All Questions
Tagged with expectation or expected-value
10,965 questions
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Expected value of $(X-Y)sgn(A-B+X-Y)$ [closed]
Continuous random variables $A$, $B$, $X$ and $Y$ are independent. $A$ and $B$ have the same distribution. $X$ and $Y$ have the same distribution. What is the expected value of $(X-Y)\operatorname{sgn}...
- 101
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How to derive the $E(X)$ from $f(x,y)$
I want to compute the expectations for the joint density
$$
f(x, y)= \begin{cases}12 y^2 & 0 \leq y \leq x \leq 1, \\ 0 & \text { otherwise } .\end{cases}
$$
Here the underlying probability ...
- 43
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Expectation of an absolute value
Let $X,Y$ be two i.i.d.
I am trying to bound the expectation of how afar from one another they can get?
That is, $E[|X-Y|]$. I know that:
$$ E[X-Y] = E[X]- E[Y] = 0$$
But what about $|X-Y|$?
One ...
- 1,059
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Why does expectation project onto constant functions?
Let $X \in L^2$. Then $Z = E[X|G]$, for some sub $\sigma$-algebra $G$, is the orthogonal projection of $X$ onto $L^2(G)$. That is $Z$ is the random variable such that for every $G' \in G$:
$$\int_{G'} ...
- 969
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In the limit, what probability of tails maximizes the expected number of shorter runs of heads followed by a longer run of tails.
We flip a coin $n$ times. The probability of tails is $p$.
Let $X$ be the number of occurrences of a run of heads followed by a longer run of tails. We can also agree that if the sequence begins with ...
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How to derive the distribution of a 2D random walk?
A few years prior, an acquaintance of mine tackled a problem inspired by something in our statistics class, which basically was the idea of "what is the expected distance from the starting point ...
- 3,692
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Expected value of the logarithm of a sum of exponentially-growing values.
I have a problem related to the idle game Clicker Heroes, in finding the expected $\log_{10}$ value of the rewards gained from defeating bosses throughout an ascension. For boss number $k$, the reward ...
26
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How should a string be laid on a disk to minimize the expected shortest distance between a random point on the disk and the string?
On a unit disk, how should a string of length $2$ be laid in order to minimize $E$, the expected shortest distance between a uniformly random point on the disk and the string?
For example, the ...
- 40.3k
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the limit of variance and expectation
Let $\varepsilon_1, \dots, \varepsilon_n$ be independent random variables with $E(\varepsilon_i) = 0$.
Let $f: [0,1] \to \mathbb{R}$ be a Lipschitz function with constant $K > 0$, i.e.,
$$|f(x) - f(...
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Check my understanding of Law of Iterated Expectation proof from definition please
I need help understanding law of iterated expectation in its most basic form. I want to make sure my understanding is correct.
We have $E(E(X|Y)) = E(X)$.
We know $E(X) = \sum xP(X=x)$ for discrete ...
- 33
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1
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80
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Making sense of Stieltjes integral for distributions with jumps
Assume I have a random variable $X$ with distribution function $F$.
Its expectation would be the Stieltjes integral wrt the distribution function:
$$E[X]=\int_{-\infty}^{\infty} t d F(t)$$
Assume now ...
- 232
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2
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271
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How to make a decision with probability (expected value)?
A decision must be made as to whether to do nothing and accept a cost of X or eliminate the cost by enaging a decision that costs nominal amount Y, where Y is approximately 1/3 of X.
Without getting ...
- 129
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180
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Expected number of rolls until all dice are removed, with special rules at one die
Starting with a set of $n$ dice, roll all of them together, and remove any that come up 1, 2, or 3. Repeat this until all the dice are removed.
However, if at any point you have one die left, you can ...
- 31
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Expected time to exit first quadrant in 2D random walk
Let $W \subseteq \mathbb{R}^2$ be a finite set of vectors, $ P$ be a probability distribution on $W$, and $V_0\in \mathbb{R}^2$ (for simplicity it suffices to consider $V_0$ where both coordinates are ...
- 101
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Many dice expected sum after removing sets [duplicate]
You roll 100 dice onto a table. You then repeatedly remove sets of 6 distinct dice (i.e. the set 1, 2, 3, 4, 5, and 6) until it is no longer possible to do so. What is the expected value of the sum of ...
