Questions about maps from a probability space to a measure space which are measurable.

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0
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2answers
24 views

Which probability is greater, given minimal info

which probability is greater, given that $X$ and $Y$ are independant, positive random variables? There is also the option that it's impossible to know as we don't have enough information. I'd ...
3
votes
2answers
52 views

Determine whether a random binary sequence was generated by human or natural process

Given a binary sequence, how can I calculate the quality of the randomness? Following the discovery that Humans cannot consciously generate random numbers sequences, I came across an interesting ...
1
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0answers
16 views

$\lim X_n = 0$ iff $b > 0$

Probability with Martingales: approach 1: Assuming $$\lim E[\exp\{aS_n - bn\}] = E[\lim \exp\{aS_n - bn\}]$$ I can't seem to be able to prove $$\lim E[\exp\{aS_n - bn\}] = 0$$ with just $b &...
-1
votes
0answers
17 views

$\mathbf{E}\left[\frac{(U_1+c)^2}{\max((U_1+c)^2, U_2^2)} \right] \ge \mathbf{E}\left[\frac{U_2^2}{\max((U_1+c)^2, U_2^2)} \right]$ [on hold]

We consider two i.i.d. random variables $U_1$ and $U_2$ such that $\mathbf{E}[U_1] = \mathbf{E}[U_2] = 0$ and $\textrm{Var}[U_1] = \textrm{Var}[U_2] < \infty$. Prove that for any $c > 0$ the ...
2
votes
2answers
67 views

If we've got 10 coupons, what is expected number of different ones if there are 25 different types

I can't figure out this problem : There are 25 different types of coupon, all equally probable to get. If we have got 10 coupons, what is expected number of different coupons between them? ...
1
vote
2answers
59 views

Variance of the sum of correlated variables

If the variance of two correlated variables is: $$Var(r_1+r_2)=\sigma^2_1+\sigma^2_2+2\textrm{cov}(r_1,r_2)=\sigma^2_1+\sigma^2_2+2\rho\sigma_1\sigma_2$$ where $r_1$ and $r_2$ are vectors, then what ...
0
votes
1answer
27 views

Independence of linear combinations of random variables

This seems like a straightforward question, but I'm having trouble finding anything on it. Suppose we have a set of random variables, $X = (X_1,...,X_p)$ (the components of which may not be ...
3
votes
1answer
35 views

Maximize sum with no two consecutive variables

Random variables $x_1,x_2,\dots,x_{100}$ are drawn independently from the uniform distribution over $(0,1)$. After knowing the values, we are allowed to choose a subset of them as long as no two ...
0
votes
3answers
27 views

How to create a variable which changes randomly and smoothly?

I want to create a variable which is assumed to be the acceleration of a car. I assume it should has zero mean and normal distribution. But the acceleration cannot change rapidly. How do I make it ...
0
votes
1answer
28 views

Are $X_1$ and $X_2$ independent?

Let $X=(X_1,X_2)$ be an absolute continues random vector with the density function $f_X(x_1,x_2) = \left\{ \begin{array}{ll} \frac{2}{3}x_1+\frac{4}{3}x_1 x_2+\frac{2}{3}x_2, & \mbox{for } (...
1
vote
0answers
40 views

Transformations of two Laplace distributions resulting in a Laplace distribution

Suppose we have two independent identical random variables $X_1$ and $X_2$ with Laplace distribution \begin{align} f_X(x)=\frac{1}{2b}e^{-\frac{|x|}{b}} \end{align} I am looking for a non-...
3
votes
2answers
2k views

Assumption of a Random error term in a regression

In one of my recent statistics courses, our teacher introduced the linear regression model. The typical $y=\alpha + \beta X + \epsilon$, where $\epsilon$ is a "random" error term. The teacher then ...
0
votes
1answer
22 views

Relationship Between $\mathbb{E}$(time) and $\mathbb{E}$(Repetition)

Consider aa Stochastic Process with Expected value of time of occurring =T (less than infinity). Can we deduce that Expected value of number of occurrences until time T is equal to 1?? If not, in ...
0
votes
2answers
27 views

Expected value and variance of a random variable, defined as the largest of $6$ randomly drawn numbers

Let each of the numbers from $1$ up to $49$ be written on a ball, and let all these balls be contained in a box. From this box, we randomly draw exactly $6$ numbers (without putting them back, so we ...
2
votes
1answer
36 views

Transformation of random variables that preserves the distribution

Suppose we have a random variable $X$ with distribution $F_X$. Let $X_1$ and $X_2$ be two independent copies of $X$. My question: can we find a transformation $Z=g(X_1,X_2)$ such that the ...
1
vote
0answers
27 views

Game theory: how is law of large number applied here?

This is a claim rephrased and lifted from from Herbert Gintis' book "Game Theory Evolving" Pg187 Consider an evolutionary game with $n$ pure strategies $i = \{1, \ldots, n\}$, and time periods $t ...
1
vote
0answers
18 views

Weak Law of Large Numbers, biased expectation?

