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
31 views

closed form for $p(B_1>x>B_2)$ where $[B_1, B_2]'$ follows a bivariate lognormal dist?

Is there a closed form for $p(B_1>x>B_2)$ where $[B_1, B_2]'$ follows a bivariate lognormal dist: $$[B_1, B_2]' \sim \text{lognorm} (\boldsymbol \mu, \boldsymbol \Sigma)$$ where $\boldsymbol ...
1
vote
1answer
314 views

convolution square root of uniform distribution

I need to find a probability distribution function $f(x)$ such that the convolution $f * f$ is the uniform distribution (between $x=0$ and $x=1$). I would like to generate pairs of numbers with ...
1
vote
1answer
125 views

Random Point Sampling From a Set with Certain Geometry

There's a famous puzzle that goes like this: Suppose we pick 3 points at random from a given circle. What's the probability that the triangle they form is obtuse? The answer happens to be 3/4 and it's ...
0
votes
1answer
204 views

the derivative of an integral involving absolute value

Given a random variable Z with a continuous and compact support, and a function $g(x,y)=E|x-y-Z|$, my aim is to find $\frac{\partial g}{\partial x}$. I found the following after writing $g$ using an ...
2
votes
2answers
180 views

Law of Total Probability/Independence question

Hey first I want to start of by saying that this is homework. I'm just looking for guidance. These problems involve using a deck of cards. I want to find the conditional property that the second card ...
0
votes
1answer
76 views

Countable Sequence of Events

My Question: Let $(\Omega,\mathcal{F},\textbf{P})$ be a probability triple such that $\Omega$ is ${countable}$. Prove that it is impossible for there to exist a sequence $A_1,A_2,\ldots \in ...
1
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1answer
276 views

expectation and variance problem

$n$ married couples (a total of 2n people) are randomly arranged in a line. Let $X$ be the number of married couples that are standing next to each other. Compute the expectation and variance of X ...
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votes
3answers
132 views

Devise a combinatorial problem in which it is easier to solve via probability theory than by counting methods

We easily find probabilities by counting the number of ways we can meet a condition and dividing it by the number of total possible outcomes. This is using combinatorics to solve probabilities. Im ...
0
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2answers
59 views

independent rv expectation

If $X, Y, Z$ are independent random variables, then $E[XYZ] = E[X] E[Y] E[Z]$ But how can I say that $E[X^2Y^2Z] = E[X^2]E[Y^2]E[Z]$? That being said, how can I prove that $X^2, Y ^2, Z$ are ...
1
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2answers
145 views

Joint probabilities and conditional independence

I'm going through a revision paper and looking at the solutions and I come across this. Given a Bayesian Network (sorry I cannot post images): $A$ and $B$ are parents of $C$. And $C$ is parents of ...
1
vote
1answer
915 views

Bayes' Rules: The probability of at least one event occurring?

There is a 60 percent chance that the event $A$ will occur. If $A$ does not occur, then there is a 10 percent chance that $B$ will occur. (a) What is the probability that at least one of the ...
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votes
1answer
97 views

Expectation probability question

How can I solve this problem? For the 1st problem, I assumed that $E [x] = E [x^2]$, and got $P( X = 0 ) = \frac{1}{3} $ For (b), I got $p = 0.6, n = 10$ so $P(x = 6) $ then use binomial r.v ...
2
votes
2answers
167 views

Math symbol for approximation of probability distribution by arbitrary function?

I want to use a symbol between two functions; $$p\text{ (symbol) }f$$ such that $p$ is a probability function and $\text{(symbol)}$ implies: we do not have access to $p$ but we approximate it with ...
1
vote
0answers
29 views

Coin tossing - Two tosses, one is a head, probability other is a tail? [duplicate]

A friend of mine tossed a fair coin twice. Suppose instead that I happen to see the result of one of his tosses, and it is a head. What is the probability that the other toss is tail?
0
votes
1answer
636 views

Discerning The Set Of Values For A Random Variable

The question is: For each random variable defined here, describe the set of possible values for the variable, and state whether the variable is discrete. a. $X=$the number of unbroken eggs ...
0
votes
2answers
230 views

Variance of a function of a normal random variable

I want to define a new random variable $f$ as a function of a normal random variable $v$: $$f(v)=\begin{cases}C&\text{if } v\ge C\\ \gamma v &\text{otherwise}\end{cases}$$ where $v\sim ...
0
votes
1answer
210 views

