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

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1answer
28 views

Generating normal random variables with mean and variance [on hold]

I wish to generate normals $X$, $Y$, and $Z$ with the correlation matrix $R$ but with means $0$, $1$, and $2$, as well as variances $4$, $16$, and $25$, respectively. How would you do this?
2
votes
1answer
25 views

If $n\cdot Z_n$ has geometric distribution with parameter $\frac \lambda n$, to what random variable does $Z_n$ converge in distribution to?

If $\frac \lambda n$ is the parameter of the geometric distributions then $$P(nZ_n\le x)=\sum_{i=1}^{[x]} \frac \lambda n \left(1-\frac \lambda n \right)^{i-1} $$ then $$P(Z_n\le x)=P(nZ_n\le ...
0
votes
1answer
16 views

Distribution Theory - bivariate normal distribution

Question: Let X and Y have a bivariate normal distribution with E(X) = 5, E(Y ) = −2, var(X) = 4,var(Y ) = 9, and cov(X, Y ) = −3. U and V are defined as U = 3X + 4Y and V = 5X − 6Y .Determine the ...
0
votes
2answers
31 views

Find the CDF of a function given its PDF

The probability density function of the random variable X is as follows $f_{X}(x) = \begin{cases} 1/4, & \text{if 0 < x < 1} \\ 1/4, & \text{if 2 < x < 4}\\ 1/4, & \text{if 6 ...
-1
votes
0answers
19 views

n is the number of Bernoulli trials with success p. Let $X_i$ be the number of attempts until success. What is the joint probability function? [on hold]

n is the number of Bernoulli trials with success p. Let $X_i$ be the number of attempts until success. What is the joint probability function where $i=1,2$. Well let's figure them out separately ...
0
votes
2answers
20 views

Probability: Reading tables and using the data from them?

Alright probability is not as hard as I imagined yet I strugle with reading tables and applying them to the formulas. The question bellow has a table with 3 rows and 3 collumns and I am asked to see ...
0
votes
1answer
17 views

2 Cards are picked from a deck without replacement. Let X= number of aces, and Y= number of kings. Find the joint probability function.

2 Cards are picked from a deck without replacement. Let X= number of aces, and Y= number of kings. Find the joint probability function (in a 3x3 table) X and Y are both discrete random variables ...
1
vote
1answer
53 views

A silly question regarding a badly written exercise for probability equations. [on hold]

I am doing some exercises and this silly question is bothering me even though I am familiar with probability theory and Bayes law but this question is written in a rather peculiar manner I have no ...
0
votes
2answers
30 views

Central limit theorem on packs of variables

I'm trying to solve the following exercise: Let $\mu$ be a probability distribution on $\mathbb{R}$ having second moment $\sigma^2<\infty$ such that if $X$ and $Y$ are independent with law ...
1
vote
0answers
32 views

Optimal choice for the values of money units

I just thought about how to find the optimal values for money units, given that you want your currency to come in $n$ different values (e.g. Euros come in 7 values for bills and 8 values for coins, so ...
3
votes
0answers
32 views

What is the limit distribution of $\frac{S_{N_n}}{\sigma \sqrt{a_n}}$ as $n\rightarrow \infty$.

Let $X_1, X_2, X_3,...$ be iid with $\mathbb E[X_i]=0$ and $\operatorname{Var}[X_i]=\sigma^2>0$, and let $S_n = \Sigma_{i=1}^{n} X_i$. Let $N_n$ be a sequence of integer valued random variables ...
0
votes
2answers
20 views

Given a Poisson distribution, $2f(0) + f(2) = 2f(1)$, what is the mean of the distribution?

If for a Poisson distribution $2f(0) + f(2) = 2f(1)$, what is the mean of the distribution? I know that for X ~ POI($\lambda$), then the pdf for the random variable X is \begin{equation} ...
0
votes
2answers
12 views

Does this follow a binomial distribution?

Q. Four roads start from a junction. Only one of them leads to a mall. The remaining roads ultimately lead back to the starting point. A person not familiar with these roads wants to try the different ...
1
vote
1answer
41 views

Word Problem: Probability of Y books Fitting in Book Case

Problem: You have $4600$ cm of book case. The thickness of the books are independently distributed with $X \sim N(1.8$ cm$,0.7^2)$. Approximately determine what the probability of ...
2
votes
1answer
22 views

Derivation of the Negative Hypergeometric distribution's expected value using indicator variables

I'm trying to understand how to derive the Negative Hypergeometric's expected value using indicator variables. Note, in the problem below, we are only interested in the expected value before the first ...
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votes
0answers
22 views

Multivariate Gaussian distribution [closed]

I done parts (a) and (b) but I am stuck on part (c). I think the joint distribiution of R^2 is a chi squared r.v but I am not sure
2
votes
1answer
25 views

Probability - Finding the Support of a Joint Transformation

$$ f(x,y) = \left\{ \begin{array}{ll} 12xy(1-y) & \quad 0< x < 1, 0<y<1 \\ 0 & \quad \text{elsewhere} \end{array} \right. $$ ...
-1
votes
0answers
15 views

Finding joint pdf from marginal pdf's

I have $N$ samples $(X_1,\cdots X_N)$ of exponential random variables with parameter 1. The samples are ordered such that $X_N \geq X_{N-1} \geq \cdots X_1$. I know the individual pdf's of $X_N$ and ...
0
votes
1answer
16 views

How to prove $2d_H(\{XY\},\{X\}\{Y\})^2 \le I(X,Y)$?

