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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Challenging question about probability

A manufacturer of plastics claims that its waste is managed in such a way that benzene, a harmful chemical, cannot get into the local ground water. People living near the factory are not so sure. A ...
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3answers
17 views

Finding probability of a random point

Consider a square with sides of length 1 and the bottom left corner at (0;0). Choose a point P randomly within the square. Show that the probability that P is closer to (0;0) than to (0.5, 0.5) is ...
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Computing area/ space of intersection between a pair of Beta distribution/ Dirichlet distributions

I need to compute the area/ space of intersection between a pair of Beta distribution/ Dirichlet distributions. As I am a non mathematics guy, it will be great if someone helps me out with the ...
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1answer
10 views

Proving negative natural log of Beta($\alpha$, 1) distribution is an exponential distribution

I'm looking to prove that taking the negative natural logarithm of a Beta distribution with parameters $\alpha$ and $\beta=1$ is an exponential functions. I've found two different proofs, both of ...
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0answers
13 views

Probability density functions (setting limits of integrals)

The joint density of Y1 , the proportion of the capacity of the tank that is stocked at the beginning of the week, and Y2 , the proportion of the capacity sold during the week, is given by ...
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2answers
20 views

probability: arithmetic with random variables.

I have a question of using arithmetic on random variables. Please refer to the following question, to which I will present my solution using the arithmetic (which I thought it's correct but actually ...
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0answers
23 views

Expectation and variance of seats in a movie theater

A movie theater has a row of 10 seats. Each seat is occupied with probability $1/2$ independently of the other seats. A visitor is "comfortable" if he or she is not at the edge of the row and both ...
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2answers
30 views

Question on Binomial Probability

The Grunters and the Screamers are playing for the Grand Championship, which is a best of 7 series. The first team to win 4 games wins the Championship. Each team has a $\frac{1}{2}$ probability of ...
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0answers
13 views

Average minimum distance between two random vectors

Let $\mathbf{y_1} =\begin{bmatrix}g_1x_1 & g_2x_1 & \dots & g_Nx_1 \end{bmatrix}$ and $\mathbf{y_2} = \begin{bmatrix} f_1x_2 & f_2x_2 & \dots & f_Nx_2\end{bmatrix}$. All the ...
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1answer
14 views

Prove the probability of n events intersecting

I'm trying to write a proof for this, but I don't know how to get started. Would proof by induction be the easiest way? If you could break it down into general steps I could wrap my head around, I ...
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1answer
21 views

Continuous-time Markov Question

I have a question about a continuous-time Markov process on the discrete space. I am given the generator and asked for find the expected time the Markov process needs to get back to state 3, given ...
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1answer
21 views

Tricky Marble Probability

A bag contains $3$ red marbles, $3$ green ones, $1$ lavender one, $6$ yellows, and $4$ orange marbles. How many sets of five marbles include either the lavender one or exactly one yellow one but not ...
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2answers
46 views

Show that $E[(X-E[X])^2] = E[X^2]-E^2[X]$

I'm trying to use linearity of expectation, but I seem to be failing somewhere. So if we expand the squared expression: $$(X-E[X])^2 = (X-E[X])(X-E[X]) = X^2 - 2XE[X] + E^2[X]$$ So we have: $$E[X^2 ...
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2answers
39 views

Why is the probability of drawing a king and then a heart the same as drawing the king of hearts?

I calculated that the probability of drawing a king and then a heart from a deck of cards is $\frac{1(12) + 3(13)}{\text{Permutation}(52,2)}=\frac{1}{52}$ However, I also noticed that this is the ...
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0answers
17 views

Is there a standard proof for $\mathbb P(S^X_n\text{ hits }A\text{ before }B) >\mathbb P(S^Y_n\text{ hits }A\text{ before }B)$?

