$E[E(ZX\mid Y)\mid Z,Y]$ can be simplified? Any comment would be much appreciated.
121523
I know that $E[E(X\mid Y,Z)\mid Y]=E(X\mid Y)$.  What if $E[E(X\mid Y)\mid Y,Z]=?$ It has any meaning?
91112
Given that $X,Y\stackrel{\text{i.i.d.}}{\sim}\text{Exp}$ with mean $\lambda(&gt;0)$ and $Z=\mathbf{1}_{X&lt;Y}$, what is $\mathbb{E}(X\mid Z=z)$ ?   We have $Z=1$ with probability $\Pr(X&lt;Y)$ and $Z=0$ with probability $1-\Pr(X&lt;Y)$.   But $\Pr(X&lt;Y)=\frac{1}{2}$ as $X$ and $Y$ are i.i.d. random variables.   Does this mean I can simply say that $Z\sim\text{Ber}\left(\frac{1}{2}\right)$ ?   Then, $\mathbb{E}(X)=\mathbb{E}(X\mid Z=0)\times\frac{1}{2}+\mathbb{E}(X \mid Z = 1) \times \frac{1}{2}$  $\implies2\lambda=\mathbb{E}(X\mid Z=0)+\mathbb{E}(X\mid Z=1)$.   But I cannot conclude anything from here. I couldn't find the conditional distribution of $X\mid Z$ either. Maybe I have to condition on another variable $U$ so that $\mathbb{E}(X\mid Z)=\mathbb{E}\,[\mathbb{E}(X\mid Z,U)\mid Z]$.
24993
I don't really know how to start proving this question.  Let $\xi$ and $\eta$ be independent, identically distributed random variables with $E(|\xi|)$ finite.  Show that $E(\xi\mid\xi+\eta)=E(\eta\mid\xi+\eta)=\frac{\xi+\eta}{2}$  Does anyone here have any idea for starting this question?
32208
Suppose $y$ is a continuous random variable and $d$ is a binary random variable that takes the value $1$ with probability $p$ and $0$ with probability $1-p$.   How do I show that $\text{Cov}(y,d)=(E[y|d=1]-E[y|d=0])p(1-p)$?
26817
\begin{multline*} \displaystyle \left \| f  \right \|_{M\dot{K}^{\alpha(\cdot),q}_{p(\cdot), \lambda}(\mathbb{R}^{n})} \approx \max \left\{ \sup_{L\leqslant0,L\in  \mathbb{Z}}2^{-L\lambda}\left ( \sum_{k=-\infty}^{L}2^{k\alpha(0)q}\left \|  f\chi_{k} \right \|^{q}_{L^{p(\cdot)}} \right )^{1/q},\\\sup_{L&gt;0,L\in \mathbb{Z}}\left[ 2^{-L\lambda}\left ( \sum_{k=-\infty}^{-1}2^{k\alpha(0)q} \left \| f\chi_{k} \right \|^{q}_{L^{p(\cdot)}} \right )^{1/q}\\ +2^{-L\lambda}\left ( \sum_{k=0}^{L}2^{k\alpha(\infty)q}\left \|  f\chi_{k} \right \|^{q}_{L^{p(\cdot)}} \right )^{1/q}\right] \right\}  \end{multline*}
9295
I'm trying to follow a line in a derivation for $P(Z&gt;X+Y)$ where $X,Y,Z$ are independent continuous random variables distributed uniformly on $(0,1)$.  I've already derived the pdf of $X+Y$ using the convolution theorem, but there's a line in the answer that says:  $P(Z&gt;X+Y) = \mathbb{E}[\ P(Z&gt;X+Y\ |\  X+Y )\  ]$ where $\mathbb{E}$ is the expectation.  I'm not familiar with this result. Could anyone give a pointer to a similar result if one exists?  Thanks.
95925
Let $X$ be a geometric random variable with parameter $p$, find the expectation of $E[1/X]$.  I need help simplifying the series.
109109
I am reading about the proof of law of expectation for the discrete type,   $$ \begin{align} E[E[X|Y]] &amp;=  E \left[ \sum_{x} x \cdot P(X = x | Y) \right] \\ &amp;= \sum_y \left[ \sum_{x} x \cdot P(X = x | Y = y) \right] P(Y = y) \tag 2) \\ &amp;= \sum_x x \sum_y  P(X = x | Y = y) \cdot P(Y = y) \tag3) \\ &amp;= \sum_x x \sum_y P(X = x \, \text{and} \, Y = y) \tag4) \\ &amp;= \sum_x x \cdot P(X = x) \tag5)\\ &amp;= E[X] \end{align} $$     Question: Why does it assume Y=y? And can someone give me justification of each step?
