Expected Ratio of Boys to Girls?
Assume that a couple keeps having children until they have $2$ consecutive girl children. What is the expected ratio of boys to girls? Assume that probability of having a girl child is $p$.I tried...
View ArticleExpected size of "Minkowski reflection" of a random subset of $w \times w$...
For $U$,$V$ sets of points on Cartesian plane, define the "Minkowski reflection" $U \star V$ of $U$ about $V$ as the set of positions occupied by reflections of every point in $U$ about every point in...
View ArticleWhy can I consider the values of n uniform(0,1) i.i.d's evenly spaced for...
Essentially this idea came up while I was doing two separate questionsa. A bee wants to fly on the real line from the point $0$ to the point $1$, visiting $n$ flowers which are positioned at the...
View ArticleExpectation of the norm of a random vector multiplied by a matrix
Lets say we are given random (row) vector $x \in \mathbb{R}^n$, and a non-random matrix $M \in \mathbb{R}_{nxn}$I came across a claim (which does not impose any assumptions on the distribution of $x$ a...
View ArticleWhy $E[X^n]$ calculated as $\int x^n pdf(x)\,dx$? Why is it not $\int x^n...
Why E[X^n] calculated as integral of x^npdf(x)? Why is it not integral of x^npdf(x^n) ? Wont't the integral limits and pdf change when x is changed?
View ArticleTerminating Sequence expected length
I was preparing for a quant interview and I came across a puzzle on QuantGuide(named Sequence Terminator):A fair 6−sided die is rolled repetitively, forming a sequence of values, under the following...
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Derivation of Relations in Expected Hitting Times
i'm working through the book Markov Chains by Norrishow can one establish the following relations within its...
View ArticleExpected Number of Flips for a Sequence of 4 to Repeat
I recently had this question in an interview:You are flipping a fair coin until a sequence of four flips repeats itself. The sequences are allowed to overlap. What is the expected number of flips?For...
View ArticleCan one bound a moment of a random variable with bounded support if given the...
If $X$ is a random variable whose support is a bounded interval, say, $[0, 1]$, and the 1st through $n$th moments $\mathbb{E}[X] = m_1, \mathbb{E}[X^2] = m_2, \dots, \mathbb{E}[X^n] = m_n$ are known,...
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Number of tries in key guessing game
That's the game (from Brilliant.com): The key is hidden in this $8\times 8$ matrix. If we looked for the key under a tile, we will not look there again. So after 64 tries, we definitely will have found...
View ArticleThe difference between the expectation of the inverse of a matrix and the...
Suppose we have a random matrix $\mathbf{A}=\mathbf{a}\mathbf{a}^{\top}+\lambda\mathbf{I}$ ($\mathbf{A}$ a symmetric matrix), and we want to know the difference between $\mathbb{E}(\mathbf{A}^{-1})$...
View ArticleExpected number of cards greater than at least one neighbor
Six cards numbered from $1$ to $6$ are randomly shuffled and then arranged in a circle. Given that $1$ and $6$ are adjacent to each other, what is the expected number of cards larger than at least one...
View ArticleExpectation of Smallest Card in Half Deck
We are given a standard 52-card deck, where ace represents 1, jack represents 11, queen represents 12, and king represents 13. If we randomly shuffle this deck then split it into two decks of 26 cards,...
View ArticleProving $\int^{\infty}_{-\infty} \frac{p(x-\mu)}{0.5p(x-c) + 0.5p(x+c)} p(x)...
Let $p$ be a Lebesgue density function with infinite support (i.e. $p(x)>0 \forall x\in \mathbb{R}$ and $\int p(x) dx = 1$). Moreover, assume that $p$ is even (i.e. $p(x) = p(-x)$) and that $p(x)...
View ArticleRadon-Nikodym conditional expectation [closed]
If $X$ is integrable, $G \subset F$ then $E(X|G)$ exists and it is unique.Proof:Assume $X\ge0$ and define a probability $P_1$ on the $\sigma$-field $G$ by$P_1(A)=\frac{1}{E(X)}\int_AXdP$By...
View ArticleHölder's inequality Exapectation for $n$ random variables. [closed]
I am trying to show that, if $X_1,X_2,\dots,X_n$ are positive random variables, then$$E[X_1 X_2 \dots X_n] \le E[X^{n}_1]^{\frac{1}{n}}E[X_2^{n}]^{\frac{1}{n}}\dots E[X_n^{n}]^{\frac{1}{n}}.$$Like...
View ArticleCollectiong Toys-II [closed]
Every box of cereal contains one toy from a group of $5$ distinct toys, each of which is mutually independent from the others and is equally likely to be within a given box. How many distinct toys can...
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Calculating expected value in baby probability vs measure-theoretic probability
How does the baby probability definition of expected value agree with the measure-theoretic definition?I am currently learning measure-theoretic probability using the book by Athreya and Lahiri (a...
View ArticleIf for $E(X)$: $E[X] = \sum_{m=1}^{\infty} P(X \geq m)$ then for $E(X^2)$?
$E[X^2] = \sum_{m=1}^{\infty} m \cdot P(X \geq m)$ ?need to prove: $\text{Var}(X) = 2 \cdot\sum_{m=1}^{\infty} m P(X \geq m) \ - E(X)(E(X) + 1)$I did the easy part (I hope that's right) : $E(X^2) +...
View Articlefinding lower bound on expectation of $X_i*X_j$ given a lower bound on the...
let $$X_0,X_1,...,X_n$$ be a non-independent RVs such that each R.V is uniform on $\{1,-1\}$let $z \in (0,1)$, show the existence of $y \in (0,1)$ (y depends solely on the choice of z)s.t that if...
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