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

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32 views

Why are absolute values involved in functions of random variables?

From a textbook: If $X$ is a continuous random variable, then so too is the new random variable $Y = Y (X)$. The probability that $Y$ lies in the range $y$ to $y + dy$ is given by $$g(y)=\int_{...
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2answers
29 views

Why is $P(X\in[a,b])=P(X\in[a,b))=P(X\in(a,b])=P(X\in(a,b))$

I saw, for any continuous random variable $X$, $P(X\in[a,b])=P(X\in[a,b))=P(X\in(a,b])=P(X\in(a,b))$, where $a,b\in\mathbb{R}$, in my textbook. I don't quite understand why the openness/closeness of ...
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2answers
25 views

How to generate a random variable $r_i$ such that $\sum_{i=1}^n |\frac{r_i}{\sigma_i}|^2\leq\chi^2_{n,\alpha}$

How can I generate $r_i$ for $1 \leq i \leq n$, such that $\sum_{i=1}^n |\frac{r_i}{\sigma_i}|^2\leq\chi^2_{n,\alpha}$, where $\sigma_i^2$ is the variance of $r_i$ and, $\chi^2_{n,\alpha}$ is a chi-...
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0answers
33 views

Best estimator for 'expected goals' in a soccer game with known outcome

Consider a soccer game between team $A$ and an irrelevant opponent. We know the outcome of the game: Team $A$ scored $G_A$ goals, had $ST_A$ shots on target and $S_A$ shots. Assume that the expected ...
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1answer
33 views

Expected value is equal to $\infty$?

Given the following set: $$ M=\{2^k | k \in\mathbb{N}_{>0}\}=\{2,4,8,16,...\} $$ Now we chose randomly some elements. For this we have to define a distribution: $$ P(m\in M)=m^{-1} $$ As can be ...
2
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1answer
17 views

Find the PMF for number of heads following the first tail on a four consecutive coin toss expriment

Suppose a fair coin is toss four times consecutively. Find the PMF for random variable of number of heads following the first tail. My take: Let random variable $X$ be the number of heads in this ...
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0answers
27 views

If $\mathbb{E}[f(X)]=\mathbb{E}[f(Y)]\,\,\,\forall$ continuous $f$, then $X, Y$ have same distribution

Let $(\Omega, \mathcal{F}, P)$ be a probability space and $X, Y:\Omega \to [0,1]$ be random variables. Prove that if $$\mathbb{E}[f(X)]=\mathbb{E}[f(Y)] \text{ for all continuous }f:[0,1]\to\...
3
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1answer
43 views

Variance of sum of linear combination

I want to calculate the variance of a sum of linear combinations, so $$\operatorname{Var}\left(w'R_1 + w'R_2\right)$$ where $w$ is a $N\times 1$ vector and both $R_1$ and $R_2$ are $N\times 1$ ...
1
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1answer
25 views

Computing the distribution function

I have troubles solving this task: Let $U_1,U_2,\dots$ be an i.i.d. sequence of random variables with uniform distribution on $[0,1]$. We set for every integer $n\geq 1$ $$M_n=\max\{1/\sqrt{U_1},\...
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1answer
26 views

Compute expected received balls from boxes

I have 6 boxes: $A,B,A',B',C \text{ and } D$. The box $A$ has $n_1$ red balls that are numbered from $1, \cdots, n_1$. The box $B$ has $n_2$ green balls that are numbered from $1, \cdots, n_2$. Make a ...
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1answer
21 views

Sufficient condition to conclude joint pmf equals the product of marginal pmf's?

Consider random variables $X$ and $Y$ with marginal probability mass functions $f$ and $g$ respectively and with joint pmf $h$. Also, $E(\cdot)$ denotes the expected value. It is true that if $f(x) ...
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2answers
37 views

given a graph of density function, what can we conclude about expected value

given the following graph (the density function), what can we conclude about the expected value? I got stuck a little bit with that question and I would appreciate your help! I found out that C must ...
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0answers
23 views

What is the intuition behind Uniform Integrability?

A definition of Uniform Integrability I am currently working with is that: A sequence $X_1, X_2, \ldots$ of random variables is Uniformly Integrable if: $$ \sup_n \mathbb{E}\left(|X_n|\cdot \mathbb{...
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0answers
38 views

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

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 ...
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2answers
34 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 ...
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1answer
37 views

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

Probability with Martingales: It looks like $$\lim \exp\{aS_n - bn\} = 0$$ if $b > 0$ because $$\lim aS_n - bn = -\infty \tag{*}$$ but how to prove $(*)$?
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1answer
34 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 ...
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1answer
40 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 ...
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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 ...
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2answers
69 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 ...
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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 } (...
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2answers
69 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? ...
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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-...
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0answers
30 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 ...
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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 ...
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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 ...
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1answer
19 views

Expected value conditioned [closed]

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
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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,...
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2answers
28 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 ...
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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 ...
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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{...
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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 ...
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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 ...
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1answer
48 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 ...
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2answers
32 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 ...
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3answers
73 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., $$\mathbb{E}(X)=\mathbb{E}(Y)\mathbb{E}\left(\frac{X}{Y}\right).$$ ...
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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 ...
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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{\...
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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.
2
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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, \...
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1answer
32 views

limit superior and law of large numbers [closed]

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 $(\Omega, \mathcal{F}, \mathbb{P})$ ...
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0answers
67 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 ...
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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\...
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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 ...
2
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1answer
26 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 $...
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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 ...
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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}...
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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?
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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 ...
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 $aX+ bY$ ...