# Questions tagged [random-variables]

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

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### Characteristic function of gaussian $X + X^2$

Assume $X$ is a standard normal random variable. Are there any known results for the form of the characteristic function of $X+X^2 = X(X+1)$? Essentially I'm interested in the c.f. of $a + bX + c X^2$ ...
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### Describe the following spaces

I'm trying to practice my understanding of $L^1$ and $L^2$ spaces with the problem below. Given $\Omega = \{\omega_1,...,\omega_7 \}$ let's consider the probability space $(\Omega, \mathcal{F},P)$ ...
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### With random probability

I have a question about the possibility of talking about this problem :\ Let $X,Y$ two random variables, then : X = a with probability Y and X = b with probability 1-Y. Thanks
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### Does a non-stationary source has an entropy rate $H_X$?

Let $\{X_k\}_{k\in \mathbb{N}_+}$ be a source and we assume that $H(X_k )<∞$ for all $k$. Then we deﬁne the entropy rate of an information source, which gives the average entropy per letter of ...
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### $45$ percentage of population of a large community is in favor of a proposed rise in school taxes.

$45$ percentage of population of a large community is in favor of a proposed rise in school taxes. $i)$ Approximate the probability that a random sample of $1000$ people will contain at least $500$ ...
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### Is there a general method for generating correlated random variables from independent random variables?

For generating correlated normal random variables from independent normals, I know that you can use Cholesky/SVD. Is there a general method that applies for other random variables, e.g., uniformly ...
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### Let $Y$~$Gamma(a,b)$ with PDF $f(y)$= $y^{a-1}e^{-y/b}/\gamma(a)b^a$; $y>0$ For $a>1$, show that the mode of $Y$ is $(a-1)b$.

Suppose that $Y$ follows a gamma distribution with parameters $a$ and $b$. That is, $f(y)$= $y^{a-1}e^{-y/b}/\gamma(a)b^a$; $y>0$ For $a>1$, show that the mode of $Y$ is $(a-1)b$. My working: I ...
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### Distribution of square root of positive normal random variables [closed]

Could you teach me what is pdf of square root of positive normal random variables if the distribution of normal variables follows f(x)=1/(σ√2π) exp⁡(-(x-μ)^2/(2σ^2 )), (x - μ)> 0 where μ and σ are ...
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### Random variable estimation stochastic [closed]

I am totally lost with this proof. Can someone please help me? Show that if a random variable $X\geq 0$ satisfies, for some constants $c,C>0$ and $r\in\mathbb{R}$, $P(|X-r|\geq t)\leq C\exp(-ct^2)$ ...