Let $X_i$ and $Y_i$ be two continuous random variables on $\mathbb{R}$ having distribution functions $F$ and $G$, respectively satisfying $G(y)>F(y)$ for all $y$. Let futhermore $S^X_n=\sum_{i=1}^n ...
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0answers
10 views

Borel-Cantelli lemma: convergence of series [on hold]

Consider a sequence $(X_{n} )$ of r.v.'s and $M_{n} = \max_{k \leq n} |X_k|$. If $\sum_n \frac{1}{n} P(M_n > \varepsilon) < \infty$ then $\sum_n P(X_n > \varepsilon) < \infty$?
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0answers
15 views

Markov Chain Geometric Convergence

Perhaps due to my non-mathematician background (lack of Measure Theory knowledge), I have some difficulties about the Markov Chain Theory. Given an ergodic (irreducible and aperiodic) Markov Chain ...
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0answers
7 views

Projected multivariate normal distribution

Assuming a set of points (x_i,y_i,z_i) is distributed according to a multivariate normal distribution with a given mean and an isotropic covariance matrix, i.e. ...
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0answers
27 views

Poisson process: Has my book used a necessary condition, when it should have used a sufficient condition?

My book says that if we know that if we are viewing a poisson process with length $t$, and know that $n$ events happened in that interval, than the time that any of those events happened is uniformly ...
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2answers
27 views

Probability distribution morphing from Gaussian to heavy tail

I require a probability distribution which morphs from a something similar to a Gaussian (image 1) to something with a heavy tail on one side (image 2) based on some parameter. Can someone give me a ...
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1answer
19 views

Probability of item distribution with a restriction

I'm having a hard time analyzing my research data, and was wondering if anyone had any suggestions? I've reworded the question so it is presented more like a statistics problem. There are $x$ number ...
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0answers
15 views

Formula for a Skewed Distribution

I'm trying to create a model of some data, using a skewed normal distribution. I have the following data: Mean Median Standard Deviation from the mean Standard Deviation from the median I've been ...
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0answers
14 views

How to calculate the distribution within a series

The probability of Event A happening is 50.7%. The probability of Event B happening is 49.3%. What is the probability that in a series of 100 trials, there will be at least one point where more ...
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1answer
21 views

calculating variance of a random variable

Suppose you have a playlist consisting of four songs that you play in a smart shuffle mode. In this mode, after the current song is played, the next song is chosen randomly from the other three ...
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0answers
25 views

CDF of the random variable $U$ defined by $U=Y^n$

I was working on a practice exam today and I got stumped on the following question: The continuous random variable $Y$ has probability density function $f$ given by $$f(y)=\cos(y),\quad 0\le ...
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2answers
35 views

Finding conditional probability [on hold]

Two floppies are selected at random without replacement from a box containing $7$ good and $3$ defective floppies. Let $A$ be the event that the first floppy drawn is defective, and let $B$ be the ...
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2answers
21 views

Probability of begin cut

Any athlete who fails the Enormous State University's women's soccer fitness test is automatically dropped from the team. Last year, Mona Header failed the test, but claimed that this was due to the ...
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0answers
18 views

House edge when playing blackjack with perfect memory

If one plays single-deck blackjack with standard rules, what edge does the house have assuming the player remembers exactly how many 2s, 3s, etc. have been seen and plays optimally according to this ...
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1answer
21 views

Expectation and variance of the number of couples sitting at the same table

There are $2n$ people at a party, making $n$ couples. These $2n$ people are seated randomly at $n$ tables that have $2$ seats each. Let $X$ be the number of couples that sit at the same table. Find ...
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2answers
8 views

Meaning of the expectation of a conditional variance

I'm reading page 2 of conditional probability. The author states that if $\text{Var(X|Y)}$ is treated as a random variable then the expectation is $\text{E[Var(X|Y)] = E[E[X^2|Y]] - E[E(X|Y)]^2}$ My ...
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0answers
13 views

Variance of exit time for simple symmetric random walk

For a simple symmetric random walk starting at 0 (that is, a Markov chain on the integers starting at 0 with equal probabilities of going to the left and right at each step), I want to compute the ...
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0answers
12 views

Non-Linear Transformation of two R.V.

Suppose I want to take some action if an object is with in some distance from me with some probability. I have an x and y measurement of its position with noise ...
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1answer
14 views

Time Series Analysis.Calculate the variance mean and autocorrelation of the time series below.

For the following time series, calculate the mean, varia nce and autocorrelation function: (a) Y_t=5+Z_t+ 0.6Z_t-1
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1answer
32 views

Calculating bounds with multiple random variables.

