Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.

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0answers
30 views

Truchet tiles on a cube [duplicate]

We randomly place copies of the tiles into faces of the flattened cube. 1.Find the probability that the circular arcs on the Truchet tiles will form one loop, two loops, three loops and four loops? ...
1
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1answer
61 views

An Easter game related to the geometric distribution

Consider the following game. You have a bowl of 10 eggs. With your 10 eggs, you want to compete in an egg-braking contest for Easter. In other words, you take your egg and bang it against your ...
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2answers
97 views

finding the number of circles we get when randomly placing given patterns into a grid of squares

We have an 11$\times$11 table of squares (consist of 121 squares of dimension 1$\times$1). we have 3 tiles shown in the picture. Each tile has dimension 1$\times$1. we now randomly pick 3 tiles into ...
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1answer
349 views

Trouble understanding sum and product of probability distributions

Having trouble understanding where can we use the sum and product of probability distributions. Could someone please provide me with a real-life scenario? I think this is what I need to understand the ...
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2answers
91 views

Summation of independent discrete random variables?

We have a summation of independent discrete random variables (rvs) $Y = X_1 + X_2 + \ldots + X_n$. Assume the rvs can take non-negative real values. How can we find the probability mass function of ...
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1answer
414 views

Simple geometric distribution (solution verification)

The question is: In a hockey competition, a player scores $80\%$ of his shots. What is the probability that the player will not miss until his $10^{th}$ try? So I did the following ...
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1answer
214 views

Finding probability that a person gets $7$ when rolling a pair of dice

*I STILL DON'T GET THE ANSWERS PROVIDED. PLEASE HELP! In a game, the participant rolls a pair of dice. If the result is a $7$, he wins. If the outcome is a number $n$ different from $7$, he continues ...
3
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1answer
155 views

Truchet tiles on a flattened cube

We randomly place copies of the tiles into faces of the flattened cube. 1.Find the probability that the circular arcs on the Truchet tiles will form one loop, two loops, three loops and four loops? ...
1
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1answer
38 views

Question on Poisson Processes

Good evening, I am sort of stuck in one problem of Poisson Processes and I hope I could get some help (no it is not a homework). Suppose that the customers arrive at the ticket booth independently. ...
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3answers
108 views

How I can find the expected value of $G$?

Suppose two teams play a series of games, each producing a winner and a loser, until one team has won two more games than the other. Let $G$ be the total number of games played. Assuming each ...
1
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1answer
67 views

Moment generating function of two non-independent Brownian increments

I am writing to ask if it is possible to get closed-form solution to the expression to the following expression: $\mathbb{E}[e^{\sigma^2(W_t-W_u)(W_s-W_u)}]$ where $W$ is a standard Brownian motion, ...
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1answer
36 views

Estimating a probability with converging moments

Let me rephrase my question. If you look at the random variable $X$ which simply picks a random integer between $1$ and $N$ (distributed uniformly) and now look at the inequality $$ t^k \cdot ...
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1answer
276 views

Generalized chi distribution

Let $v\in\mathbb{R}^n$ follow a multivariate Gaussian$(0,I)$ distribution, and $M\in\mathbb{R}^{n\times n}$ a matrix. Has the distribution of the Euclidean norm $\|Mv\|$ been studied? I know that its ...
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1answer
1k views

How do I sum two Poisson processes?

If we have a Poisson Process $Y$ with intensity $\lambda$ and a Poisson Process $X$ with intensity $\mu$, where $X$ and $Y$ are two independent Poisson processes. How can I find the process ...
2
votes
1answer
313 views

Maximum likelihood estimators, hypergeometric and binomial

I'm trying to solve a two part problem. The set up is as follows: consider a bag with $\theta$ red marbles and $7-\theta$ blue marbles, with $\theta$ being unknown. Let $x$ denote the number of red ...
6
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1answer
475 views

How was the normal distribution derived?

Abraham de Moivre, when he came up with this formula, had to assure that the points of inflection were exactly one standard deviation away from the center, and so that it was bell-shaped, as well as ...
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1answer
74 views

A modified Buffon's needle

A needle 2.5cm long is dropped on a piece of paper that has a very fine parallel lines 2.25cm apart drawn on it. What is the probability that the needle lies between the two lines? I can see how ...
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1answer
62 views

this is about prabability [duplicate]

Suppose a multiple choice test consists of $100$ questions and each question has $5$ possible answers, only one of which is correct. Four points are awarded for each correct answer, and $1$ point is ...
2
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2answers
158 views

Other way to express $e^{|x|+|y|}$

I have a joint PDF with $e^{|x|+|y|}$. I know I can separate the function in two functions, $e^{|x|}$ and $e^{|y|}$. The limits for $x$ and $y$ are from $-\infty$ to $\infty$. Can I integrate from $0$ ...
0
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0answers
31 views

What is the measure of all probability distributions with finite variance?

