This question is an additional question to the given posted here:   Let $X, Y$ be independent $N(0,1)$ and let $M=Max(X,Y)$. In the previous problem, what was being asked was the covariance between $M$ and $m=Min(X,Y)$. My question is what if the only requirement is to find $E(M)$? Is it necessary to define $Min(X,Y)$ and use it to eventually solve for $E(M)$?
122759
Let $X$ and $Y$ be iid $\sim Normal(0,1)$  Let $A=max(X,Y)$ and $B=min(X,Y)$  What are $Var(A)$ and $Var(B)$?  From simulation, I get $Var(A)=Var(B)$ approximately 0.70.  How do I get this analytically?
71659
I'm studying associated variables now and in one proof the following Lemma is used:  Let $X$ and $Y$ be random variables with distribution $F_X$ and $F_Y$ and joint distribution $F$. Then \begin{align} &amp; \mathbb{E}[XY] - \mathbb{E}[X] \mathbb{E}[Y] \\[8pt] = {} &amp; \int_{- \infty}^\infty \int_{- \infty}^\infty (F(x,y)-F_X(x) F_Y(y)) \,dx\,dy, \end{align} if $\mathbb{E}[XY]$, $\mathbb{E}[X]$ and $\mathbb{E}[Y]$ exist.  The proof begins with introducing to vectors of independent variables:  Let $(X_1,Y_1)$ and $(X_2,Y_2)$ be independent, each distributed to $F$.   Define $I(u,x)=\begin{cases}1&amp; \text{if } u \leq x \\ 0 &amp; \text{otherwise} \end{cases}$.  Then it is said, that  $$ 2 (\mathbb{E}[X_1Y_1] - \mathbb{E}[X_1] \mathbb{E}[Y_1]) = \mathbb{E}[(X_1 - X_2)(Y_1-Y_2)] $$ $$ = \mathbb{E} \left[\int_{- \infty}^\infty \int_{- \infty}^\infty [I(u,X_1)-I(u,X_2)][I(v,Y_1)-I(v,Y_2)] \, du \, dv \right].$$ The first equation is clear for me, but I don't get the second one. Some help would be nice here.  Furthermore the proof ends with the argument, that we can take the expectation under the integral sign since the expectation is finite, thats clear. Then we are ready, but I don't get the right result by calculating.To make the proof fit, we would have to end up at 2 times the right handside of the statement.  What I also observed is that this inequality gives a simple proof for $\mathbb{E}[XY] \geq \mathbb{E}[X] \mathbb{E}[Y]$.
64136
Let $X,Y$ be independent $Uniform(0,1)$ random variables. Find $E(|X-Y|^a)$ where $a&gt;0$.  My working:  Define $W=1$ if $X&gt;Y$ and $W=0$ if $X&lt;Y$. We seek $E(|X-Y|^a)=E[E(|X-Y|^a|W)]=E(|X-Y|^a|W=1)P(W=1)+E(|X-Y|^a|W=0)P(W=0)=E((X-Y)^a)P(X&gt;Y)+E((Y-X)^a)P(X&lt;Y)$. Now $P(X&lt;Y)=\dfrac{1}{2}$ and $P(X&gt;Y)=\dfrac{1}{2}$ by symmetry. $E((X-Y)^a)=\int_{0}^{1}\int_{0}^{x}(x-y)^af_{X,Y}(x,y)dydx=\int_{0}^{1}\int_{0}^{x}(x-y)^adydx=\dfrac{1}{(a+1)(a+2)}$. Similarly $E((Y-X)^a)=\dfrac{1}{(a+1)(a+2)}$ by symmetry when $Y&gt;X$. Required expectation=$\dfrac{1}{(a+1)(a+2)}\dfrac{1}{2}+\dfrac{1}{(a+1)(a+2)}\dfrac{1}{2}=\dfrac{1}{(a+1)(a+2)}$  But correct answer is $\dfrac{2}{(a+1)(a+2)}$. Where am I going wrong?
1225
I know that the variance of the difference of two independent variables is the sum of variances, and I can prove it. I want to know where the covariance goes in the other case.
51403
Let $X\sim N(0,1)$ and let $Z$ be a random variable independent of $X$ such that:  \begin{equation} \Pr(Z=z) = \begin{cases} \frac{1}{2} &amp; \mbox{if $z = -1$ or $z=1$}, \\ \\ 0 &amp; \mbox{otherwise}. \end{cases} \end{equation}   Define $Y = ZX$. Show that $Y \sim N(0,1)$. Determine $\operatorname{Cor}(X,Y)$. Are $X$ and $Y$ independent?   Progress:  I'm basically stuck at the beginning. I got as far as $P(ZX &lt; y) = P(Z &lt; y/X)$, which I'm not even sure is correct. I'm also not clear on what it means by defining $Y = ZX$ - does it mean I need to state the probability density function or something? And how do I show that $Y \sim N(0,1)$?
54639
Let $X_i \sim N(\mu,1)$, i.i.d. We aim to find UMVUE for $p(\mu) = P_{\mu}(X_1 \leq u)$ for some fixed $u$.  I have shown that $\bar{X}$ and $X_1 - \bar{X}$ are independent. ($\bar{X}$: sample mean).  Now I want to use this to compute $E_\mu[1\{X_1 &lt; u\}\mid\bar{X}]$ and if I am not mistaken, by Lehmann-Scheffé, it is UMVUE.  I am quite new in statistics, maybe my arguments are wrong, or I do not know how to compute. Any hint appreciated.
