4 votes
Accepted

A few questions regarding random points on a disk.

$a$.) Let the initial point chosen be $p=(x_0, y_0)$. This gives us a distance from some other randomly chosen point on the disc of; $$D(x,y)=\sqrt{(x-x_0)^2+(y-y_0)^2}$$ Integrating this over the ...
Volk's user avatar
  • 1,643
3 votes
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Probability distribution of a random variable - Interview

Interviewer: Assume $a\sim U[0,1]\,,$ and conditional on $a\,,$ $$b\sim U[a,1]\,.$$ What is the unconditional distribution of $b\,?$ Candidate (sweating): \begin{align} &\mathbb P\big\{b\le y\big\}...
Kurt G.'s user avatar
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3 votes

Probability distribution of a random variable - Interview

Initially this problem seems a little confusing. I initially mistook $P(B=b|A=a)$ for $P(B=b)$ and thought $P(B=b)$ was already uniform! In such a situation to get a handle on the problem it can be ...
Joseph's user avatar
  • 387
3 votes

A few questions regarding random points on a disk.

I suspect that if $z=\sqrt{x^2+y^2}$ then, using the cosine rule, the density for the distance $d=PP'$ is proportional to $\dfrac{2d}{R^2}$ when $0 \le d\le R-z$ $\dfrac{2d}{\pi R^2} \cos^{-1} \left(\...
Henry's user avatar
  • 157k
2 votes

Difference between random variable $2X$ and the sum of two independent observations of $X$?

$X_1+X_2$ is not like $2X$, because extreme values are less likely. For example: $$\Pr(2X=8)=\Pr(X=4)=\frac16,$$ but$$\Pr(X_1+X_2=8)=\Pr(X_1=4\text{ and }X_2=4)=\frac{1}{6}\times\frac{1}{6}=\frac{1}{...
Especially Lime's user avatar
2 votes

A probability question over multiple questions test.

This is a hypergeometric probability. In the question pool, there are $100$ questions you have prepared for, and $40$ questions you have not. If $31$ questions are selected without replacement and $...
heropup's user avatar
  • 135k
2 votes

Coin flip puzzle, prove $P(X>Y) > P(Y>X)$

This is not a complete answer, but a partial attempt too long for a comment. I hope someone can turn it into a full answer, as I have to give up thinking about it for the rest of the day: The natural ...
K.Power's user avatar
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1 vote

Density of power with random variable

Note that if $X \sim \operatorname{Uniform}(0,1)$, then $$-\log X \sim \operatorname{Exponential}(1).$$ The proof of this is left as an exercise. Next, consider $$\log (XY^Z) = \log X + Z \log Y.$$ ...
heropup's user avatar
  • 135k
1 vote

Difference between random variable $2X$ and the sum of two independent observations of $X$?

Too long for a comment. You seem to be interested in the intuition behind expectation and variance and why it behaves the way it does with sums of random variables. To understand this intuition, ...
William M.'s user avatar
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1 vote
Accepted

Related to random variables

I try to reformulate the variables at play to make sure I got it right: $$ h_0 \sim \text{Nak}(m, \Omega_0) \\ h_1 \sim \mathcal{N}(0,\sigma^2) $$ Assuming the variables $h_0$ and $h_1$ are ...
finch's user avatar
  • 1,586
1 vote
Accepted

Compute the probability density of a function of a random variable

Let's say that $Y=g(X)$. Formally speaking (here $g$ has to be strictly increasing) the CDF of $Y$ is defined by $F_Y(y)=P(Y\leq y)=P(g(X)\leq y)=P(X\leq g^{-1}(y))=F_X(g^{-1}(y))$, hence the density ...
Stéphane Mottelet's user avatar
1 vote
Accepted

Find the copula to minimize KL-divergence

Long comment. Assuming $P$ has positive density $p$ which is $N$-times continuously differentiable, a heuristic computation shows that the minimizer $C_*$ has the density $c_*$ satisfying $$ \frac{\...
Sangchul Lee's user avatar

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