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

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15 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
22 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
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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 ...
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1answer
21 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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2answers
26 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
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1answer
34 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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0answers
21 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
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 })...
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0answers
18 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
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
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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,...
1
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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 ...
1
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0answers
62 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
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2answers
634 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.)=...
2
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1answer
393 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 ...
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1answer
43 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 ...
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
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1answer
42 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
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 ...
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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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1answer
21 views

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

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.
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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, \...
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0answers
32 views

sigma algebra generated by fraction of random variables [on hold]

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
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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 $...
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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 $(\Omega, \mathcal{F}, \mathbb{P})$ ...
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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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1answer
20 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
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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 ...
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0answers
17 views

Given the joint density function of the variables $ X,Y$ [closed]

Given the joint density function of the variables $ X,Y$ both variables are given values interval $ [0,1]$. $$f_{x,y}\begin{cases} & \frac{2}{3}\text{ if } 0\leq x\leq 1,0\leq y\leq 1,0\leq x+...
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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}...
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
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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?
2
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2answers
36 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
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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 ...
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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 “...
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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
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1answer
35 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 $k_i$ ...
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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
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3answers
65 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 ...
2
votes
1answer
41 views

Probabilistic constraint implying deterministic constraint?

Suppose $X$ is an $N$-dimensional random variable $X := [X_1 \; X_2 \; \cdots \; X_N]$ such that all entries can either be 0 or 1 while satisfying the following: (i) $\mathbb{P}(X_i = 1) = p_i \; \; ,...
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0answers
24 views

Is there a formula for the MGF of $Y=g(X)$?

Let $X$ be a real valued random variable with cumulative distribution function (CDF) $F_X$ and probability density function (DF) $f_X$. Suppose $g\colon\Bbb{R}\to\Bbb{R}$ is a differentiable, strictly ...
4
votes
3answers
542 views

Negative Variance

I have two independent variables $X$ and $Y$. $W=X-Y$ when $X\sim $Bernoulli$(1/2)$ and $Y\sim N(0,1)$. This puts $Var(x)=1/4$ and $Var(Y)=1$, but I have to be misunderstanding something because if $...
3
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1answer
64 views

Are there general methods that can be applied when using the Borel-Cantelli Lemma, to get a statement about a sequence of random variables?

I hope the title in itself is clear, if not allow me to give an example. In Class my Professor did the following: Given a sequence $(X_n)_{n \in \mathbb{N}}$ of non-negative i.i.d. RV $X_n \sim X$...
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1answer
47 views

Prove $A_{\infty} < \infty$?

From Williams' Probability with Martingales How do we know that $A_{\infty} < \infty$? If $T = \infty$, then $$E[A_{T \wedge n}] \le (K+c)^2$$ $$\to E[A_{n}] \le (K+c)^2$$ $$\to \lim ...
6
votes
2answers
69 views

Why does Slutsky's Theorem Fail to Generalize? [on hold]

What is a counterexample to the claim that $X_n \rightsquigarrow X$, $Y_n \rightsquigarrow Y$ implies that $X_n + Y_n \rightsquigarrow X + Y$? I know that Slutsky's Theorem guarantees the case that $...
1
vote
2answers
26 views

Uniform Distribution Problem on $X, Y, Z$

Problem: Let $X \sim \text{Uniform}(0,1)$. Let $0 < a < b < 1$. Let $$ Y = \begin{cases} 1 & 0 < X < b \\ 0 & \text{otherwise} \end{cases} $$ ...
2
votes
1answer
503 views

Variance stabilization for Poisson data

Intro Let $Z > 0$ be a random variable with the mean and variance defined as $\mathbb{E}\{ Z \}$ and $\operatorname{Var}\{ Z \}$, respectively. The variance stabilization transform (VST) $f(z)$ ...
13
votes
5answers
2k views

Are we guaranteed that the harmonic series minus infinite random terms always converge?

Consider the known harmonic series $\sum_{n=1}^\infty \frac{1}{n}$ and modify it as follows $$\sum_{n=1}^\infty a_n\frac{1}{n}$$ where $$a_n \sim \operatorname{Bern} \left({\frac{1}{2}}\right)$$ i.e. ...