Expected Number of Steps To Reach 0 in Circle
You are in a circle with 100 points labelled 0 - 99 clockwise. You start at 1 and can move left or right with equal probability. What is the expected time to reach $0$?I attempted this by setting up a...
View ArticleIndependent coin tosses , double or halve current sum
(Quant job Interviews - Questions and Answers - Joshi et al, Question 3.5)Suppose you have a fair coin. You start with 1 dollar, and if you toss a H your position doubles, if you toss a T your position...
View ArticleExpected value of fastest career hat-trick?
This is a sports variation of line segment (rope, stick...) divided into $n+1$ subsegments by $n$ uniformly distributed points.Assuming goals are uniformly distributed among player's career, what's the...
View ArticleOrder statistics vs Integration to calculate expectation
I have the following problem -$a$ and $b$ are independently drawn from uniform distribution. $a$ is drawn from uniform distribution $\{m, 1+m\}$ and $b$ is drawn from uniform distribution $\{0,1\}$....
View ArticleHow to get this formula for expectation of continuous-time urn process
We define the continuous-time, multi-type branching process $(X(t))_{t\ge0}$ as follows: $(X(0))=\alpha\in\mathbb{R}^d$, where $\alpha$ is the urn initial composition, meaning that, at time $0$, there...
View ArticleProbability of a Markov Chain value conditional on an observation of the next...
Let $X=(X_i)_{i \ge 1}$ be a Markov chain and $Y=(Y_i)_{i \ge 1}$ be such that $Y_i=\phi(X_i)$ for an arbitrary function $\phi$.In attempting to derive a recurrence relation for the expectations...
View ArticleA dime is tossed repeatedly until a head appears. Let N be the trial number...
A dime is tossed repeatedly until a head appears. Let N be the trial number on which this first head occurs. Then a nickel is tossed N times. Let X count the number of times that the nickel comes up...
View ArticleCalculating the expected value for a complicated pdf
I am given the following cdf (cumulative distribution function) for the random variable $T$ with support $x\in(0, \theta)$. $$F_{T}(x)=\Bigg(\frac{x-\frac{1}{n}}{\theta}\Bigg)^{n}$$ I am asked to...
View ArticleYou have a bucket of unfair coins. Calculate the expected value.
I have recently found the following question."You have a bucket of unfair coins. Each coin has a probability of getting heads, $p$, which is uniformly distributed between zero and one (inclusive). You...
View ArticleJane Street: Cutting a cake n-times - expected number of pieces
I'm trying to solve a brainteaser (from Jane Street t-shirt):A 2D circle (cake) is cut randomly with straight lines n-times. What's the expected number of pieces?Could you give me a hint on how to...
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Expectation as Integral: Visualizing the area under the curve
For continuous variables we can define expectation as $E(X) = \int_\Omega X \, dP$hopefully where dP = p(x) dx and $\Omega$ being the range in which X can fall.Area under which curve are we trying to...
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Expected value and average value problem
One has a very large batch of tablets. The weight (unit: gram) of a randomly selected tablet can be accurately considered as a normally distributed random variable X with mean μ and standard deviation...
View ArticleExpected payout riddle: How to keep signature collectors from forging?
I made up this riddle, but I don't know how to solve it.Marissa owns a company that collects signatures. Marissa can only hire perfectly rational self-interested agents (as defined in game theory), and...
View ArticleExpected Value of Children With Mutation
The probability that a child in a given family will inherit a certain genetic mutation is 5%, independently of their siblings. Consider a family with 4 children.(a) If it is known that the first born...
View ArticleSimple question about expectation
I have a question whether given a random variable $X$ such that $P(X ≥ 0) = 1$ and $P(X > 0) > 0$ it is true that $E[X] > 0$? It was mentioned as a remark during a lecture without proof, but I...
View ArticleCan Chernoff Bound Theorem be applied to functions of independent random...
Let $X_1, X_2, \ldots X_n$ be independent random variables, can we apply the Chernoff Bound Theorem to bound $E\left[\sum_i \Theta( \widehat{X_{i}}))\right]$ if $\Theta( \widehat{X_{i}}) :...
View ArticleProve approximate equality
Now, I know$\mathbb{E}\left\{\log_{2}\left(1+\frac{X}{Y}\right)\right\}\approx\log_{2}\left(1+\frac{\mathbb{E}\left\{X\right\}}{\mathbb{E}\left\{Y\right\}}\right) \tag{1}$if $X$ and $Y$are...
View ArticleI roll a fair die 4 times. Let $X$ be the number of different outcomes that I...
I roll a fair die 4 times. Let X be the number of different outcomes that I see. Find $\mathbb{E}[X]$.My attempt:I know that I can write X as a sum of indicator random variables and then I can use the...
View Articleshow that the fish number expected value is e, if you stop fishing only if...
A man goes fishing in a river (with a lot of fishes!) and only will stop when the last fish caught has less weight than the one before. Show that the fishes caught expected value is e(2,71828...).The...
View Articlelinear regression, expectation and mean squared error
Let us assume that data is generated according to a true model $$y_i = \beta_{true}x_i + \epsilon_i$$for $i = 1, ..., n$Assume that $x_i$ are fixed, and $\epsilon_i$~ N(0, $\sigma^2$) independently.Let...
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