# Tag Info

### Probability that Mercury is the nearest planet to Earth.

Per the comments, here's some MATLAB code that runs a simulation using both random planetary positions and deterministic positions based on orbital velocities. For both, I've set up polar coordinates ...
• 2,071

### Probability that Mercury is the nearest planet to Earth.

Lifting my comments to an answer basically to try and more verbosely explain how I interpreted the given details, and what my simulations did. My interpretation of the probabilities is the following. ...
• 133k
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\}...
• 14.2k

### 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 ...
• 361
Accepted

### Order statistics vs Integration to calculate expectation

All things reconsidered, I take back all the words that I said and am about to introduce the solution which works fine and satisfies Python simulations I've run. To calculate the conditional ...
1 vote
Accepted

### Uniform distribution problem solving

Consider one end (left end which I'll consider to be the back end) of the car to be at $x$ meters away from $0$. if say $3<x<5$, then the remaining space is $x$ meters behind the car(which is ...
1 vote

### What is the formula that connects the average distance to the nearest point and the average number of points per unit volume?

Your formula is pretty good approximation. Even better approximation can be achieved in the following way. Assume that distribution of points is uniform and uncorrelated, so that it obeys the Poisson ...
• 26.3k
1 vote

If $U <\frac 1 2$ then $\left\lfloor 2U\right\rfloor=0$ and if $U >\frac 1 2$ then $\left\lfloor 2U\right\rfloor=1$. This gives $$P(2U-\left\lfloor 2U\right\rfloor \le x)=P(U<\frac 1 2, 2U-... • 36.9k 1 vote ### Random vector (X, Y) has a uniform distribution on the unit circle. Regardless of whether we're talking about just the boundary or including the interior of the circle, it is very easy to see that the components are not independent by considering a region between the ... • 24.3k 1 vote ### Density of power with random variable Note that if X \sim \operatorname{Uniform}(0,1), then$$-\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. ...
• 136k

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