Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [expectation]

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

3734 questions
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Does $E(X-E(X))=0$?

In a book I was reading, it seemed to imply that $E(X-E(X))=0$. My intuition tells me this is true, because if $E(X)$ is the "centre", then the average displacement from this centre should be 0. However, can someone show me a formal proof (assuming…
someguy
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Expected Number of coin flips for 2 consecutive heads for first time.

I was working on problems on expectation and found this one as a question from a well-known exam Assume that you are flipping a fair coin, i.e. probability of heads or tails is equal. Then the expected number of coin flips required to obtain two…
user3767495
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$8$ cards are drawn from a deck of cards without replacement

I draw $8$ cards randomly from a shuffled deck without replacement. What is the expected value of the sum of the largest $3$ cards? The ace is given a value of 1 and the jack, queen and king are all given a value of $10$. I can generate a simulation…
bobbym
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Expectation of sum of independent random variable

Let, for $m\neq 1$, $X_1, X_2\ldots$ be independent random variables with $E(X_n) = m^n, n \ge 1$, let $N \sim \text{Poisson}(\lambda)$ be independent of $X_1, X_2\ldots$ and set $$Z = X_1 + X_2 + \ldots + X_N$$ Determine $E(Z)$. It is trickier than…
user133140
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Derivative of an Expectation over an Indicator function

I have a small question on how to compute the derivative of an expectation when an indicator function gets in the way. Let $x$ be a random variable. We are interested in computing the derivative wrt A of $E_x [(A-x)1_{(x \leq A)}]$ where $1_n$ is…
fox
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Does a second moment of zero imply a mean of zero?

I was wondering about this. I have a second moment that is zero. Can I conclude that if $E[x^2]=0$ then $Var[x]=E[x^2]-E[x]E[x]$ implies $E[x]=0$ since the variance must be non-negative? (If not, can I imply anything else from that, in particular,…
Majte
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Given $l$ points in unit $n$-sphere, expectation of smallest radial distance

Consider the unit $n$-sphere and $l$ points distributed uniformly randomly inside its hypervolume. What is the expectation of the smallest distance from a point to the sphere centre? Steps done so far are: Find distribution function of distance from…
NotImplemented
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Is $E\left[\frac{1}{\sum_{i=1}^{n}X_i}\right]$ = $\frac{1}{\sum_{i=1}^{n} E\left[X_i \right]}$?

If $X_i$'s are i.i.d. random variables then is this statement true? $$E\left[\frac{1}{\sum_{i=1}^{n}X_i}\right] = \frac{1}{\sum_{i=1}^{n} E\left[X_i \right]}$$ Here $E\left[X\right]$ is the expected value of a random variable $X$ Edit - I was…
Banach Tarski
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Inequality involving Expectation 3

How do we prove $\mathbf{E} \left(\frac{1}{X} \right) \geq \frac{1}{\mathbf{E}(X)}$ for random variable $X$? Can we use Jensen's inequality ?
Milan Amrut Joshi
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How to find expectation of geometric distribution 2?

Alec and Bill take alternate turns at kicking a football at a goal, and their probabilities of scoring a goal on each kick are $p_1$ and $p_2$ respectively, independently of previous outcomes. The first person to score allows the other person one…
Leon
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Why $E(X\mid X^2+Y)=0$ for $(X,Y)$ standard normal?

Please help me explain why $E(X\mid X^2+Y)=0$ for $(X,Y)$ standard normal, i.e $(X, Y) \equiv (0, I_2)$.
bankrip
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Expected matching number

I have this question: Let's say we have $n$ students sitting in $n$ seats, Now we get the students out of the seats and randomly assign them to the $n$ seats, what is the expected students sitting in their original seats? For example, with $2$…
faz
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Quick question for covariance

according to the definition Cov(X,Y)=E[XY]-E(X)E(Y) I happen to get a negative value, I guess there is a problem? When I tried to get the correlation since the E(X) and E(Y) are really big I ended up with another negative value which cannot be…
natsu
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Taking the expectation of a random variable. With

Let the probability measure Q be $Q(A)=E[X^p1_A]/E[X^p]$ Then the expectation of Z with respect to probability measure Q is $E_Q[Z]=E[ZX^p]/E[X^p]$ Why is this? Shouldn't taking the expectation involve integrating over the support of Z? Is that…
JJ chips
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Find joint expectation value $\mathbb E(\sqrt{X^2+Y^2})$

Let the pair (X,Y) be uniformly distributed on the unit disc, so that $f_{X,Y}(x,y)=\begin{cases}\frac{1}{\pi}&\text{if }x^2+y^2\leq1,\\0&\text{otherwise}.\end{cases}$ Find $\mathbb E\sqrt{X^2+Y^2}$ and $\mathbb E(X^2+Y^2)$. We are not familiar…
Sha Vuklia
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