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Questions tagged [probability]

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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Probability Density Function of X with a Uniform Distribution on a Unit Sphere

Given that $f_\Phi(\phi)=\frac{\sin(\phi)}{2}$ and $f_\Theta(\theta)=\frac{1}{2\pi}$ for $\Phi\in[0,\pi]$ and $\Theta\in[0,2\pi]$, what is the PDF of $X=\sin(\Phi)\cos(\Theta)$? $\Phi$ and $\Theta$ ...
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2answers
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State probabilities of a markov chain with multiple absorbing states

Consider a gambler who starts with a fortune of 1 dollar. He keeps flipping a coin and gets one more dollar if he gets heads and loses one dollar if he gets a tails. He stops playing if he either ...
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1answer
16 views

$5$ prizes are distributed among $20$ students. What is the probability that a particular student receives $3$ prizes?

There are $5$ prizes that are to be distributed among $20$ students. What is the probability that a particular student receives $3$ prizes ?
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spaghetti hoops combinatorics variation

You may have heard about the classic spaghetti hoops combinatorics problem, which has been stated like this: "You have N pieces of rope in a bucket. You reach in and grab one end-piece, then reach in ...
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1answer
22 views

Proof using chebyshev's inequality

Consider the sequence of random non-negative values $X_1...X_n$ where $E[X_n], Var[X_n] > 0$ for all $n \in \mathbb{N}$ and: $$\lim_{n\rightarrow \infty} \frac{Var[X_n]}{E[X_n]^2} = 0$$ Prove:...
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1answer
25 views

Covariance proof

Let the covariance random values X,Y be : $$Cov[X, Y]= E[(X- E[X])(Y - E[Y])] = E[XY]-E[X]E[Y]$$ Prove the following for random values $X_1...X_n$ : $$Var[\sum_{i = 1}^n X_i] = \sum_{i = 1}^n ...
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0answers
16 views

Finding the joint mgf of two random variables

Let the joint pdf of $(X, Y)$ be given by $$f(x, y) = \frac{1}{\sqrt{2\pi}} \text{exp}\left(-y-\frac{(x - y)^{2}}{2}\right) \hspace{1cm} \text{ for } y > 0, -\infty < x < \infty$$ ...
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1answer
18 views

Computing marginal PDF given joint pdf

Let the joint PDF of $(X, Y)$ be given by $$f(x, y) = \frac{2}{x} e^{-2x}, \hspace{1cm} x>0, 0 \leq y< x. $$ a) Find the marginal PDF of $X$ and find $\mathbb{E}[X]$ b) Find the ...
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1answer
24 views

joint PDF of max and min of $n$ iid standard uniform random variables

Let $U_1, ... U_n$ be iid standard uniform variables. Let $X = max(U_i)$ and $Y = min(U_i)$. The goal is to compute the joint PDF of $X, Y$! I have already computed the PDFs of $X$ and $Y$ separately....
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0answers
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On Random Rearrangements

Let $\text{Sym}(\mathbb{N})$ denote the group of bijections $\mathbb{N} \to \mathbb{N}$. It is well-known that this has cardinality $\mathfrak{c}$. Suppose $a_n$ is a conditionally convergent series ...
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24 views

Converges to $0$ in probability

Suppose $\chi_n \to \chi:=\mathrm{exp}(1)$ in distribution, and set $\rho_n = 1 - \chi_n/n$. Given that $\frac{1}{n \log \rho_n} \to -\frac {1}{\chi}$, show that \begin{align*} \frac{1}{n\sqrt{\...
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3answers
39 views

Sumin rolls three distinct dice and gets a, b, and c. Find the probability of 2a + b + c = 10.

Sumin rolls three distinct dice and gets a, b, and c. Find the probability of 2a + b + c = 10. My answer: a b c 4 1 1 3 2 2 2 3 3 1 4 4 3 3 1 3 1 3 2 1 5 2 5 1 2 4 2 2 2 4 1 5 3 1 3 5 total ...
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1answer
29 views

Calculate $E(Y)$ where $Y=X^{1.5}$

Let $X$ be a rv which is $Exp(\lambda=2)$. The pdf of $X$ is given by $f_X(x)=2e^{-2x}, x\geq 0$ (and $0$ otherwise). We define $Y=X^{1.5}$ and ask $E(Y)$. It does not look like the moment ...
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0answers
11 views

Including random variables in differential equations

I am having trouble finding information regarding the inclusion of random variables in differential equations. For example I have a simple model describing the growth of some population $X$ over time: ...
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2answers
26 views

Why uniform discrete pdf does not converge to continuous?