I want to show that: $$\hat{\sigma^2}=(1/n)\sum^{n}_{i=1} ( X_i-\bar{X} )^2$$ is a consistent estimator of $\sigma^2$. I was using the Weak Law of Large Numbers in the sense that: $$E(X_i-\bar{X })...
1
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0answers
20 views

Shifting the mean of a composite function of deterministic and random variables

For a project I am involved in relating to communication, I have the following model: $L = f(r).X$ where $X$ is a lognormal random variable with zero mean in the logarithmic scale and standard ...
-1
votes
1answer
19 views

Expected value conditioned [on hold]

Given $X_1, \ldots, X_n$ r.s.s. from a random variable with probability function $$f_{\theta}(x)=\frac{1}{\theta}\text{ for }x=1, \ldots, \theta$$ Let $T_1=2X_1-1$ and $T_2=X_{(n)}$ (maximum of $X_1, \...
2
votes
1answer
19 views

Distribution of the minimum

I have the following problem, given a random variable $X$ with density $$f(x)=2x\text{ for }x\in(0,1)$$ and a r.s.s. $X_1, X_2, X_3$. I have to calculate the probability that $X_{(1)}=\min\{X_1,X_2,...
1
vote
1answer
38 views

I am trying to prove the distribution function for the 'birthday problem' can anyone help?

Let $Y_1, Y_2, . . .$ be i.i.d. and uniformly distributed on the set ${1, 2, . . . , n}$. Define $X^{(n)} = \min \{k : Y_k = Y_j \,\,for \,\,some \,\,j < k\} $, the first time that we see a ...
1
vote
0answers
65 views

Sum of random variables that are independent but not identical [closed]

For a real number $t$, let $X_t$ be the random variable that is uniformly distributed in the interval $[t/2, 3t/2]$. If $\{t_n\}$ is a sequence of positive real numbers, is there anything we can say ...
1
vote
2answers
635 views

Tossing a coin with at least $k$ consecutive heads

Toss a coin with $\Pr(\text{Heads})=p$ repeatedly. Let $A_k$ be the event that $k$ or more consecutive heads occurs amongst the tosses numbered $2^k,2^k+1,...,2^{k+1}-1$. Show that, $\Pr(A_k\ i.o.)=...
3
votes
1answer
394 views

Function of stationary processes

Suppose we have stationary processes $X_1(t), X_2(t),..., X_n(t)$ and let $f_t(X_1(t), X_2(t),..., X_n(t))$ be a continuous function of these stationary processes. Will $f_t(\cdot)$ also be stationary ...
-1
votes
1answer
44 views

In the answer to the question attached below, I don't quite see how step-3 is derived from step-2, Can anyone explain [duplicate]

Calculating the expected values of the min/max of 2 random variables Consider two fair $k$-sided dice with the numbers 1 through $k$ on their faces, obtaining values $X_1$ and $X_2$. What is $\mathbb{...
0
votes
0answers
32 views

Function of random variable: Two ways to find the pdf

Suppose $X$ is a r.v with pdf $f_X(x)$. Let $Y = g(X)$. To find the pdf of $Y$ - $f_Y(y)$. I use one of two ways and I assume g to be a monotonically increasing function. Method I first using the ...
5
votes
3answers
72 views

Can $Y$ and $\frac{X}{Y}$ be uncorrelated if neither $X$ or $Y$ is constant?

Suppose I have two variables $X$ and $Y$ with $Y>0$. Can the random variables $Y$ and $\frac{X}{Y}$ ever be uncorrelated, i.e., $$\mathbb{E}(X)=\mathbb{E}(Y)\mathbb{E}\left(\frac{X}{Y}\right).$$ ...
5
votes
1answer
46 views

Is it incorrect to call the probability mass function by the name “discrete probability density function”?

Commonly, the probability density function (pdf) is used when dealing with continuous random variables, while the probability mass function (pmf) is used for discrete random variables. This also ...
0
votes
2answers
31 views

Give an example of two discrete random variables X and Y on the same sample space such that X and Y have the same distribution,

Give an example of two discrete random variables X and Y on the same sample space such that X and Y have the same distribution, with support {1, 2, . . . , 10}, but the event X = Y never occurs. If ...
0
votes
1answer
20 views

Question regarding probability density function?