I have this problem: Suppose there are 4 students (who we'll refer to as A, B, C, and D) in a class and each student is equally likely to have been born in any of the twelve months of the year. For ...
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2answers
18 views

Probability of 2 independent events and 1 mutually exclusive

Let $\Omega$ be the sample space for an experiment $E$ and let $A,B,C\subset \Omega$. If events $A,B$ are independent, events $A,C$ are disjoint, and events $B,C$ are independent, find $\Pr(B)$ if ...
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1answer
18 views

Marble Probability

A bag contains 3 red marbles, 3 green ones, 1 lavender one, 6 yellows, and 4 orange marbles. How many sets of five marbles include either the lavender one or exactly one yellow one but not both ...
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0answers
20 views

$\{Q^{(n)}\}$ is tight? [on hold]

Let $Q^{(n)}$ , $n\ge 1$, are probability meaures in $E^{Q_+}$, where $E\subset \mathbb{R}^d$ ($d\ge 1$) is compact, $Q_+=Q\cap [0,\infty )$ and $E^{Q_+}$ denotes the collection of function from $Q_+$ ...
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2answers
32 views

question about probability problem

How is the last column calculated? I don't understand, and I don't understand the explanation. $P(A \cap B)$ is calculated by $P(A)P(B\mid A),$ right? How is $P(A\mid B)$ calculated? Thanks
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2answers
22 views

Question on Conditional Probability and dependence

If a card is chosen randomly from a standard deck, show that the events 'face card' and 'diamond' are independent. I know to do this you need to show that $P(\text{Face Card}) = P(\text{Face ...
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1answer
23 views

Sequence of letters of length 5

How many sequences of letters are there of length 5 with exactly 2 vowels? Don't count "y" as a vowel. Pretty lost on this one. I know it involves 5 choose 2 has part, but I feel like that's not all
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1answer
23 views

calculating X, Y, Z random variables

Suppose X, Y, and Z are random variables that each take the value 0 or 1. If P(X=0,Y=1,Z=0)=1/3 and P(X=0,Y=1,Z=1)=1/4, what is the value of P(X=0,Y=1)? I am trying to calculate this but I am really ...
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0answers
12 views

arbitrage free price in martingale measures

Consider a one-period market with $S^1_t,\cdots,S^n_t$, with $t=0,1$ the price process of $n$ assets, where $S_1$ is a risk-free asset: $S^1_0=1$,$S^1_1=1+R$. Assumes that this market satisfies the ...
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1answer
19 views

Understanding $Po(np)\{A\}$ probability notation

I am trying to read a textbook on probability and am already stuck on what must be basic notation. It says Thus, for example, if $A$ is any subset of $\mathbb{Z}^+$, it follows that for some ...
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0answers
16 views

Odds of winning more than 50% of many bet of different %

I made a bet with a friend and I would like to know if I'm ahead or not. We have a package of 6 games. Each game have a different probability of a team to win. If it's a tie 3-3, it's a push. If it's ...
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0answers
30 views

Probability that a number falls within a specific range

Suppose we have a set of numbers $A$, what is the probability that some random $x \in A$ lies between the $25^{th}$ and $75^{th}$ percentile? Assume that it takes time in $|A|$ to check. ...
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1answer
21 views

How are the waiting times distributed, poisson process.

I am wondering how the waiting times are distributed for the poisson process, conditioned on a number of events by time t. Look at this theorem: Here, the S's are the sum of the waiting time to ...
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1answer
17 views

Ping Pong Winning Probability (World Series)

You are playing ping pong with a friend and your chance to win any point is P. This is a world series. Find the probability that you score 4 points before your friend has a score of 4. Evaluate this ...
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1answer
20 views

probability for two vectors to lie on different regions created by hypeplane

Suppose we have two vectors $v_i,v_j$ and there is one hyperplane whose normal is chosen uniformly from the unit sphere. Then what will be the probability that $v_i$ lies on one side and $v_j$ lies on ...
2
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1answer
20 views

Probability of swapping elements in an array

Given an array with n numbers (n>30). Perform the following steps: Compare the first element with the second element. If the first one is greater than the second one, swap their positions in the ...
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2answers
20 views

Covariance of dependent random variables from a Poisson process

Question: Given a Poisson process $N(t),t≥0$ with rate $λ$, calculate the covariance of $N(2)$ and $N(3)$. Attempt: So clearly $N(2) \sim Po(2\lambda)$ and $N(3) \sim Po(3\lambda)$. So, ...