67065
Suppose that we observe the discrete random variable $ X = (X_1, \dotsc , X_n)$ with state space S, whose distribution we do not know but we are assuming that its joint p.m.f. belongs to a known family $\{f_θ : θ ∈ Θ\}$. We derive a sufficient statistic $T(X)$.  Is it correct to write $\Bbb E[X]$? If not, why? Would it be more correct to write $\Bbb E_{\theta}[X]$?
31081
So I need to compute $$E\left[\frac{\min\{|X|,|Y|\}}{\max\{|X|,|Y|\}}\right]$$ given $X,Y \sim$ Normal$(0,1)$ and independent.   What I am having trouble seeing is    whether $\min\{|X|,|Y|\}$ and $\max\{|X|,|Y|\}$ are independent. If so I can factor the expectation into $$E\left[\min\{|X|,|Y|\}\right]\cdot E\left[\frac{1}{\max\{|X|,|Y|\}}\right]$$ how I should find the expected value of $1/\max\{|X|,|Y|\}$.
119522
I understand the proof of this first formula.  I am confused about where he got the first E[min(X, k)] formula. It seems to have just come out of nowhere. And also, can someone explain why the two formulas are correct please?
114331
Let $X : \Omega \to \mathbb{N} $ be a r.v. Then $$ \mathbb{E} \{ X \} = \sum_{n=0}^{\infty} P(X &gt; n) $$ Attempt: $$ \mathbb{E} \{ X \} = \sum_j j P(X = j ) = \sum_j j(1 - P(X &gt; j) - P(X&lt;j ) )$$ But here it gets complicated. Am I on the right track?
114030
I have up until Cov(X+Y,X-Y) = E[((X-E(X))+(Y-E(Y)))((X-E(X))-(Y+E(Y)))] and now I am stuck and do not know how this turns into Var(X)-Var(Y)
95095
I have a seemingly trivial question regarding conditional expectation. Consider $x$ and $y$ be two integrable random variables on Probability space (X, $\Sigma$, $P$) such that $$E(X|Y) =_{a.s} Y$$ and $$E(Y|X) =_{a.s} X$$ Show that $$X=_{a.s} Y$$  The thing looks like a simple proof result, but I don't know how to move on after showing that$$\int_{B} Y = \int_{B} X$$ for all $B \in \sigma(x)$ and $B \in \sigma(y)$  Now any hint on how to proceed? Thank you so much in advance
56322
Given a random variable $X\sim\mathcal N(0,\sigma^2)$, how can we prove that $E[X^4]=3\sigma^4$? I am having trouble even starting with the proof.
21440
$X_1, X_2 \dots X_n$ represent independently observed Bernoulli random variables.  $Z_1, Z_2, \dots Z_n$ are unobserved.  $Z_i | \theta_i \sim N(\theta_i,1)$  $\theta_i \sim N(\epsilon, \sigma^2)$  $X_i = \begin{cases} 0 &amp;  Z_i \leq u \\ 1 &amp; Z_i &gt; u \end{cases}$ where $u$ is known.  Show that the EM sequence for estimation of $\epsilon$ is given by:  $\epsilon^{t+1} = \frac{1}{n} \sum_{i=1}^n E[Z_i|X_i, \epsilon^t, \sigma^2]$  $\Phi$ represents CDF of standard normal.  Attempt  $Z_i \sim N(\epsilon, \sigma^2+1)$   \begin{align*} f(X,Z| \epsilon, \sigma^2) &amp;= \prod_{i=1}^n p_i^{x_i} (1-p_i)^{1-x_i}\\ p_i &amp;= P(Z_i \leq u)\\ &amp;=\Phi(\frac{u-\epsilon}{\sqrt{\sigma^2+1}}) \end{align*}  Complete log likeihood = $\sum_{i=1}^nx_i \log p_i + (n-\sum_i x_i) \log(1-p_i)$  I am not able to proceed further.
62382
I'm less than a novice in analysis, I don't even know how to approach this. Thanks in advance.