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 ...
3
votes
Accepted
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\}...
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 ...
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(\...
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$$\Pr(X_1+X_2=8)=\Pr(X_1=4\text{ and }X_2=4)=\frac{1}{6}\times\frac{1}{6}=\frac{1}{...
2
votes
A probability question over multiple questions test.
This is a hypergeometric probability. In the question pool, there are $100$ questions you have prepared for, and $40$ questions you have not. If $31$ questions are selected without replacement and $...
2
votes
Coin flip puzzle, prove $P(X>Y) > P(Y>X)$
This is not a complete answer, but a partial attempt too long for a comment. I hope someone can turn it into a full answer, as I have to give up thinking about it for the rest of the day:
The natural ...
1
vote
Density of power with random variable
Note that if $X \sim \operatorname{Uniform}(0,1)$, then
$$-\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.$$
...
1
vote
Difference between random variable $2X$ and the sum of two independent observations of $X$?
Too long for a comment.
You seem to be interested in the intuition behind expectation and variance and why it behaves the way it does with sums of random variables. To understand this intuition, ...
1
vote
Accepted
Related to random variables
I try to reformulate the variables at play to make sure I got it right:
$$
h_0 \sim \text{Nak}(m, \Omega_0) \\
h_1 \sim \mathcal{N}(0,\sigma^2)
$$
Assuming the variables $h_0$ and $h_1$ are ...
1
vote
Accepted
Compute the probability density of a function of a random variable
Let's say that $Y=g(X)$. Formally speaking (here $g$ has to be strictly increasing) the CDF of $Y$ is defined by $F_Y(y)=P(Y\leq y)=P(g(X)\leq y)=P(X\leq g^{-1}(y))=F_X(g^{-1}(y))$, hence the density ...
1
vote
Accepted
Find the copula to minimize KL-divergence
Long comment. Assuming $P$ has positive density $p$ which is $N$-times continuously differentiable, a heuristic computation shows that the minimizer $C_*$ has the density $c_*$ satisfying
$$ \frac{\...
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