1
vote
2answers
161 views

We have $100$ balls (numbered $1,2,\cdots,100$) and $50$ boxes (numbered $1,2,\cdots,50$).

We put the balls, independently and randomly, into the boxes. Let $X$ be the number of boxes that receive no balls. Find $E(X) =$ ? Notes: I am a little confused as to what type of probability ...
0
votes
1answer
23 views

Distributions of local times of a single excursion of 1D random walk

Consider Simple Random Walk in one dimensions, starting from $x \in \mathbb{Z}^+$. The walker jumps to the right with probability $p$ and to the left with probability $1-p$. Assume $p \leq ...
2
votes
1answer
59 views

Example of non continuous random variable with continuous CDF

Can someone provide an example of $X$ being a non-continuous random variable with continuous cumulative distribution function? For instance: $X$ is discrete if it takes (at most) a countable number ...
2
votes
1answer
45 views

Expectation related to Normal distribution and its density

Given $\sigma^2>0$. Let $Z\sim N(0,1)$ and $\Phi$ be the cumulative standard normal with density function $\phi$. I wish to show that $$ E\left(\frac{Z^2}{[\phi(\sigma Z)]^2}\Phi(\sigma ...
1
vote
2answers
54 views

let $X$ be a standard Gaussian random variable. Show that $(X,X)$ is not absolutely continuous.

i'm trying to understand a proof of the following statement: let $X$ be a standard Gaussian random variable. Show that $(X,X)$ is not absolutely continuous. I'll write down the proof in such a ...
0
votes
0answers
28 views

Problem calculating the average power of a vector?

I am calculating the average power of a vector. I would like to compare the final expression with the simulation. However, they are not equal. Please help me to point out which steps are wrong. Thank ...
1
vote
1answer
40 views

probability of getting 5 calls in 5 minutes

Phone calls arrive at the rate of 48 per hour at the reservation desk for Regional Airways. 1 a. Compute the probability of receiving three calls in a 5-minute interval of time. b. Compute the ...
0
votes
1answer
32 views

Box-Muller method for correlated normals

The standard Box-Muller method produces two independent normal variables given two uniform ones. Is it possible to extend the method such that given a correlation coefficient $\rho\in[-1, 1]$ and two ...
0
votes
2answers
32 views

On the definition of a random variables

Let $(O,F,P)$ be a probability space. That is $O$ is a set, $F$ is a $\sigma$-algebra of subsets of $O$ and $P$ is a probability measure. Consider a function $f:O\to\mathbb R$. Would we call $f$ a ...
0
votes
1answer
60 views

Operations on Random Variables

It is known that the equivalent resistance of a parallel combination of two resistors is equal to \begin{align*} R = \frac{R_1R_2}{R_1+R_2} \end{align*} which could be also written as ...
3
votes
1answer
56 views

Prove Number of Arrivals $N(s)$ up to time $s$ follows $\mathrm{Poisson}(\lambda s)$ Distribution

This comes from my self-study of Durrett's "Essentials of Stochastic Processes" book, page 97. Definition Let $\tau_1,\tau_2,\ldots$ be independent $\mathrm{exponential}(\lambda)$ random variables. ...
0
votes
0answers
50 views

More on transformations and convolution on continuous random variables

This question is related to my last question but I've done some more exploring and then got stuck again. I decided to modify the problem a little bit and use a transformation of a random variable that ...
0
votes
0answers
25 views

Distributions with infinity variance.

I'm looking for a list (or something like that) of distributions with infinity variance (or infinity second moment), like non-gaussian Stable Distributions. I have an important warning: Some ...
0
votes
2answers
37 views

Finding mean from die probability

Example 4.4.5: Suppose that there is a 6-sided die that is weighted in such a way that each time the die is rolled, the probabilities of rolling any of the numbers from 1 to 5 are all equal, ...
3
votes
1answer
71 views

[Probability]need help to understand the following expression

So assume $Y$ and $X$ are exponentially distributed with parameters $y_1$, and $x_1$ respecitively. assume c is a constant. I am having huge trouble to understand the integration of the following ...
0
votes
0answers
19 views

Kullback-Leibler or Jensen-Shannon divergence between two distributions.

i would like to understand well what Kullback-Leibler or Jensen-Shannon divergence between two distributions will tels us about two distribution,for instance let us consider following code ...
0
votes
1answer
45 views

Summing dependent random variables with unknown joint cdf

Suppose that X_1, X_2,... X_5000 are discrete and dependent non-identically distributed random variables, whose marginal distributions are known, but whose joint distribution is not known. Is there ...
1
vote
3answers
74 views

Finding expected value??