I'm in over my head here, but I am wondering about the probability that a distribution has finite variance? (or a finite mean?) By this, I don't mean that there is some set of data, just over the set ...
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0answers
24 views

Distribution of the entries of a matrix, which is the product of two normally distributed matrices

This is not a homework question, but a small part in understanding a research paper. Let $A$ and $B$ be two matrices, where $A_{ij} \sim \mathcal{N}(\mu_1, \Sigma_1)$ and $B_{ij} \sim ...
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2answers
60 views

pdf of a simple random variable calculated two different ways.. with two different answers

so we have a random variable Y with a uniform pdf on the interval [0,1]. the question is, what is the pdf of W=Y^2. method 1 using the transformation of variables formula: ...
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2answers
48 views

standard deviation of a certain distribution

If I have a list of N outcomes of drawing a number from the set {-1\$,+1\$}, and I know that the probability of getting (in a single draw) (-1\$) is p, and probability of getting (in a single draw) ...
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1answer
86 views

Calculating the pdf and cdf of $X^2$ and $X^3$ with the pdf of $X$ given

Let $X$ be a random variable with the density funtion: $$\phi_X(x)= \begin{cases} 6x(1-x) & \text{if } 0<x<1, \\ 0 & \text{otherwise.} \end{cases}$$ Find the density and ...
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2answers
229 views

binomial distribution transformation

Let X and Y be independent random variables with X having a binomial distribution with parameters 5 and 1/2 and Y having a binomial distribution with parameters 7 and 1/2. Find the probability that |X ...
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1answer
109 views

Conditioning on a Discrete Random Variable

Could anyone help me out with this problem? I am preparing for an exam and this problem was given to us as practice, but I am not sure how to start. Suppose we observe a Poisson process with ...
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1answer
288 views

Integral of product of normal cdf and pdf

What do you think, is there a closed form solution of the following Integral $\textbf{ }$ $$\int_{-\infty}^{a-y}n(x)\, N(b-2y-x)\, dx,$$ where $N(x)=\int_{-\infty}^x n(z)\, dz\quad$ and $\quad ...
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2answers
107 views

Probability that $x<V<U^2$ when $V$ and $U$ are uniform on $[0,1]$

I have a question asking for $P(x<V<U^2)$ when $U$ and $V$ are independent, identically distributed uniform variables on $[0,1]$. I started with: $$ \begin{align} P(x<V<U^2) &= ...
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1answer
454 views

Calculating the expected value of a weird random variable

Let $Z$ is a standard normal random variable, and, for a fixed $x$, set $$X = \left\{ \begin{array}{l l} Z & \quad \text{if $Z>x$}\\ 0 & \quad \text{otherwise} \end{array} ...
2
votes
1answer
124 views

weak/vague convergence

I am trying to understand 'vague/weak convergence' and need to decide whether or not a measure converges vaguely or weakly. Weak convergence implies vague convergences. However, I don't really ...
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1answer
116 views

Probability mass function

Why is the probability function of a discrete random variable called a probability "mass" function? What does the word "mass" mean here?
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1answer
133 views

Probability that multi-dimensional random variable is positive?

If we have some multi-dimensional normal probability distribution, $X \sim \mathcal{N}(0,\Sigma)$, with zero mean and a known covariance matrix then what is the probability that every component of ...
3
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0answers
92 views

A question about the stability of a property of the normal distribution

Recall that the standard normal distribution can be characterized as the unique standardized (having mean zero and unit variance) distribution $P$ on $\mathbb{R}$ with the property that with $X$, $Y$ ...
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1answer
92 views

Finding conditions on unspecified CDF

Let $F(\theta)$ be an arbitrary, strictly increasing and twice differentiable CDF that is defined on the interval $[0, \overline{\theta}]$, where $\overline{\theta}$ may be infinite. Moreover, let ...
0
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1answer
89 views

how to compute expectation and variance of r.v.(geometric d.)?

how to compute expectation and variance of r.v.(geometric d.) straight forward using definition and using MGF 1/(1-z)^(2) i have tried using the expectation of geometric d. E(x)=x*(N1 C x)(N2 C n-x) ...
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0answers
71 views

Choosing at random with Replacement. pmf? E(x)?

