Questions about maps from a probability space to a measure space which are measurable.
1
vote
1answer
58 views
Obtaining the pdf of the sum of two iid random variables with pdf $f_{X}(x)= \sqrt{\frac{1}{2\pi x}e^{-x/2}}$
X1, and X2 are independent with the pdf: $f_{X}(x)= \sqrt{\frac{1}{2\pi x}e^{-x/2}}$ defined for x>0
Y=X1+X2
What is the pdf of Y?
This is what I did so far:
$$f_{Y}(y) = \int_{0}^{\infty } ...
2
votes
1answer
52 views
Missing assumption? (Convergence of random variables and characteristic functions)
Here are two exercises from my Probability book ("Probabilidade: um curso em nível intermediário", by Barry James). (I have translated them from Portuguese to English and modified them a bit.)
...
3
votes
1answer
72 views
I.i.d. Geometric Random Variables
Let $ X_{1},X_{2},X_{3},\ldots $ be i.i.d. geometric random variables, where $ \mathbf{Pr}(X_{i} = k) = p(1 - p)^{k - 1} $.
Define $ \displaystyle Z_{r} \stackrel{\text{def}}{=} \sum_{i=1}^{r} X_{i} ...
1
vote
2answers
204 views
Expected value of the inner product of two random vectors
$X=[x_1, x_2,...x_n] Y = [y_1, y_2,...y_n]$
If $x_i, y_i$ are both random variables with
$P(x=1) = .5$
$ P(x=2) = .5 $
$P(y=1)=.5$
$P(y=2)=.5$
How would I find the expected value of the inner ...
1
vote
1answer
81 views
Can we say that 2 random variables $(X, Y)$ are independent if $P(X \mid Y=y)$ is a function without $y$
Discrete Case:
Random variable $X, Y$ are independent if $P(X \mid Y=y)$ is a function (which can be deemed as pmf for random variable $X \mid Y$) without $y$.
Continuous Case:
Random variable $X, ...
0
votes
0answers
29 views
how to compare correlation between random variables?
Suppose I have a random variable, S(k) for starting date of callable bonds, M(k) for the maturity date of the bonds, and C(k) for the called date of the bonds.
$$S(k) < C(k) < M(k)$$
C(k) is ...
2
votes
2answers
54 views
What is the exact nature of messages in belief propagation on a factor graph?
The recursive nature of the definition of messages in belief propagation on a factor graph makes it quite confusing for me what the messages exactly correspond to: ...
1
vote
1answer
90 views
Markov's Inequality(?) Example
Let $c>0$ and $X$ be a real random variable such that for any $\lambda\in\mathbb R$:
$\displaystyle \mathbb E\left(e^{\lambda X}\right)\leq e^{c\frac{\lambda^2}{4}}$.
Prove that, for any $\delta ...
4
votes
2answers
301 views
Expectation of the min of two independent random variables?
How do you compute the minimum of two independent random variables in the general case ?
In the particular case there would be two uniforms variables with difference support, how should one proceed ?
...
3
votes
2answers
43 views
$X,Y$ are I.I.D R.V and $P(X+Y\in(0,2))=1$. How to prove $P(X<0)=P(X>1)=0$?
I am trying to solve an old exam in my probability course and I stumbled
on a question that I don't have an idea on how to start:
Let $X,Y$ are I.I.D R.V and denote $U=X+Y$. Assume that
...
2
votes
1answer
117 views
Computing the Expectation of the Square of a Random Variable: $ \text{E}[X^{2}] $.
What is the rule for computing $ \text{E}[X^{2}] $, where $ \text{E} $ is the expectation operator and $ X $ is a random variable?
Let $ S $ be a sample space, and let $ p(x) $ denote the probability ...
2
votes
2answers
116 views
Find the distribution of X, EX, and VarX.
Suppose that the random variable $X$ is uniformly distributed symmetrically around zero, but in such a way that the parameter is uniform on $(0,1)$; that is, suppose that $$X\mid A=a\in U(-a,a) \text{ ...
1
vote
2answers
62 views
How to find condtional expectation given 2 random variables with joint density
Given two random variables, X and Y with joint density
$$ f(x,y)= c*cosx$$for $0<y<x<\pi/2$ (and zero otherwise),
how do you find the conditional expectation $E(Y|X=x)$?
A general method ...
1
vote
0answers
18 views
Estimating the likelihood of independence of two discrete variables using the co-occurrence count matrix.
I have some data about users from different regions visiting different directories of some website. Aggregating that data I get the co-occurrence frequency matrix (for regions and directories). Now I ...
1
vote
2answers
122 views
Expectation of inverse of sum of random variables
Let $X_i$'s ($i=1,..,n$) be i.i.d. random variables with mean $\mu$ and variance $\sigma^2$.
Is there a method that can be used to compute $\mathbb{E}[1/(X_1+...+X_n)]$?