In a game, a man wins a rupee for a six and loses a rupee for any other number when a fair die is thrown. The man decided to throw a die thrice but to quit as and when he gets a six. Find the ...
1
vote
0answers
30 views

Probability: NEED HELP to Understand with the follow [duplicate]

I need help to understand the probability derviation of a paper. Please help me. For the following, please only treat $|h_{R,B}|^2$ and $|h_{A,R}|^2$ as random variables (other parameters can be ...
1
vote
1answer
26 views

Product of 2 random variables:domain of integration

I am trying to compute the PDF of the product of two ind. random variables: $Z=XY$, where $0\leq x \leq d$ and $ 0\leq y \leq 1 $. ($0<d<1$) I found this formula : $ f_Z(z) = ...
0
votes
1answer
53 views

Prove that E(X) exists if and only if E(|X|) exists.

I found this theorem in a book, but there is no proof there: If X is a random variable, then Prove that E(X) exists if and only if E(|X|) exists. where $E(X)$ is the expected value of $X$ I know ...
1
vote
1answer
21 views

Creating a bivariate distribution from two independent variables

If you have two random variables that are independent say $X\sim f_X (vars)$ and $Y \sim f_Y (vars)$. Is this a way to produce a bivariate distribution $f_{(X,Y)}$? $f_{(X,Y)} = p(X=x \cap Y=y) = ...
1
vote
0answers
24 views

Independent of random variables.

When reading Shiryaev's Probability. In the chapter 1, section 4. problem 11: Show that the random variables $\xi_1,\cdots,\xi_n$ are independent if and only if ...
1
vote
1answer
21 views

Sampling on Axis-Aligned Spherical Quad

Given spherical coordinates on a unit sphere, imagine a spherical quad defined by two ranges $[\phi_0,\phi_1]$ and $[\theta_0,\theta_1]$. If you have a globe, for example, the grid formed by the ...
0
votes
4answers
38 views

Generate random numbers in a random fashion

How can I generate 9 random numbers between 1 to 9,without repetition, one after another. Its like: Lets assume that the first random number generated is 4, then the next random number has to be in ...
0
votes
1answer
32 views

Normalizing constants for Extreme value distributions

I have a question regarding the normalizing constants $\mu$ and $\sigma$ that appear in the following problem. Let the random variable $Y_n$ be $Y_n=max(a_1,a_{2},\cdots, a_n)$ and $X_{n}$ be ...
1
vote
2answers
88 views

Expectation of random variables ratio

Let $X_1, X_2, \dots, X_n$ be $n$ positive iid random variables. Then show that $$E\left(\frac{\sum_{j=1}^k X_j}{\sum_{i=1}^{n} X_i}\right) = \frac{k}{n}.$$ Because of the linearlity of the ...
3
votes
1answer
88 views

If $X_{i}$ are I.I.D and $n^{-1}\sum_{i=1}^{n}X_{i}$ converges a.s/in-distribution to a constant $a$ is it true that $a=\mathbb{E}[X_{i}] $?

The question itself is in the title. It is immediate by the strong law of large numbers that if $X_{i}$ had a finite first moment then we would have a.e convergence (and thus in probability and in ...
6
votes
1answer
52 views

Finding tight upper/lower bounds for $\mathbb{E}[\frac{1}{1+X^{2}}]$ where $X$ is a RV with $\mathbb{E}[X]=0$ and $\mbox{Var}(X)=\nu<\infty $

The question is pretty much in the title. My first thought was using Jensen's inquality to get some sort of lower bound. Since $\frac{1}{1+x^{2}}$ is convex on ...
0
votes
1answer
25 views

$X_i\sim \operatorname{Ber}(\theta_i)$ and $Y = \sum X_i$, sum of independent Bernoulli trials with different $\theta_i$. Find $\operatorname{Var}(Y)$

$X_i\sim \operatorname{Ber}(\theta_i)$ and $Y = \sum_{i=1}^n X_i$, sum of independent Bernoulli trials with different $\theta_i$. So this is something like we have a collection of $n$ possibly ...
0
votes
1answer
39 views

Show that $\Pr(S_N\in A\mid N=n)=\Pr(S_n\in A)$

Let $X_1,.\ldots,X_n$ be i.i.d. random variables and $N$ be a positive integer-valued random variable, which is independent from the sequence. If $S_n=\displaystyle\sum\limits_{i=1}^{n} X_i$, then ...
0
votes
2answers
51 views

X and Y are independent random variables and their distributions are..