Why a discrete probability distribution does not converge to a continuous when the number of support points grow to infinity? Consider a distribution with support in $[0, 1]$, if that is continuous, ...
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1answer
36 views

Verifying solution: Twelve fair coins are flipped

I need to know if I did this problem correctly or incorrectly. Twelve fair coins are flipped. (a) What is the expected number of heads that will be obtained? if a coin is tossed 12 times, the ...
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1answer
17 views

Computing the joint moment generating function for two functions of two random variables

Let $X$ and $Y$ be i.i.d random variables in the plane with a pdf $$f(x) = \frac{1}{\sqrt{2\pi}} \cdot \text{exp}(-x^{2}/2) \hspace{1cm} -\infty < x< \infty.$$ Let $U = X + Y$ and $V =...
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0answers
18 views

Choosing a parameter for a Poisson distribution

This question is kind of weird, and it raised from an even weirder question (in the bins and balls model), but I've tried to simplify it as much as I can let $n,k\in \mathbb N$ be 2 numbers such that ...
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0answers
15 views

Conditional Expectation of stopping time given hitting one point of simple random walk 1D

Let $S_n$ be a simple random walk started at $0$, i.e., $S_n =X_1+\ldots+X_n$ where $X_i$ are i.i.d. with values in $\{\pm 1\}$ with probabiliy 1/2. Fix $a,b\in \mathbb{Z}_{\geq 0}$ and let $\tau = \...
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1answer
28 views

calculate CDF from given PDF

I am trying to understand the calculate the CDF from the given PDF $f(x) = \begin{cases} 0.5& 0\le x<1\\ 1& 1\le x<1.5\\ 0& \text{otherwise}\end{cases}$ The CDF is $F(x) = \...
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1answer
29 views

Eight fair coins are flipped. Find the probability that 5 heads and 3 tails are obtained

I am struggling to solve this probability question. I don't know which approach to take with regards to finding the equation for calculating the given amount. ...
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0answers
20 views

Probability and averages dice

If you know the odds of something happening, say a particular set in a roll of 3 dice is 1/64 odds then how does one calculate the average number of rolls to produce this outcome?
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2answers
82 views

Conditional expectation, what is my mistake

From SOA sample #238: In a large population of patients, $.20$ have early stage cancer, $.10$ have advanced stage cancer, and the other $.70$ do not have cancer. Six patients from this population ...
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3answers
61 views

Do random experiments actually exist?

I am studying probability and in most of the books that i have read they mention that for an experiment to be random-- (1)there should be more than 1 possible outcome--(2)even when the experiment is ...
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1answer
26 views

$\operatorname{supp}(f_{X,Y})=\{(x,y)\in\mathbb{R}^2||x|+|y|\leq1\}$ then $X,Y$ are not independent

Let $Z=(X,Y)$ be a absolutely coninuous random variable such that $$ \\\operatorname{supp}(f_Z)=\{(x,y)\in\mathbb{R}^2||x|+|y|\leq1\} \ $$ Show that $X,Y$ are not independent. I don't have a good ...
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1answer
17 views

Convergence in square mean

If $X_n$ is a sequence of random variables such that $E(|X_n|)\to0$ but $E(X_n^2)\to 1$, does it imply that $X_n$ doesn't converge in square mean? I can find a sequence that satisfies the two first ...
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2answers
24 views

Determine the Expected Value of Uniformly random elements of sets

Answer is D The way I attempted this was that for X = MAX(a,b), the random variable X is equivalent to the max value of a and b. So, from the 2 sets, the probability of getting k from set {1,2...100} ...
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2answers
253 views

Determining probability of a rainy day

I have the following problem: If today is a sunny day, a probability that it will rain tomorrow is $0.2$. If today is a rainy day, a probability that it will be sunny tomorrow is $0.4$. I need to ...
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0answers
25 views

Failure rate problem trying to prove

$X$ and $Y$ are independent non-negatively distributed. I am trying to prove $P(X<Y|\min(X,Y)=t) = \dfrac{r_1(t)}{r_1(t)+r_2(t)}$ where $r_1(t)$ and $r_2(t)$ are failure rate of $X$. The ...
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0answers
18 views

Assigning charges to elements in a set [on hold]

Let S be a finite set, and let S1, S2... Sk be subsets of S, and each of them has the the same cardinality s. Assume that each element in the set S may be assigned one of two charges, +1 or -1. Show ...
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4answers
32 views

What is the probability of passing through a node in a directed graph

Say I have a directed graph with no cycles like this one. And say someone travels along it choosing a random edge to go down at every node. We know that the person walking starts from node 0 and is ...
2
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1answer
26 views

Estimate relationship between two Bernoulli random variables

$X$ and $Y$ are Bernoulli random variables $X$ and $Y$ are not independent $x_{t} = P(X_t = 1)$ and $y_{t} = P(X_t = 1)$ for time $t$. Is it possible to estimate $P(Y = 1 | X = 1)$ from many pairs of $...
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0answers
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Find the variance of $X^{T}Y$ where $X,Y$ are multivariate normal vectors

Let $X \sim N_p(\mu,\Sigma)$ and $Y \sim N_p(0,\Sigma^{-1})$ where $N_p$ refers to the $p$ variate normal distribution with dispersion matrix $\Sigma$. Show that $\text{Var}(X^{T}Y)=p+\mu^{T}\Sigma^{-...
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1answer
8 views

Convergence locally uniformly VS $L^1$ convegence for probability density functions

Let $f_1, f_2, \ldots$ and $f$ be probability density functions on $(0, \infty)$ - so $\int_{(0,\infty)}f_n(x)dx=1$, $\int_{(0,\infty)}f(x)dx=1$. Assume that for every $x \in (0, \infty)$ there exists ...
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0answers
14 views

Is this Bayesian Network Probability calculation correct?