There are $30$ balls in a box. $6$ of them are red, $10$ are white and $14$ are blue. $10$ balls are chosen at random with replacement. Let $X$ be the the number of red balls in the sample. 1) ...
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1answer
616 views

finding Moment generating function and CDF with pmf

The random variable X has the pmf f(-1)=1/4, f(0)=1/8, f(1)=1/4, f(2)=3/8 a) How would you draw the c.d.f with points (-2,F(-2)), (-1,F(-1)), (0,F(0)), (1,F(1)), (2, F(2)), (3,F(3)) b)Write the MGF of ...
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0answers
139 views

Replace a continuous probability distribution with a discrete one

Say one wants to fit a curve $f(x)$ to a set of noisy data points $(x_i, y_i)$. If the error for each point $y_i$ is assumed to be normally distributed with variance $\sigma_i^2$, one wants to find ...
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1answer
376 views

Conditioning a Poisson process on the number of arrivals in a fixed time

Let T1 and T5 be the first and fifth arrivals in a Poisson process with rate lambda. (a) Find the conditional density of T1 given that there are 10 arrivals in the time interval (0,1) (b) Find the ...
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0answers
41 views

canonical form of dyadic martingales

Let $(X_k)_{1\leq k \leq n}$ be a Walsh-Paley $L^p$-martingale (a dyadic martingale) with values in a Banach space $X$. Why does there exist a dyadic martingale $(Y_k)_{1\leq k \leq n}$ with the ...
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2answers
150 views

Conditional Probability: Joint Density and Expectation

Suppose $X$, $Y$ are random variables with joint density: $$f_{X,Y}(x,y)=\frac{e^{-y/2}}{2\pi\sqrt{x(y-x)}}$$ where $0< x< y$. (a) Find the distribution of $Y$. Hint: for ...
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1answer
87 views

Question regarding fair ticket price for a game with 3 dice

I have the following exercise: Consider a game where three (fair) dice are rolled. You pay $1 to enter the game. If there are no sixes for the 3 rolls, you ...
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0answers
77 views

Computing Conditional Probability (Poisson Random Variables)

The number of red and blue cars that go through a given intersection in an hour is a Poisson-distributed random variable with $\lambda$ = 10. What is the probability, conditionally, that at most ...
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0answers
101 views

Method to show independence of discrete random variables

I thought I would be able to do these problems, all I can find are proofs that if random variables are independent that the product of distributions is the joint distribution of the two. I cannot ...
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1answer
111 views

calculating cumulative distribution function $F_X(x)$ of the problem

Suppose we have a street intersection. It consists of a center point and 4 one-mile-long streets from the center. Street 1 points up, street 2 points down, street 3 points to the right and street 4 ...
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0answers
35 views

Testting independence probability given a joint density function [duplicate]

Two continuous random variable A and B have a joint density function f(A,B)(u,v)=m⋅min{a,1-a,b,1−b}for 0≤a≤1 and 0≤b≤1 m is a positive number. For example, f(A,B)(0.3,0.4)=m⋅min{0.3,0.4,0.7,0.6}=0.3m. ...
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2answers
96 views

Conditional Probability with marginal densities

X and Y have the joint denstiy: $f(x,y) = 2x+2y-4xy$ for $0< X< 1$ and $0< Y< 1$ and 0 otherwise. . (a) Find The marginal densities of X and Y I got both marginal densities equal to 1 ...
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1answer
133 views

Conditional Probability Proof

Suppose that X and Y are independent discrete random variables. Let h(x,y) be a bounded two-variable function. Show that: E [h(X,Y)|X = x] = E [h(x,Y )] Explain why this is usually not true if X and ...
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1answer
64 views

Conditional Expectations: Calculating E(Y|X=x) and E(X|Y=y)

X1 and X2 are independent and uniformly distributed on {1,2,...,n}. Let X be the minimum and Y the maximum of X1 and X2. Calculate: (a) E(Y|X=x) (b) E(X|Y=y) I tried making distribution tables for ...
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1answer
147 views

Conditional Probability Given N=n

suppose that $N$ is a Poisson$(μ)$ random variable. Given $N=n$, random variables $X_1,X_2,X_3,\cdots,X_n$ are independent with uniform∼$(0,1)$ distribution. So there are a random number of $X$'s. ...