For questions about the expectation of a random variable: computations, upper/lower bounds, etc.

# Questions tagged [expectation]

3755 questions

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### Determine mean and standard deviation given -

Find the mean and standard deviation for 30 objects given
∑(x - 40) = 315
∑(x - 40)2 = 4022
Normally, questions like these have to deal with expectation and variance, however, I don't really understand how to deal with the additional number within…

lessaint

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### Another balls game

I have an infinite bag of balls. Balls can be black and white.
Well, infinite, but I will get black ball from there with probability $p=0.6$ and white with $1-p=0.4$.
I have two friends. One chooses black color and another one -- white color. With…

user324463

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### if $X\sim \operatorname{Bin}(n,p)$ and $Y={X \choose 2 }$ What is the intuition behind $E(Y) = {n \choose 2}p^2$

A possible motivation for what $Y$ represents is:
given a graph with $n$ vertices and no edges, each vertex in the graph is painted red with probability $p$. After all vertices are either painted red or not painted, an edge is drawn between every…

Adi

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### A maximal inequality for random variables with finite $L^{p}$ norm

I would appreciate any help with the following question from Kosorok's Introduction to Empirical Processes and Semiparametric Inference (Q10.5.2):
Show that for any $p>1$ and any real i.i.d. $X_{1},\ldots, X_{n}$ with
$E |X|^{p}<\infty$, we have…

möbius

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### Setting up questions relating to expected number

I'm having difficulty setting up questions relating to finding the expected number of something. So, for example, this question "Eight cards are drawn with replacement from a standard 52 card deck. Find the expected number of different ranks (Ace,…

Gerald Tan

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### Random Walks with independent symmetric relations

Let ($S_n^{(1)}$)n≥0, . . . ,($S_n^{(1)}$
(1))n≥0 be independent symmetric random walks on the integers,
each starting at 0. Consider the RW Sn = ($S_n^{(1)}$ , . . . , $S_n^{(1)}$ ) on the lattice $Z^d$. Show the dimensions d in which this walk is…

james black

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### Median of Random Variable

Let X be an integrable random variable. Show that the function $a \mapsto E|X −a|$ attains its minimum at $a = \mathrm{Med}(X)$.
I think this means as $a$ approaches median of X. This makes intuitive sense since the expected value of a random…

TomWangzilin

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### renewal process with underlying distribution

Consider a renewal process with underlying distribution function $F(x)$.
Let $W$ be the time when the interval duration from the preceding renewal event first exceeds $\xi > 0$ (a fixed constant). Determine an integral equation satisfied by
$$ V(t)=…

Rosa Maria Gtz.

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### Variance of the real part of a complex random variables

Let $S = \{ x : x^3 - 1 = 0, x \in \mathbf{C}\}$, denote the 3rd roots of unity. Let $c \in \mathbf{R}^n$ be a vector, and let $a^Tb = \sum_{i=1}^n a_i b_i$, for $a$, a real vector, and $b$ a complex vector. Let $X$ be the random vector with $X_i…

Drew Brady

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### How to calculate expectation for the variables with given density

Let $a_i \in\mathbb R^N,\, i=1,\ldots, N$ and let $x_i, i=1, \ldots, N$ be independent random variables with density with respect to the Lebesgue measure given by $\dfrac{e^{-|t|^q}}{\Gamma(1+1/q)}, \, t\in\mathbb R, q\geq 1$.
My question is how to…

Anna

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### Uncorrelated Random Variables

If $X_1, X_2$ are $2$ random variables such that $(X_1, X_2)$ and $(-X_1, X_2)$ have the same joint distributions then show that $X_1$ and $X_2$ are uncorrelated.
I know that to be uncorrelated the $Cov(X_1, X_2) = E(X_1X_2)-E(X_1)E(X_2) =…

Note

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### Why is it bad to multiply two expectations of the same variable?

In Sutton & Barto's book: Reinforcement Learning (chapter 11.5) they say that it is bad to multiply two expectations of the same variable, as otherwise the sample of the product will be biased.
Why is that the case?
Excerpt from the book: (part…

Toke Faurby

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### Moments of products of independent random variables: $E[ X^kY^k ]$

If we are given that two random variables $X$ and $Y$ are independent, I'm wondering if the rule: $E[XY] = E[X]E[Y]$ applies for any integer $k>0$, such that:
$E[X^kY^k] = E[X^k]E[Y^k]$.
Is this a straight forward result? or am I missing something…

kentropy

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### Product of the three numbers obtained by rolling three dice. Expectation and Variance

What is the expected value and variance of X, the product of the three numbers obtained by rolling three fair die?
I tried solving this problem by dividing the numbers $\{1,...,216\}$ into primes and composites, but realized there were too many. The…

Sally G

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### Convergence in distribution for nonnegative R.V.s

Problem. Let $Z, Z_1, Z_2, \cdots, Z_n$ be non-negative R.V.s with integer values. Prove that $Z_n \to Z$ in distribution, i.f.f $Pr(Z_n=t) \to Pr(Z=t)$ for all non-negative $t \in N$.
I have read about the Portmanteau theorem which states that if…

james black

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