I was going through the concept of probability density functions and had a small confusion about the notation that a pdf can take values greater than one.I found this How can a probability density be ...
0
votes
1answer
21 views

sigma algebra generated by fraction of random variables (continued) [closed]

Suppose $X_1,X_2,X_3$ are positive i.i.d. random variables. Let $S=X_1+X_2+X_3$. Is this true that $$ \sigma(X_1,X_2)\subset\sigma(X_1/S,X_2/S)? $$ Any hint of this will be appreciated. Thanks.
1
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0answers
41 views

Understanding an equation

I am trying to understand an equation from the paper "Dynamic Model for generating Synthetic ECG signal" (http://web.mit.edu/~gari/www/papers/ieeetbe50p289.pdf). The equation is: $$S(f) = \frac{\...
2
votes
0answers
31 views

Sum of two logarithmic random variables

I would like to compute the PDF of the difference of the logarithms of two shifted Rayleigh laws ($Z$): \begin{equation} Z = \log{X_{1}} - \log{X_{2}} \end{equation} where $X_1 \sim R(\alpha_1, \...
-1
votes
0answers
32 views

sigma algebra generated by fraction of random variables [closed]

Suppose there are two positive random variables $X$ and $Y$. Is this true that $$ \sigma(X/Y)=\sigma(X,Y)? $$ Any help will be appreciated. Thanks a lot.
2
votes
1answer
25 views

Confusion in Calculating Conditional Probability mass function

Question: If $X_1$ and $X_2$ are independent binomial random variables with respective parameters $(n_1,p)$ and $(n_2,p)$, calculate the conditional probability mass function of $...
-1
votes
1answer
31 views

limit superior and law of large numbers [on hold]

I am wondering whether the following result is true: Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of i.i.d. real-valued random variables on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ ...
0
votes
1answer
28 views

Understanding the solution of finding the number of red balls drawn before the first black ball is chosen

Question: An urn contains $n + m$ balls, of which n are red and m are black. They are withdrawn from the urn, one at a time and without replacement. Let $X$ be the number of red balls removed ...
0
votes
1answer
21 views

Since the variance matrix is the expected value of a dyadic tensor, why is it not singular? Which is the probabilistic property behind that?

I will try to explain better my annoying doubt. The variance matrix (or covariance matrix, according to an alternative notation) $\Sigma_v \in \mathbb{R}^{n\times n}$ of the vector random variable $v\...
1
vote
1answer
27 views

Joint density function of $T_1,T_2$ and expectation of $E[T_1 ^2 +T_2 ^2 ]$

Given that $T_1,T_2$ are random variables representing the useful life (in hours) of two electrical appliance. The joint probability function of two variables distributed uniformly in the domain ...
-1
votes
2answers
32 views

Continuous random variables with joint density function

Given that X, Y are continuous random variables with joint density function $$f_{x,y}=x-y+1$$ And: $$0 \leq x \leq 1$$ $$0 \leq y \leq 1$$ Need to calculate this: $$P(y\geq \frac{1}{4}|x=\frac{1}{4}...
4
votes
3answers
74 views

Series of independent Bernoulli variables

Let $X_1, X_2, \ldots$ be independent, identically distributed random variables with distribution $\text{Ber}(\frac{1}{2})$. Define the random varible: $$Y:=\sum_{n=1}^\infty \frac{X_n}{2^n}$$ ...
0
votes
1answer
40 views

Is the difference of two i.i.d random variables symmetric around 0?

Let $X, Y$ be i.i.d random variables. Is $\mathbb{P}(X \le Y) = \mathbb{P}(Y \le X)$? This looks 'obvious' to me. I see no reason why symmetry should not hold. But how can I prove it?
2
votes
2answers
38 views

How to calculate probability distribution of a function of two independent Poisson random variables?

I can't figure out how to determine the probability distribution function of $$aX + bY,$$ where $X$ and $Y$ are independent Poisson random variable. Basically, I want to check whether $aX+ bY$ ...
0
votes
2answers
24 views

Calculate the Covariance of random variables that distribute normally

$X_1$ and $X_2$ are two independent random variables that distribute normally with mean $μ$ and variance $σ^2$. $Y_1 = X_1 + 2X_2$ $Y_2 = X_1 - 2X_2$ Calculate $Cov(Y_1,Y_2)$. Well, I ...
1
vote
3answers
39 views

Distribution of ages of 3 children in a family

Please consider the following problem: A family has 3 children, creatively named A,B, and C. (a) Discuss intuitively (but clearly) whether the event “A is older than B” is independent of the event “...
1
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0answers
44 views

Probability: finding the expectation of “overlapping events” [duplicate]

Suppose there are 666 coins with 6 different colors in a non-transparent box. 111 of them are white coins. 111 of them are black coins. 111 of them are yellow coins. 111 of them are red coins. 111 of ...
2
votes
1answer
36 views

Expected number of duplicates

Suppose I have $m$ bins and throw $n\ll m$ balls into the bins uniformly at random. (In my application $n\sim m/\log m.$) What is the expected number of duplicates? In other words, if there are $k_i$ ...
0
votes
0answers
30 views

In what sense does does linear dependence correspond to random variable dependence?

In linear algebra, there is a theorem that states that $\langle v, w \rangle = 0$ implies that $v$ and $w$ are linearly independent. Now let $V$ be a vector space of real-valued random variables on ...
2
votes
3answers
66 views

Finding the density for $\min\{X, Y\}$

Problem: Let $X$ and $Y$ be independent and suppose that each has a $\text{Uniform}(0,1)$ distribution. Let $Z = \min\{X, Y\}$. Find the density $f_Z(z)$ for $Z$. Hint: It might be easier to first ...