Markov Chains Probability

A Markov chain $X_0$, $X_1$, $X_2$, ... has the transition probability matrix $$ P = \left[ \matrix { 0.3&0.2&0.5 \\ 0.5&0.1&0.4 \\ 0&0&1 } \right] $$ and is known to ...
0
votes
1answer
116 views

Mean Duration of Stochastic/Markov Game

An urn contains five red and three green balls. The balls are chosen at random, one by one, from the urn. If a red ball is chosen, it is removed. Any green ball that is chosen is returned to the urn. ...
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votes
2answers
121 views

marriage probability problem

$N$ marriged couples are lined up randomly.( so there'are 2n people). What's the probability that A and B (who are a couple) sit next to each other? What's the probability that A and B (who are a ...
1
vote
1answer
232 views

Size of intersection between sets

Let sets $A_i, i\in [1,\ldots,k]$, be sets of $O(\sqrt{n})$ arbitrarily chosen numbers without repetitions in range $[1,\ldots,n]$. Consider a set $B$ with randomly chosen $O(\sqrt{n})$ distinct ...
0
votes
1answer
55 views

Convergence in distribution sequence of r.v.'s

If I have a sequence of random variables $Y_n$ which converges in distribution to the exponential distribution with mean 1 How to conclude that $n Y_n$ does NOT converge to any random variable in ...
3
votes
1answer
173 views

Binomial-Like Distribution with Changing Probability

The Question Assume we have $n$ multiple choice questions, the $k$-th question having $k+1$ answer choices. What is the probability that, guessing randomly, we get at least $r$ questions right? If ...
1
vote
1answer
156 views

5 cards / Joint Probability Function

From a deck of playing cards, you take out 5. The random variables X and Y denote the number of "aces" and "queens" in the sample, respectively. Find the joint probability function of X and Y, and ...
0
votes
1answer
134 views

Beta Coin Toss Experiment

I've run across this tricky little problem that I just can't seem to figure out. Say we flip a coin N times, but we don't know the true value of the success parameter $\theta$. Instead, we have a ...
0
votes
1answer
102 views

The Probability of an event occuring an exact amount of times

Suppose there are 40 people in a room and I want to know the probability of exactly 2 people having the same birthday (excluding leap years). The probability of two people having the same birthday is ...
0
votes
1answer
47 views

Calculate Expectation : $150$ questions each with $4$ options

In exam , there are $150$ questions each with $4$ options for each correct response $+1$ mark is awarded for each incorrect response $-0.25$ mark is deducted If $1000$ students appear for exam and ...
0
votes
1answer
53 views

normal distribution curve figure interval not at figure bars: get number of instances within

I have a normal distribution curve figure. It is vertically chopped into equal-width segments based on the distance from the mean value my (in the middle of the ...
1
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1answer
34 views

Combination problem?

How many way can UNC, Duke, and Florida State finish 1-2-3 in the AAC regular season rankings? Would I have to find out the AAC'S regular season rankings?
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3answers
60 views

Probability Rules

The probability that coal fired electric generator 1 is working is .10. The probability that coal fired electric generator 2 is working is .20. What are the probabilities that: Both work? Neither ...
0
votes
2answers
186 views

Random Variable

I was wondering if I correctly understand what a random variable is. Is a random variable's domain the set of numbers that are reasonable, when considering how the random variable is defined. For ...
0
votes
1answer
30 views

Probability of large Unions

Given a large number of independent events, how does find the probability that exactly one occurs, in terms of the individual probabilities.
0
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1answer
52 views

Prove that $(X_1 X_2\cdots X_n)^{1/n} \to c$ as $n\to\infty$ where $c$ is a constant

This is a assignment question, a part of my homework. So I need hints to start towards the solution. I was thinking that under the given conditions of the problem the random variables $\log X_1$, ...
0
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1answer
33 views

Trying to find limiting of binomial probability

So how to approximate $c_3=\ {\infty +a-1 \choose \infty-1}$ $c_2=\ {\infty \choose \infty-2-a+b}$ $c_1=\ {\infty +a-b +1 \choose \infty-1}$ $p=\ {(c_1-c_2)/c_3}$ I'm trying to get the limit of ...
1
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0answers
48 views

Optimal distribution with moment conditions

Basically, I want to find a probability distribution which maximizes a convex objective function and satisfies two moment constraints. For given $\bar x$, $m_{n-1}$, $m_n$ $$ \max_{f(x)} ...
2
votes
0answers
85 views

Prove that $\mathrm{E}[X\mid A] = \mathrm{E}[X]$ for an event $A$ independent of random variable $X$

I am a novice at statistics. I have approached the problem in the following way: $$\mathrm{E}[X\mid A] = (x_1\mid A \cdot P[x1\mid A] + x_2\mid A \cdot P[x_2\mid A] + \ldots + x_k\mid A \cdot ...
0
votes
1answer
931 views

Four married couples, eight seats. Probability that husband sits next to his wife?

