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

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

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