# Questions tagged [expected-value]

Questions about the expected value of a random variable.

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### Expectation of a random variable related to expectation of its indicator function

Let $X^*$ be a random variable and let $X = 1[X^*>0]$. Where $1[.]$ is the indicator function. I am interested to express the expected value of $X^*$ in terms of the expected value of $X$. In ...
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### Relating expectation of a difference of random variables with expectation of their indicator function

Let $X^*$, $Y^*$ as two random variables; respectively with pdfs $P_{X^*}$ and $P_{Y^*}$ Let $X = 1[X^*>0]$ and $Y = 1[Y^*>0]$. Where $1[]$ is the indicator function. I am interested in finding ...
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### Dominance of exponential moment due to Stochastic dominance

Suppose $X$ and $Y$ are positive random variables such that $P(X\geq x)\leq P(Y\geq x)$ and let $\lambda >0$ Then can we say $$E(e^{\lambda X})\leq \int_{0}^{\infty}e^{\lambda x}P(Y\geq x)\,dx$$ ...
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### Density of normal c.d.f. of a normal random variable

Let $a$ be a constant, X is a standard normal and $\Phi$ is the c.d.f. of standard normal. what is the density of $Y$ where $$Y = \Phi(aX)$$ what is the density of $$Y=\Phi(X + a)$$ The second case ...
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### Upper bounds for E(f(x))

Given a discrete random variable X, with finite mean $\mu$ and variance $\sigma^2$ and a convex function f(x), what is the tighter upper bound one can give for E(f(x))? I am looking for something in ...
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### Find variance and expectation with given probabilities of variable values

I've got random variable $x$ and probabilities: $P(x=0)=1/4, P(x\in[a,b])=\cfrac{b-a}{2}, P(x=3)=1/4$ where $1\leq a < b \leq 2$. What I did: First I tried to calculate the mathematical expectation ...
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### prove $E\frac{X^k}{\omega X^{2r}+\sigma}<\infty$ for $\omega,\sigma>0, k\in\{0,1,...,2r\}$, $r\in \mathbb{N}$

I have found a statement here equation 23 without explanation that $E\frac{X^k}{(\omega X^2+\sigma)^r}<\infty$ for $\omega,\sigma>0, k\in\{0,1,...,2r\}$, $r\in \mathbb{N}$, where we don't know ...
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### Find conditions on $f,g$ $E[ \langle g(X), f(X) \rangle ] \ge \langle E[ g(X)], E[f(X)] \rangle$ (i.e., FKG inequality for multivariate functions))

The classical FKG inequality states that for two non-decreasing function $f,g: R \to R$ we have that \begin{align} E[ f(X) g(X) ] \ge E[f(X) ] E[g(X)] \end{align} I am interested in understanding ...
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### Marginalization of the conditional expectation

Let $Y, Z, X_1, X_2, T$ be random variables. Let $\mathbb{E}(Z\mid X_1, X_2) = \mathbb{E}[Y\mid T=t, X_1, X_2]$. I need a property $$\mathbb{E}(Z\mid X_1) = \mathbb{E}[Y\mid T=t, X_1].$$ Does it hold ...
### Lower-bound for $E(\log((x+1)!))$? [closed]
I have this random discrete variable $X$ such that $E(X)=\mu$. Is there a closed form for the value of $E(\log((x+1)!))$? If not, is there a lower bound for it, for example using Jensen inequality? I ...