Let $X$ and $Y$ be discrete random variables. Denote the joint distribution of $X$ and $Y$ by $\{XY\}$ and their marginal distributions by $\{X\}$ and $\{Y\}$. Let $\{X\}\{Y\}$ denote the product of ...
0
votes
1answer
22 views

Let $X$ have pdf $f(x) = e^{-x}$. Find the pdf of the integer part of $X$.

A continuous random variable has a pdf defined by $$f(x) = e^{-x} , x > 0.$$ The discrete random variable $Y$ is defined as the integer part of $X$, that is the largest integer less than or ...
-1
votes
0answers
30 views

Generating a Uniform R.V with specified correlation [closed]

I understand that it involves copulas, but I'm looking for a specific methodology for a specific correlation. I want to generate $U$ and $V$, random variables that are $~Uniform (0,1)$ with ...
0
votes
1answer
19 views

Calculating the magnitude of random numbers from normal distribution

Statement: Given an array of 80 random numbers, normally distributed between 0 and 1, we can expect that the numbers are all of similar magnitude, on the order of $80^{-1/2} \approx 0.1$. Question: ...
0
votes
2answers
21 views

Show that this MC is ergodic?

Suppose I have a Markov Chain with States, $S = {1,2,3,4}$ and a PTM given by $P =$ $\begin{pmatrix} .25 & .25 & .25 & .25 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\\ 0 ...
-1
votes
1answer
43 views

Finding the conditional probability

enter image description here Let $(X,Y)$ be a two-dimensional stochastic vector with density $$ f_{X,Y}(x,y) = \begin{cases} \dfrac{e^{-y}} y & \text{if } 0<x<y, \\[4pt] \,\,\,\, 0 & ...
0
votes
1answer
33 views

Heavy tailed discrete distribution infinite mean

I'm looking for an example of a discrete distribution with infinite mean $f_n = P(X = n)$ for $n=1,2..$ such that the sequence $r_n = \sum\limits_{k=n+1}^{\infty}f_k$ satisfies the relation $$r_n = ...
0
votes
0answers
16 views

Similarity measure for uncertain point sets

Imagine that we have two sets of points $M=(x_{1}, x_{2},...x_m)$ and $N=(x_{1}, x_{2},...x_n)$. These are actually lists of $x$, $y$ (and $z$) in 2D (or 3D) space so $x_i\in\ \mathbb{R}^2$ (or ...
0
votes
0answers
22 views

Distribution of Double Stochastic Integral

Assume that $f(s)$ is a $C^\infty$ univariate function and that $\{ (W_{1,t}, W_{2,t})\}_{t \geq 0}$ is a two-dimensional, correlated Wiener process. Then, does the random variable $X_T \equiv ...
0
votes
1answer
24 views

Convergence a.s. and convergence in $L^1$ don't imply each other [closed]

I'm trying to get two examples that convergence a.s. and convergence in $L^1$ don't imply each other. Now, I only know the examples that convergence a.s can't implied by convergence in probability, ...
0
votes
1answer
24 views

Uniform distribution on $\{\frac{i}{n}: 1 \leq i \leq n \}$

I am trying to do a problem in which there is a type of distribution I am not familiar with, the exercise says: Find the limit in distribution of the sequence $(Z_n)_{n \in \mathbb N}$, where for each ...
0
votes
1answer
16 views

GRE Quantitative problem on distributions

I was doing some problems on this .Can some one please help me with the following: Here the given answer is that quantity B is grater than Quantity A. How is this obtained? Do we know anything ...
0
votes
0answers
37 views

Difference of dependent central Chi-Square random variables with 2 degrees of freedom

Suppose we have $X$ and $Y$, both are dependent and complex Gaussian random variables with zero means and the same variance $\sigma^2$. The real and imaginary parts of $X$ and $Y$ are independent, ...
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0answers
94 views

Mathematics Homework 2 Question 8d :What is the probability you and your partner are now able to meet the new deadline? [closed]

You are working on a programming project with your partner for a computer science course. The project is due in 48 hours. Together, you are to produce a computer program and each of you are assigned ...
1
vote
1answer
29 views

Probability that X is greater than the mode of X?

How do I solve this problem: Let X be a continuous random variable with density function \begin{equation} f(x) = \begin{cases} \hfill ax^2e^{-10x} \hfill & \text{for x $\geq$ 0} \\ ...
0
votes
0answers
23 views

Expectation of the trace of An Inverse Wishart random matrix

Assume that Σ~IW(Α,Τ,Ν) with T>N. Σ,Α are positive definite symmetric matrices and IW stands for Inverse Wishart. What is the following Expectation? $E(tr((Σ^{-1}B)$=? let B=bb', where b is a ...
3
votes
2answers
41 views

Finding the median given the PDF $f(x) = cx^2$.