There are four married couples and eight seats. When they sit, what is the probability that husband sits adjacent to his wife? The answer to this problem is 12/35 I can arrive at this answer when I ...
0
votes
2answers
290 views

A probability problem - probability of one card being red and the other one being black.

Consider a deck of 50 playing cards (2 cards missing). What is the probability that one of them is red and the other one is black? I've got two solutions which one is correct ? Let $R$ represent red ...
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votes
1answer
56 views

Normal distribution function

Given Data : Normal Distribution function $$P(x \leq -1) = P(y \geq 2)$$ x : mean= 1 variance= 4 y : mean= -1 variance= to be found..?. Ans: 9 How to solve..?
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votes
1answer
84 views

Probability and Z-Scores

I'm new here and was hoping you guys could help me with a statistics problem that I don't quite understand. I'm not sure if it's proper etiquette to ask for help on a specific homework problem here, ...
0
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0answers
49 views

A tough counting question: Specified order

Assume that there are $3n$ people in the population. These people are ranked randomly and uniformly. ( that is, each permutation is equally probable) There is a competition and $m+1+k$ people among ...
0
votes
1answer
53 views

Upper bound of integral of conditional probabilities

Given X as a random variable with distribution p(x). Then $w_1$ and $w_2$ are conditioned on X. How to prove: $$\int_{-\infty}^\infty 2P(w_1|x) P(w_2\mid x) P(x)\,dx > \int_{-\infty}^\infty ...
0
votes
1answer
453 views

Find the probability density function of the following random variables

Suppose there $\mathbf{2n}$ i.i.d exponential random variables $X_{1},\ldots, X_{n},Y_{1},\ldots,Y_{n}$ with probability density function $$f(x)=\left\{\begin{matrix} e^{-x}, &x\geqslant 0 \\ 0, ...
0
votes
2answers
54 views

Probability Problem choosing $6$

If you have $10$ red marbles and $15$ green marbles in a bag, and you randomly choose $6$, what is the probability that $5$ red marbles will be selected?
1
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1answer
99 views

Statistics and growth

Formula $f(n)=k-k(1-\frac{1}{k})^n$ has $k$ as a constant. Function $f(n)$ shall be a measure of confidence. How can we best describe the growth of confidence with every $n$ in a few words? A plot of ...
0
votes
1answer
278 views

CDF and conditional probability

The cdf of $T$ is given by $F_T(t)=1-e^{-\lambda t}$ for $x>0$. Show that $P(T>t+s\mid T>s)=P(T>t)$. Find the MGF of $T$. I don't really know what's going on. Do I substitute $t+s$ into ...
2
votes
1answer
126 views

stationary and ergodic

Let $p\in \mathbb N$ and $a_1,\ldots,a_p\in \mathbb R $. Denote by $x$ the sequence $$x=(x_k)_{k\in\mathbb N }=(a_{k \bmod p})_{k\in\mathbb N }$$ where $(k \bmod p)\in{1,\ldots,p}$ is the remainder ...
1
vote
1answer
216 views

Link between Standard Normal R.V. transofrmation to polar coordinates and orthogonal matrices

I'm trying to solve this problem which has 2 parts: 1) Let $X, Y$ be independent standard normal r.v. define $R, \theta$ by $X=R\cos\theta$ $Y=R\sin\theta$ and find the joint distribution of ...
0
votes
1answer
145 views

Bins in balls where bin size grows exponentially

I have $k$ bins. The first bin can fit $1$ ball. Each subsequent bin can fit two times more balls than the previous one. In other words, the $i$th bin can fit $2^i$ balls. We randomly assign $U = ...
0
votes
1answer
86 views

Clasification of parameter estimation method

Consider that $P$ is the water pressure coming out from a valve A, therefore, the population is all the valve A pressure values. Let $P_{dif}$ be defined as the difference between the maximum and the ...
1
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2answers
1k views

Why do non-Decreasing Functions have countable discontinuities [duplicate]

I was reading some notes and one of the results in it implicitly used a result which fell along the lines of "non-decreasing functions have countable discontinuities". I don't completely understand ...