49656
I have been trying to compile conditions for when characteristic polynomials equal minimal polynomials and I have found a result that I think is fairly standard but I have not been able to come up with a proof for.  Any references to a proof would be greatly appreciated.  Let $M\in M_n(\Bbb F)$ and let $c_M$ be the characteristic polynomial of $M$ and $p_M$ be the minimal polynomial of $M$.     How do we show that $p_M = c_M$ if and only if there exists a column vector $x$ such that $x, Mx, \ldots M^{n-1} x$ are linearly independent ?
60383
If $X \sim \mathrm{Normal}(\mu,\sigma^2)$ and $Y \sim \mathrm{Normal}(\mu,\sigma^2)$ are independent random variables, how do I prove that $X+Y$ and $X-Y$ are also independent?  What happens with the independence between $X+Y$ and $X-Y$ when $X \sim \mathrm{Normal}(\mu_x,\sigma_x^2)$ and $Y \sim \mathrm{Normal}(\mu_y,\sigma_y^2) $  Thank you
80741
I'm trying to understand my teacher's solution to this problem:     Let $X_1,X_2,...$ sequence of iid random variables with expected value   $\mu$. Define $Y_n = \frac{2}{n(n+1)} \sum_{j=1}^{n} jX_j$. Prove that   $Y_n \to_\mathbb{P} \mu$.   His solution:  Let $\sigma^2 = \rm{Var}(X_i)$.  $E(Y_n) = \frac{2}{n(n+1)} \sum_{j=1}^{n} jE(X_n) = \mu$.  $\displaystyle\rm{Var}(Y_n) = \frac{4}{n^2(n+1)^2} \sum_{j=1}^{n} j^2 \rm{Var}(X_n) = \frac{4\sigma^2}{n^2(n+1)^2}\bigg(\frac{n(n+1)(2n+1)}{6}\bigg) = \frac{2\sigma^2}{3}\bigg(\frac{2n+1}{n(n+1)}\bigg) \leq \frac{2\sigma^2}{3}\bigg(\frac{2}{3}\bigg) = \sigma^2 ,\forall n$  Then it follows by Tchebychev's Weak Law of Large Numbers.  My doubt is in the last step. Isn't  Tchebychev's WLLN only true for uncorrelated random variables? In this case, $\mathbb{E}(Y_n Y_m) \neq \mu^2$, since they aren't independent, so $\rm{Cov}(Y_n,Y_m) \neq 0$, right?  Thanks.  Edit: As u/clarinetist noted, there's a problem in this exercise, since the WLLN states the convergence of the arithmetic mean of random variables, whose expected value isn't $\mu$.
12502
If $X_1, ..., X_n$ are independent identically-distributed random variables, what can be said about the distribution of $\min(X_1, ..., X_n)$ in general?
71503
If we have 3 random variables, with the first two with a covariance of 0.1 and the second two with a covariance of 0.1, what is the minimum covariance of the first and third? Is there a generalized solution on bounds on covariance matrices?
105214
I want to find the expected value of $\text{max}\{X,Y\}$ where $X$ ist $\text{exp}(\lambda)$-distributed and $Y$ ist $\text{exp}(\eta)$-distributed. X and Y are independent. I figured out how to do this for the minimum of $n$ variables, but i struggle with doing it for 2 with the maximum.  (The context in which this was given is waiting for the later of two trains, with their arrival times being exp-distributed).  Thanks!
106785
Given 3 random variables $X_1$, $X_2$ and $Y$. $Y$ and $X_1$ are independent. $Y$ and $X_2$ are independent. Intuitively I would assume that $Y$ and $X_1+X_2$ are independent. Is this the case, and how can I prove it formally?
25002
$X_1, X_2, \ldots, X_n$ are $n$ i.i.d. uniform random variables. Let $Y = \min(X_1, X_2,\ldots, X_n)$. Then, what's the expectation of $Y$(i.e., $E(Y)$)?  I have conducted some simulations by Matlab, and the results show that $E(Y)$ may equal to $\frac{1}{n+1}$. Can anyone give a rigorous proof or some hints? Thanks!
95283
Let $Y_0$ and $Y_1$ be both identically (not necessarily independent)    normally distributed with mean $\mu$ and $\sigma^2$, i.e., $Y_i \sim  N(\mu, \sigma^2)$ for $i = 1, 2$. Let $\rho$ denote the correlation    between $Y_0$ and $Y_1$. Show that the conditional expectation    $E[Y_1|Y_0 = y_0] = (1-\rho)\mu + \rho y_0$.   I know that in general if $(X, Y)$ are bivariate normally distributed, then $E[X|Y] = E[X] + \rho \frac{\sigma_X}{\sigma_Y}(Y-E[Y])$. But in this question, there is no indication that $Y_0$ and $Y_1$ are jointly normally distributed; they are simply marginally normally distributed. If they were joint normal, the result falls out simply from the result I just mentioned. Any ideas on how I can prove the result?
116254
if $x_i$ are IID  continuous random variables, with $E[x_i]=μ$, is the following correct?   $E\left[\prod_{i=1}^n(1+⍺\space x_i)^i\right] = \prod_{i=1}^nE\left[(1+⍺\space x_i)^i\right] =  \prod_{i=1}^n(1+⍺\space E\left[x_i\right])^i = (1+⍺\space μ)^n$  thanks!
53110
Could anybody help me with solving such problem:  You are given two independent random variables $X$ and $Y$ with continuous uniform distribution. You are to find expected value $E$ of random variable $Z = \max(X,Y)$.