I'm new to stats and I facing problems in finding the median of a PDF. I have to find the median of this PDF $$ f(x) = \begin{cases} cx^2, & \text{if 0 $\le$ $x$ $\le$ 3} \\ 0, & ...
0
votes
2answers
34 views

Computing the probability of waiting someone - Uniform distribution

I have the following problem and I having trouble in finding it solution. I need a hint. The problem: Two people arranged to meet between 12:00 and 13:00. The arriving time of each one is i.i.d. and ...
1
vote
1answer
43 views

Flip $n$ coins, discard tails, and continue until $k$ heads remain

Consider the following game: $n$ participants have a fair coin each, on a given round, the not already discarded participants flip their coins, those who flip a tail are discarded from the game, the ...
0
votes
1answer
20 views

Exponential distribution of 3 independent random variables

Let X be the lifespan of an electrical component in days. X is a random variable that has an exponential distribution with parameter $\frac{1}{12}$. All component lifespans are independent. ...
3
votes
1answer
48 views

Showing 2 Distributions are the Same

Let $X_1, X_2, \dots$ be i.i.d. exponentially distributed RVs. For $n = 1,2,\dots$ consider: $Y_n = \max(X_1, \dots X_2)$ $U_n = \sum_{i=1}^{n}\frac{X_i}{i}$ Show that $Y_n$ and $U_n$ have the same ...
1
vote
2answers
53 views

How do I find the cdf of $X_1 + X_2$?

$X_1$ uniform $(0,1)$ and $X_2$ uniform $(0,2)$ $$ \begin{cases} f(x_1,x_2) = \frac{1}{2}, &\quad \mbox{for} \ 0<x_1<1, 0<x_2<2 \\ 0, & \quad \mbox{otherwise} \end{cases} $$ ...
0
votes
0answers
17 views

why the uniform distirbution function F(X) equal to 1 when the X is a fixed value?

I have the following quetion: Let X be a continuous random variable with distribution function $F_X(x)$ and density function $ f_X(x)$. Consider the random variable Y dened by $Y = X $ if $X < a$ ...
0
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0answers
48 views

How to calculate the probability that $X_n$ is not the largest observation in the sample?

I am trying to solve the following problem: Let $X_1,\dots, X_n$, where $n > 4$, be independent random variables such that $X_i ∼ N(i, i)$ for $i = 1, \dots, n$. Let $\bar{X} = ...
2
votes
0answers
52 views

How to Find Expected Value from a Joint Distribution?

I am trying to solve the following problem: Let $X$ be a random variable from a contaminated normal distribution. That is, let $B ∼\text{Bernoulli}(p).$ Then $X|B = 0 ∼ N(µ, τ^2 )$ and $X|B = 1 ...
1
vote
1answer
15 views

Expectation of Covariance Matrix for MVN, got answer 0 with Cov operation?

Let $x_1, \ldots x_n$ be iid realizations of a $p$-dimensional random column vector $X= (X_1, \ldots, X_p)$ such that $X \sim N_p ( \mu ,\Sigma)$. We can show that $$ \hat{\mu} = \frac{1}{n} ...
2
votes
2answers
63 views

Distribution of $Y = \frac{1}{X} + X$

As in the title, I am trying to get the distribution of $Y = \frac{1}{X} + X$, where $X$ is uniformly distributed on the interval (0, 1). I am having trouble seeing how to move beyond what (very ...
0
votes
1answer
23 views

How to Find Probability mass function from card problem?

This is a problem with Probability. Question is like this: Recall that there are 52 cards in a standard deck, with 4 cards for each denomination. Suppose that you flip over the top 4 cards. This ...
0
votes
1answer
8 views

Lognormal distribution inverse equivalent

In Lognormal distribution if the random variable X is log-normally distributed, then Y = ln(X) has a normal distribution. Is there inverse equivalent to lognormal distribution where Y = exp(X) has a ...
2
votes
1answer
30 views

Why the probability distribution of a uniform random variable is the Lebesgue measure?

Consider the random variable $X$ defined on the probability space $(\Omega, \mathcal{F}, P)$ distributed as a uniform on $[0,1]$. The probability distribution function of $X$ is defined as a map $$ ...
6
votes
4answers
137 views

Expected number of rolls for an unfair die to get all possibile values at least once

Suppose that we have a 6-sided unfair dice, where rolling a 1 is twice as likely as rolling any other number, and the other numbers have the same likelihood. What is the expected number of rolls to ...
0
votes
2answers
41 views

How to find the probability density function?

John wants to buy a bag of candies The price of the jar is $10$\$ and the candies cost $25$\$ for 100 grams The weight of the candies in the jar has continuous uniform distribution on the ...