## Distribution of time that a flashlight can operate

The lifetimes of batteries are independent exponential random variables , each having parameter λ. A flashlight needs two batteries to work. If one has a flashlight and a stockpile of n batteries, What is the distribution of time that the flashlight can operate? What I have so far: Let Y be the lifetime of the … Read more

## Proving a Poincare inequality using Stein’s characterization

Question: Let X be a standard Gaussian r.v. Use Stein’s characterization Ef′(X)=E(Xf(X)) to prove the Poincare inequality E|f(X)−Ef(X)|2≤E|f′(X)|2. This looks like Markov’s inequality could be used to prove the inequality. However, I’m not quite sure how to piece it together. Answer You want to prove that E|f(X)−Ef(X)|2≤E|f′(X)|2. This inequality doesn’t change if you add a … Read more

## Stochastic variable exercise: People between me and my friend.

This is the exercise: $n$ people are arraged randomly in a line (not a circle), among which are yourself and a friend. Call $Y$ the number of people that are between you and your friend. Show: $E[Y] = \frac{n-2}{8}$. This is how I started: There are $P_n = n!$ ways to arrange $n$ people in … Read more

## Question about an exercise from Feller

The following is an exercise from the classical textbook of Feller on probability theory. Four girls take turns at washing dishes. Out of the total of four breakages, three were caused by the youngest girl. Was she justified in attributing the frequency of her breakages to chance? This is a variant of the balls and … Read more

## Picking Unique Balls from a Bin

Problem: We have a bin with 5 red balls, 7 green balls, and 9 blue balls. We draw 3 balls out of the bin, without replacement. What is the probability that no two of the three balls have the same color? My Answer: Note that the probability of no two balls having the same color … Read more

## What is the expected distance between these two points?

Suppose you have a straight line of length L, and a point is chosen at random along the line. Now suppose a second point is chosen at random to the left of the first point. What is the expected distance between these points? Note: I am aware that if two points are independently chosen at … Read more

## Math Problem (one-to-one correspondences)

Alex the ant starts at (0,0). Each minute, he flips a fair coin. If he flips heads, he moves one unit up; if he flips tails, he moves one unit right. Betty the beetle starts at (2,4). Each minute, she flips a fair coin. If she flips heads, she moves one unit down; if she … Read more

## Mean of the difference between uniform random variables.

I have two uniform random variables $B$ and $C$ distributed between $(2,3)$ and $(0,1)$ respectively. I need to find the mean of $\sqrt{B^2-4C}$. Could I plug in the means for $B$ and $C$ and then solve or is it more complicated than that? The original question is here: Difference between two real roots with uniformly … Read more

## Determine the expected value of $X$ using indicator random variables

Let $n\geq 1$ be an integer. Consider a uniformly random permutation $a_1, a_2, \ldots , a_n$ of the set $\{1, 2, \ldots , n\}$. Define random variable $X$ to be the number of indices $i$ for which $1 \leq i \lt n$ and $a_i < a_{i+1}$. Determine the expected value $E(X)$ of $X$. (Hint: Use … Read more

## Finding probability of intersection of events

I was reading First course in Probability by Sheldon Ross and am stuck at the understanding this simple problem [hence proved my maths is poor 🙁 ]. Problem: Celine is undecided as to whether to take a French course or a chemistry course. She estimates that her probability of receiving A grade would be 1/2 … Read more