# Tagged Questions

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

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### 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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### 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 $aX+ bY$ ...
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### 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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### 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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### 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: $$Y:=\sum_{n=1}^\infty \frac{X_n}{2^n}$$ ...
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### 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 ...
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### 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 ...
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### Negative Variance

I have two independent variables $X$ and $Y$. $W=X-Y$ when $X\sim \mbox{Bernoulli}\left(1/2\right)$ and $Y\sim N(0,1)$. This puts $\operatorname{Var}(x)=1/4$ and $\operatorname{Var}(Y)=1$, but I have ...
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### Conditional distributions allowed pdf to take on single value?

My question is about Conditional probability distributions. From what I have learned, PDF's aren't allowed to take on singular values, yet I find that this definition seems to go out the window when ...
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### Linearly Dependent Random Variables

Intuitively, what is meant to be captured by the notion of linearly dependent (real-valued) random-variables?
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### What is the Standard Way to Define a Norm on a Vector Space of Real-Valued Random Variables?

Let $V$ be the vector space of real-valued random variables over $\mathbb{R}$. How does one traditionally define the norm on $V$?
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### Finding the distribution of a random vector in a conditional probability problem [closed]

Players A and B are playing a game of drawing coins from two boxes without returning/replacing them. Box1 has three coins with values 0, 1 and 2 and Box2 has two coins with values 1 and 2. In the game,...