Questions tagged [random-variables]

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

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For the probability triple $(\Omega, \mathcal{F}, \Bbb{P})$, a random variable $X$, and a function $g$, is $g(X)$ automatically measurable?

For the probability triple $(\Omega, \mathcal{F}, \Bbb{P})$, a random variable $X: \Omega \to D$, and an arbitrary function $g: D \to E$, is $g \circ X$ also measurable and thus a random variable? I ...
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Exemple where tower property of conditional expectation is NOT verify

Question: Let $\Omega=\{a,b,c\}$. Give an example for $X, F_1, F_2$ in which $E(E(X|F_1)|F_2) \neq E(E(X|F_2)|F_1)$ My answer: I am not at all sure of my answer. If you have any shorter and nicer ...
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What positive distribution has the form $x^{-\ln(x)}$

What is the name for the continuous distribution $f(x)\propto x^{-\ln(x)}$ for $x\geq 0$? More precisely, I'd like the name for the family of distributions of the form $f(x)\propto x^{-c \ln(x)}$, ...
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Clarifying weird integral notation

I would like your help to understand the following notation for integrals which I have never seen. Consider an integral $$\int_{a}^b 3 \text{ }d X$$ where $X$ is a random variable. What does this ...
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What positive distribution has the form $e^{-x-\frac{1}{x}}$

What is the name for the continuous distribution $$f(x)\propto e^{-\left(x+\frac{1}{x}\right)}$$, where $x\geq 0$? Does it even have a name? My interest in it is because it is the simplest ...
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Probability: are realizations of random variables what is actually observed?

According to Wikipedia, yes. In probability and statistics, a realization, observation, or observed value, of a random variable is the value that is actually observed (what actually happened). https:...
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Probability Algebra With a Wordle Example

Define $X$ as the proposition "you will win today's Wordle," $A$ as expressing the information returned by the coloring of your first guess (in a vacuum), $B$ as expressing the information ...
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The covariance between the sum of N independent random variables and N.

Problem Let $X_1,X_2,...$ be independent random variables with $E(X_i)=a, Var(X_i)=b$ for $i\geq1$, and $N\geq0$ an integer-valued random variable with $E(N) = c, Var(N) = d^2$ independent of the ...
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Bernoulli trials: Probability of S consecutive successes followed by F consecutive failures

I am reading a probability text book and I am having difficulty understanding a paragraph and how the equation was setup. Can anyone please explain it in more detail or show me how this was derived? ...
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Log-normal distribution: Why is this a density function?

I want to prove that $$f(x) = \frac{1}{\sigma x \sqrt{2\pi}} \exp(-\frac{\ln(x)^2}{2\sigma^2})$$ for $x>0$ and $f(x)=0$ for $x\leq 0$ is a density function, with $\sigma > 0$. So what I have to ...
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If a random variable has an atom at zero, does it have a density?

Let $Y$ be a random variable with distribution function $$F_Y(x) = \begin{cases} 0 &\quad x<0\\ p &\quad x=0\\ p + (1-p)F_X(x) &\quad x>0 \end{cases}$$ where $X$ is a continuous ...
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What is the smallest unseen number in an iid sample? (From "A number NOBODY has thought of - Numberphile")

In this Numberphile video, the question: "What is a number nobody has thought of?" is addressed. The method is as follows: Estimate a number $N$ as the number of times humans have thought ...
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1 vote
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Example of random variable with infinite entropy

Let $X\sim P$ on $A=\{2,3,\dots\}$, where $P(k)=\frac{C}{k(\log k)^2}$ for $k\geq2$ with $C$ some normalising constant. Show that $H(X)=\infty$. My attempt so far: I have shown, by direct computation ...
1 vote
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Is this function continuous (when $U$ has no flat regions)?

Apologies for that useless modifier in the brackets in the title -I had to add that to avoid "duplicate titles". It is most natural that there will be multiple questions with the title "...
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Calculating Probabilities and Using Binomial Distribution

A recent survey of residents in Texas concluded that 55% of Austin city residents and 46% of Houston city residents broke a bone at some point during their childhood. a. Let’s say Austin has 5200 ...
1 vote
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Expected number of rounds needed until a winner is declare in Rock, Paper, Scissors.

A game of Rock, Paper, Scissors is usually played with 2 or more (n) people. If when all n people reveal their play and only 1 or all 3 options appear, that round is deemed indecisive, and they just ...
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Joint Probability Mass Function of Rock, Paper and Scissors

Problem A game of Rock, Paper, Scissors is usually played with $2$ people, but it can be extended to an arbitrarily large number of $n$ people, where $n\geq2$. If when 'all $n$ people reveal their ...
1 vote
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Expectation of Exponential Random Variable

I was a bit confused about this question. You’re clearing out your garage for a garage sale, and you want to get rid of as much stuff as possible quickly. You found a dresser and decided to sell it ...
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X1, X2, X3 are independent and uniformly distributed over the interval (0,1). Decide the distribution of the second greatest of X1, X2, X3.

This problem is from a textbook. I have no idea where to begin. The textbook gives a lead: "The second greatest (variable) is smaller than $t$ if two or three of the $X$-variables are smaller ...
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Calculating Expected Number from Exponential Distribution

I was a bit confused about this question. You’re clearing out your garage for a garage sale, and you want to get rid of as much stuff as possible quickly. You found a dresser and decided to sell it ...
1 vote
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Meaning of 'almost-everywhere constant' random variable

My question stems from page 2 of this paper by Bucy, which states: [A random variable $x$] is almost everywhere constant a.e. $P$. where $P$ is a probability measure. My interpretation of this is as ...
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On the convergence in distribution of some random variables

In this paper, they mention an equivalent result of the convergence in distribution of the random variable in page $11$. I don't understand why the convergence in law of $Arg(R_n)$ to the uniform ...
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Construction of a discrete random variable uniformly choosing random variables

Let $(R_{n,k})_{n \in \mathbb{N}, k \le n}$ a sequence of iid discrete random variables. I think that Kolmogorov proved such variables exist in a probability space. I want to construct a sequence of ...
1 vote
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Bounds for summing Continuous Variable

I have a question on how the bounds are picked for the example below: Given that $X_1$ and $X_2$ are independent random variables, and the pdf is $f_{X_i}(x_i) = λe^{−λx}, x_i > 0$ Find $W=X_1+X_2$ ...
1 vote
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Chernoff-type bounds for a sum of Poissons

Let $X_1,\dots,X_n$ be i.i.d. Poisson variables $\sim\text{Pois}(\lambda)$, with $\lambda<1$. Now, if $X=X_1+\dots+X_n$ then $X\sim\text{Pois}(\lambda n)$ with $\mathbb E(X)=\lambda n<n$. I ...
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double integral showing equal to 1

The question asked is as follows: Given that the nonnegative function $g(z)$ has the property that $\int_{-\infty}^\infty g(z)dz=1$ show that $f(x,y)=\frac{g(\sqrt{x^2+y^2})}{\pi\sqrt{x^2+y^2}}$ for ...
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