0
votes
2answers
92 views
Probability distribution of the product of two dependent variables
The following joint probability distribution exists for two dependent random variables
...
2
votes
1answer
95 views
Partial derivative respect to random variable - How does one compute this?
CLARIFICATION:
If someone could please help me understand the following: When examining the expected value in this specific situation, how is the distribution of $\theta$ relevant? What ...
0
votes
1answer
106 views
Discerning The Set Of Values For A Random Variable
The question is:
For each random variable defined here, describe the set of
possible values for the variable, and state whether the variable is discrete.
a. $X=$the number of unbroken eggs ...
3
votes
3answers
224 views
Repeated convolution of probability distributions
Question
Let $$S_k=\sum_{i=1}^k X_i$$ be the sum of $k$ independent random variables. I am interested in closed-form expressions of the pdf of $S_k$.
In general, the pdf is given by the $k$-fold ...
2
votes
1answer
81 views
Proof of a Martingale with respect to Filtration
I'm having a problem with stochastic analysis, needed in my Advanced Mathematical Finance Course. We have:
Let $(\zeta _k)_{k≥1}$ be a sequence of independent random variables with the expected value ...
0
votes
2answers
72 views
Random Variable
I was wondering if I correctly understand what a random variable is. Is a random variable's domain the set of numbers that are reasonable, when considering how the random variable is defined. For ...
0
votes
1answer
47 views
Prove that $(X_1 X_2\cdots X_n)^{1/n} \to c$ as $n\to\infty$ where $c$ is a constant
This is a assignment question, a part of my homework. So I need hints to start towards the solution. I was thinking that under the given conditions of the problem the random variables $\log X_1$, ...
0
votes
1answer
56 views
expectation of a function of two dependent random variables
I have a formula that I believe it's right but I do not know how to prove it. Could you please give me some arguments or references to show that.
Let $\{X_t,t\geq0\}$ be a gamma process. Let $\tau_M$ ...
1
vote
2answers
55 views
Sum of boolean random variables with small correlations.
Suppose $X_1,\dots,X_N$ are $0/1$-random variables and the following information is known:
$$
\Pr[X_i=1]\geq 1/n, ~ \Pr[X_iX_j=1]\leq 100/n^2.
$$
Let $X=\sum_{i=1}^N X_i$. Assume $N=n^{100}$ for ...
1
vote
1answer
24 views
N arcs on a ring, which are either blue or red, find expectation and variance of number of red arcs using indicators.
There are n distinct points marked on the ring, each of which is either blue or red
with equal probabilities independently of each other. These n points divide the ring into n arcs.
If an arc has both ...
1
vote
0answers
82 views
How to calculate the highest/smallest possible value of the variance of two random variables mean?
Two random variables $X$ and $Y$ have a common expected value $E(s)$ and a common variance $Var(s)$.
What's the highest possible value of the variance of their mean, $var
((x+y)/2)$?
What's the ...
0
votes
1answer
138 views
sum of two random variables
Can any of you help me?
I have some problem with this exercise of "Probability and Statistics" :
Calculate the probability density function (PDF) of $Z=X+Y$
where $Y$ is discrete random ...
2
votes
2answers
44 views
Expected value of a random variable with minimun
Let $M$ be a positive integer, and $X$ distributed $Geo (p)$. Calculate the expected value of $Y=min(X,M)$.
0
votes
3answers
42 views
X random variable with binomial distribution. calc E(exp(x))
i have problems with calulating
$$E(e ^X)$$
X has a binomialdistributon with parameters n,p. E is the expectation.
My approach $$ E(e^X)= \sum_{k=0}^n e^k \binom{n}{k}p^k (1-p)^{n-k} = ... ?$$
1
vote
2answers
40 views
I want to find the real number c for which we have: $P(X<c)=P(Y<c)$
Consider two normal random variables $X$ and $Y$:
$$X\sim N(m_1,s_1), \qquad
Y\sim N(m_2,s_2)$$
I want to find the real number $c$ for which we have:
$$P(X<c)=P(Y<c)$$
3
votes
2answers
50 views
Independence is preserved under taking almost sure limits
I´m not sure why this theorem is right, how can i prove it?
Let $X_1,X_2.... $ and $Y_1,Y_2...$ be random variables such that $X_n,Y_n$ are independent for every $n∈\mathbb N$ and such that X, resp ...
1
vote
1answer
111 views
If $X$ and $f(X)$ are independent, then $f(X)$ is almost surely constant
Reading some exam material, I found this property:
Let $f :\mathbb{R}\rightarrow\mathbb{R} $ a measurable function. If $X$ and $f(X)$ are independent, then $f(X)$ is almost surely constant.
Most of ...
1
vote
1answer
53 views
Uniform integrability for a single random variable
Let $X$ be a random variable. Are the following three equivalent?
$X \in L^1$, i.e. $E |X| < \infty$.
$X$ is uniformly integrable. That is, if given $\epsilon>0$, there exists $K\in[0,\infty)$ ...