X and Y are independent random variables and their distributions are.. $P(X=1) = 0.1 $ $P(X=2) = 0.2$ $P(X=3) = 0.3 $ $P(X=4) = 0.4 $ $P(Y=4) = 0.4 $ $P(Y=2) = 0.3$ $P(Y=3) = 0.2 $ $P(Y=4) = 0.1$ I ...
0
votes
1answer
39 views

Compute a conditional probability of normal random variable

Suppose $X, T$ are continuous random variables, and $X \sim \mathcal{N}(0, 1)$, $T$ have density function $f_T$. (But $X,T$ do not have joint density) Is there any way to compute the following ...
0
votes
1answer
24 views

$X $ and $Y$ are continuous $RVs$, such that$ f(x,y) = 2, 0\leq x\leq 1, 0\leq y\leq 1, 0\leq x+y\leq 1$

X and Y are continuous RVs, such that $f(x,y) = 2, 0\leq x\leq 1, 0\leq y\leq 1, 0\leq x+y\leq 1$ I'm trying to find $P(x<1/2,y>1/2)$. So i'm integrating from $\dfrac{1}{2}$ to $1$ for $y$ ...
1
vote
0answers
1k views

Problem with the expectation of a maximum of independent gamma distributed random variables

Having a problem with the expectation of the maximum among $n$ independent random variables $ X_1, X_2 \dots X_n$ all ~ the same class of distributions but not necessarily the same mean and other ...
0
votes
1answer
33 views

Find the PDF of Y given Y=X(2-X) and X's PDF

Suppose that the continuous random variable $X$ has probability density function $f_X(x)=\begin{cases}\frac{1}{2}x & \text{if } 0<x<2\\0&\text{otherwise}\end{cases}$ Let $Y=X(2-X)$. ...
0
votes
1answer
182 views

If $X' \leq X$ almost surely, is it possible to prove that $P(X = s) \geq P(X' = s)$?

With respect to my previous question, let us define $X$ as: $$ X = \sum_j^r l^j Y^j, $$ where $l^j \geq 0$ and $Y^j$, $j = 1, \ldots, r$ is a Bernoulli random variable which takes on values in ...
0
votes
1answer
72 views

Is this true: probability independent from i?

We have a set of i.i.d. random variable $X_i$ with some discrete distribution. Further we have a random variable Y, Independent from $X_i$ with a Binomial Distribution Bin(n,p). Now we are ...
0
votes
1answer
21 views

Uniform Spinner is spun twice..

A fair uniform spinner is spun twice, and the results V and W are noted. V and W are uniform RVs ∼U[0,1]. I'm trying to answer the question what is the joint pdf for V and W. I know that I have to ...
1
vote
2answers
58 views

Expected Value of Intersection of two Binomial Random Variables

Ok the problem is as follows: (I am currently studying for my first actuary exam so this isn't a specific hw question! Just trying to figure it out!) A and B will take the same 10-question exam. ...
2
votes
2answers
57 views

Find the distribution of random variable $XY+X+Y+1$

X and Y are iid with density $f(x)=\frac{1}{(1+x)^2}I_{(0,\infty)}$. Find $P(Z\le z)$ where $Z=XY+X+Y+1$ my effort: $P(Z\le z)=P((x+1)(y+1)\le z)=P(x\le ...
1
vote
0answers
49 views

Conditioning on function of random variable and random variable itself

Suppose that $Y_{i}\in\{0,1\}$ is a binary variable, and $X_{i}$ is some random vector in $\mathbb{R}^{d}$ . Why can we say the following: \begin{eqnarray*} ...
2
votes
1answer
42 views

Let X be an exponential random variable with P(X < 1/3) = 0.75. What is E(X)?

Let X be an exponential random variable with P(X < 1/3) = 0.75. What is E(X)? I don't get this. Please help.
0
votes
0answers
37 views

Law of large numbers with random weights

Let $\mu_i$ be i.i.d. RVs with mean zero, and let $a_i$ be random weights that are not independent and are not identically distributed, $i=1,...,N$. $\mu_i$ is orthogonal to $a_j\;\forall j$. Is ...
2
votes
1answer
36 views

Why is this distribution Poissonian?

Do this experiment. Draw 10000 random number in $[0,1]$ according to the uniform distribution. Order them in the increasing order. The difference between two neighbouring numbers follows a Poisson ...
0
votes
1answer
32 views

Find the distribution function F(y) [closed]

Can someone show me how to do this problem? Don't know how to format my work here.
2
votes
1answer
32 views

Convergence almost surely and B-C lemmas

Showing the expectation is straightforward. I am not sure how to use the Borel-Cantelli lemmas to show the almost surely part.
2
votes
2answers
52 views

Computing cov of 2 binomial random variables

we drop a normal cube 20 times. X - is the number of even values Y - is the number of times the cube landed on 3. As much as I understand: $$X\sim B(20, \frac{1}{2}) \\Y \sim B(20, \frac{1}{6} )$$ ...
1
vote
3answers
36 views

Relationship between Binomial and Bernoulli?

How should I understand the difference or relationship between Binomial and Bernoulli distribution?
1
vote
0answers
25 views

Random sampling and i.i.d.

Can you help me to clarify the following concepts by stating whether what I have written below is right or wrong? -random sampling: units are drawn from the population with a known probability of ...