I think I understand how to calculate BN and why it is so, but complex net still confuses me. Currently how I understand it is that, if there is any 'result' variable in the probability, it can be ...
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0answers
16 views

conditional expectation of variance process

Suppose that each asset-i satisfies the sde: $dS_t^i=S_t\sigma_t^i\sqrt{v^i}dB_t^i$, $i=1,2$ where, $u^i=\exp{\int_{0}^{t} c^ie^{-k(t-s)}dW_s^i }$, $i=1,2$, $\sigma^i$ deterministic, $c^i\in \mathbb{...
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1answer
28 views

Perfect correlation between two random variables: some clarifications [on hold]

Consider two random variables $X,Y$. $X$ can take value $x_1$ with probability $p$ and $x_2$ with probability $1-p$. $Y$ can take value $y_1$ with probability $p$ and $y_2$ with probability $1-p$. ...
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0answers
18 views

Generalization of an absorbing Markov Chain

A process moves on the integers 1, 2, 3, 4, and 5. It starts at 1 and, on each successive step, moves to an integer greater than its present position, moving with equal probability to each of the ...
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Ameba splitting probability (how many of them will be if now there are n?) [on hold]

The problem is quite common, but I cannot come up with an answer to this situation. We have n amebas. Each minute each ameba can die with probability 25%, stay alive without changes - 25%, or split ...
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0answers
6 views

Exponential probability problem [duplicate]

Machine 1 is currently working.machine 2 will be put in use at time t from now.if the lifetime of the machine i is exponential with rate K_i, i=1,2 what is the probability that machine 1 fails first ...
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30 views

Uniformly independent distributions

Let be $X_1,X_2$ independent uniformly (0,1) random variables. Define $M = \min\{X_1,X_2\}$ Evalute $P(X_1 \leq x_1,X_2 \leq x_2)$ using the random variable $M$ and the event $\{X_1 < X_2\}$. ...
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1answer
22 views

If $f_X$ is a pdf of the r.v. $X$ what would be $g(s)=\int_{\mathbb R}f_X(x)\delta (s-x)dx$?

Let $X$ a real r.v. and $f_X$ it's pdf. What would be $$g(s)=\int_{\mathbb R}f_X(x)\delta (s-x)\,\mathrm d x,$$ where $\delta $ is the Dirac distribution ? It look to be a pdf of some r.v. but which ...
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2answers
44 views

Normal Distribution in Probability

I can solve the k, but I am not really able to do the rest! The mass M of apples in grams is normally distributed with mean μ . The following table shows probabilities for values of M . Values: ...
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0answers
9 views

Probability outcome of 4 people scoring 42 (in 4 rounds) description in the body [on hold]

Probability out of 12 people, having 4 people get a score of 42's considering each individual can claim 1-10, 12, 15 points per round (4 rounds and no person can have the same score in a round)
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23 views

The number of trials which maximizes the expectation of difference between the number of red balls and blue balls?

There are $N$ balls with $K$ red balls and $N-K$ blue balls. It means that the probability that the red ball is drawn with one draw is $K/N$. There is a random variables $X$ = the number of red ...
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1answer
26 views

Divergence measured from integral of minimums: $\int_{-\infty}^{\infty} \min(p(x), q(x)) dx$

I was studying a particular machine learning algorithm (the GeoGAN) and although this isn't mentioned in the paper, it seems to be that under certain conditions, the algorithm is maximizing (over $p$) ...
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1answer
8 views

Finding odds of a prize draw

Im trying to establish 3 separate odds for the same equation. Lets use a draw for an example. 200,000 people enter a draw, there are 3 top prizes lets say for example 1st prize $1m, 2nd prize $...
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1answer
45 views

Is it possible to randomly select a single member of $\mathbb N$? [duplicate]

The title is my question and the reason for asking it is the following. Define a set $\mathbb N (≤ n) \equiv$ {1,2,3, … , n} and define a “random selection” to be a selection in which each member ...
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3answers
18 views

Independent event probability

Quoted from a popular book, For instance, if we flip a fair coin twice, knowing whether the first flip is Heads gives us no information about whether the second flip is Heads. These events are ...
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1answer
42 views

Why is an elementary Ito integral necessarily continuous?

So I am working/reading through a proof that general Ito integrals have continuous versions. So that $$I_f(t) = \int_0^t f(s,\omega)dB_s(\omega)$$ Has a continuous version in $t$. The proof I am ...