2
votes
1answer
88 views
Mean of iid random variables, problem understanding a passage in a paper
I am presently reading and studying a paper (page 11 of this document) regarding epidemiology and disease spread over a network of contacts. I am having problems understanding a passage.
Some ...
0
votes
2answers
71 views
What is this probability?
Suppose that we have $n$ independent normal random variables, $X_1,...,X_n$. What is the probability that normal random variable $Y$ is less than all normal random variables $X_1,...,X_n$ ? These ...
0
votes
2answers
103 views
if X and Y are independent Normal random variables, what would be the probability that X is less than Y?
How can I find this probability $P(X<Y)$ ? knowing that X and Y are independent Normal random variables.
$X$~$N(m_1 ,v_1)$ (with mean m1 and variance v1)
$Y$~$N(m_2 ,v_2)$
I know that
$$ \Pr ...
4
votes
2answers
79 views
Where does this equality about expectation of random variables come from?
In order to prove this Lemma in my course about Probability :
Let $X=(X_1,\dots ,X_p)$ be a gaussian random variable such that $\mathbb{E}[X_j]=0$ for all $j=1,\dots,p$. Then ...
0
votes
0answers
33 views
Correlation Coefficient dealing with discretely distributed variables
I'm a bit stuck on this practice problem I have for my HS business stats class. I'd appreciate any help to get the solutions. Thank you.
Exercise #22: Let X and Y be discretely distributed random ...
0
votes
1answer
291 views
Division of two random variables of uniform distributions
Having X ~ Uniform(0,1), Y ~ Uniform(1,3) independent what's the pdf of Z = X/Y.
This means I can write the PDFs as follows $$f_X(x) = 1$$ for $ x \in \left(0,1\right)$ and 0 otherwise
$$f_Y(y) = ...
1
vote
3answers
497 views
Correlation between three variables question
I was asked this question regarding correlation recently, and although it seems intuitive, I still haven't worked out the answer satisfactorily. I hope you can help me out with this seemingly simple ...
0
votes
1answer
74 views
Can I use $P(x_1+x_2+…+x_n<k)$ to calculate $P(y_1+x_2+…+x_n<k)$?
Suppose that we have $n$ independent Bernoulli random variables, $x_1,\ldots,x_n$ such that each $x_i$ takes value 1 with success probability $p_i$ and value $0$ with failure probability $1-p_i$ ...
4
votes
1answer
62 views
Finding E(x) from E(ln(x)).
Say you have $\operatorname{E}[\ln(x)]=\mu$, is there a way to find $\operatorname{E}[x]$?
This seems like a really simple question but I can't figure it out. Any help would be appreciated.
-1
votes
1answer
112 views
When to use likelihood ratio test? [closed]
I have a few questions regarding the use of likelihood ratio test in a logistic regression model.
Suppose we have a logistic regression model like this:
...
2
votes
2answers
32 views
generating random number in the range +/-(n to n+x)
I want to generate a random number that falls in the range 50 to 100 and -50 to -100
I am now using the following formula to achieve this:
...
0
votes
0answers
21 views
Combining function estimates
I have two piecewise linear estimates for two different realisations of the same random variable.
What are some techniques that I could use to combine these function estimates into a single ...
2
votes
1answer
130 views
If $X$ is uniform on the interval $[0, 2]$ , find the PDF of $X^2-2X$
If $X$ is uniform on the interval $[0, 2]$ find the PDF of $Y =
X^2-2X$.
I solved for $X$ and got $x=1 + \sqrt{1+y}$ or $x=1 - \sqrt{1+y}$. Not sure what to do from here.
0
votes
1answer
87 views
Give examples of not independent random variables which are uniform s.t. $P(X+Y=1)=1$ and $X+Y$ which is uniform in the interval $[0,2]$
Give examples of (not independent) random variables $X$ and
$Y$, both of which are uniform in the interval $[0, 1]$ and such that
$\mathbb{P}(X + Y = 1) = 1$
$X + Y$ is uniform in the interval $[0, ...
0
votes
2answers
215 views
Find the density function of the random variable $Z=X+Y$
Let $X$ and $Y$ be independent and uniformly distributed random variable on the intervals $[0,3]$ and $[0,1]$, respectively. Find the density function of the random variable $Z=X+Y$.
I find ...
1
vote
3answers
77 views
$\int_0^\infty 2xe^{-2x} \: dx=Γ(2)2(1/2)^2$ how to find this result of the integral?
$$\int_0^\infty 2xe^{-2x} \: dx=Γ(2)2(1/2)^2$$
I don't understand. How can we write this? Please can you explain this clearly?
1
vote
0answers
51 views
Measuring a mean variance for some number of objects observed per trial for multiple trials
I'm running a bunch of trials, $T$, and the outcome of each trial is some number of objects $k_i$ for $i = [1, T]$. I would like to say something about the average "spread" in terms of the number of ...
