query: Prove that $\mathbb{E}[e^{\mu+\sigma Z}\mathbb{I}_{\{Z > -d\}}]=e^{\mu+\sigma^2/2}Φ(d+\sigma)$
positive_passages.docid: 23246
positive_passages.text: Challenge question; How do I prove the expected value of the following?
negative_passages.docid: 73367
negative_passages.text: On $E[E[Y|X]|X]= E[Y|X]$
negative_passages.docid: 27347
negative_passages.text: Verify that $\left|\int_{\gamma} \exp(iz^2)dz\right| \leq \frac{\pi\big(1-\exp(-r^2)\big)}{4r}$ where $\gamma(t)=re^{it}$, for $0\leq t \leq \pi/4$.
negative_passages.docid: 82729
negative_passages.text: Prove that $\mathrm E\left({\max}\left\{X^2 , Y^2\right\} \right) \leq 1 + \sqrt{1 - \rho^2}$ where $\rho$ is their correlation.
negative_passages.docid: 132920
negative_passages.text: Help with proof that the expected value of $x_i$ is $\bar{X}$
negative_passages.docid: 68097
negative_passages.text: Prove $\int\limits_{0}^{\infty} \mathrm{exp}(-ax^{2}-\frac{b}{x^{2}}) \mathrm{d} x = \frac{1}{2}\sqrt{\frac{\pi}{a}}\mathrm{e}^{-2\sqrt{ab}}$
negative_passages.docid: 67489
negative_passages.text: How to show that $E[(\hat\theta -\theta)^2]<Var(\bar X)=\dfrac{1}{n}$?
negative_passages.docid: 44270
negative_passages.text: $E$ measurable set and $m(E\cap I)\le \frac{1}{2}m(I)$ for any open interval, prove $m(E) =0$
negative_passages.docid: 34693
negative_passages.text: Prove $E^{\mathbb Q}[Y]=E^{\mathbb P}[XY]$ if $E^{\mathbb P}[X]=\mathbb P(X>0)=1$ and $ \mathbb Q(A)=E^{\mathbb Q}[X1_A] $
negative_passages.docid: 42193
negative_passages.text: Show that $E(Y^p)=\int_{0}^{\infty}px^{p-1}P(Y\geq x)dx$
negative_passages.docid: 68385
negative_passages.text: Prove that $ \frac{n}{\phi(n)} = \sum\limits_{d \mid n} \frac{\mu^2(d)}{\phi(d)} $
negative_passages.docid: 42366
negative_passages.text: Prove $\mathbb{P}(\sup_{t \geq 0} M_t > x \mid \mathcal{F}_0)= 1 \wedge \frac{M_0}{x}$ for a martingale $(M_t)_{t \geq 0}$
negative_passages.docid: 64856
negative_passages.text: Finding $E(X\mid X>Y)$ when $X,Y$ are i.i.d $U(0,1)$
negative_passages.docid: 125894
negative_passages.text: Prove that $\mathbb{Q}$ belongs to the Borel $\sigma$-algebra of $\mathbb{R}$
negative_passages.docid: 147705
negative_passages.text: Show that $\mathbb{E}(T) = \sum\limits^\infty_{k=1}\mathbb{P}(T \geq k)$ for $T$ nonnegative integer valued and $E[T] < \infty$
negative_passages.docid: 24585
negative_passages.text: Proving that $\mathbf{W}$+$\mathbf{W^{\perp}}$=$\mathbb{R^{n}}$
negative_passages.docid: 157432
negative_passages.text: How to use the Mean Value Theorem to prove the following statement: