n elements has a ...
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0answers
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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
29 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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12 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
18 views

Proving an inequality involving uniformly distributed random variables

$X_1, \ldots, X_k$ are uniformly distributed random variables on the interval $[0,1]$. With $Y_{k-1,i}$ we denote the $i$-th smallest nuber in $\{X_1, \ldots, X_{k-1}\}$. How can I prove the ...
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0answers
14 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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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 ...
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1answer
13 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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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 ...
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2answers
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expectation calculation problem

I got the answers for this and i know its 1.05 but the way it explains is very difficult to understand so im seeking for some help here. A system made up of 7 components with independent, identically ...
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3answers
19 views

Applying the basic formula for binomial distribution

I'm pretty confused on how this works. In my class my teacher states that: Let $X$ be a random variable with $S_X = \{0,1\}$. $X$ follows a Bernoulli distribution if $P(X = x) = p^x(1-p)^{1-x}$ for ...
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1answer
17 views

Probability Binomial Distribution Question [on hold]

The quality control unit in a medical device company inspects 20 pacemakers each hour. Let X represent the number of pacemakers in the sample of 20 that require rework. Pacemakers are assumed to be ...
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0answers
16 views

The moment-generating function of a random variable X is given by the following formula [on hold]

$M(t) = c(e^t + e^{2t}){e^t + e^{−t}} + 0.6e^{4t}$ , $c > 0$. (a) Find the value of c where c > 0. (b) Find all possible values of $X$ and the corresponding probabilities. (c) Find $E(X)$. ...
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0answers
25 views

Is there a name for this stochastic process?

Let $(\Omega,\mathscr{F},P)$ be a probability space and $\{X_n\}_{n\geq 1}$ be a stochastic process. Assume each $X_n$ only takes two values $0$ or $1$, i.e., $X_n:\Omega\rightarrow \{0,1\}$. Of ...
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0answers
25 views

Challenge Problems [on hold]

This question might be better fit for meta, but how might I find a list of challenge problems similar to the following. In addition this question may have already been asked. ...
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1answer
19 views

Minimum of random exponential variable and time

Let $U$ and $V$ be $\sim\mathrm{Exp}(\lambda)$. Let $s \in [0, t)$. Does this reasoning from left to hold: $P(U + (V \wedge t) \leq s) = P(U + V \leq s; V < t) $ or is in this case enough to write ...
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2answers
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De Méré paradox. Show that throwing at least one “one” of 4 dice is more probable than throwing at least two “ones” of 24 tosses of a dice.

De Méré paradox. Show that throwing at least one "one" of 4 dice is more probable than throwing at least two "ones" of 24 tosses of a dice. Finding the probability of the first is easy : ...
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2answers
31 views

Calculating the expected value of a random variable that's a function of a random variable

I am working on the following problem: I'm having a hard time putting all of this information together: The cost of the maintenance is $Z = X + Y$, where $X$ is the cost of the first machine and ...
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1answer
13 views

What would the expected number of swaps in a merge sort be?

If I were given a list of random numbers say x1, x2, .........., xn and these numbers are sorted according to the merge sort algorithm. What would be the number of expected swaps/exchanges which would ...
2
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1answer
28 views

Expected Value and variance of a max randomized stocks

Hey guys I have been working on a probability and expected value/variance problem and the problem is: Each day the price of a stock in the market is a random number between 0 and 1 independently of ...
0
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1answer
37 views

How to find $E[Y|X=1]$?

A fair die is repeatedly rolled. Let $X$ and $Y$ denote, respectively, the number of rolls required to obtain a $1$ and a $2$. How do I find $E[Y|X=1]$? edit: for using this I got 1*6 = 6 total ...