Suppose X, Z are normally distributed random variables and independent ($X$ follows $N(\mu,\sigma^{2})$ and $Z$ follows $N(0,\sigma_{z}^{2})$), and that Y=X+Z.   What is E[X|Y>c], where c is just a constant - how do you derive this?
19657
Suppose I have  a bivariate normal $(x,y)\sim \mathcal N(0,\Sigma)$ .  Is there an easy formula for the expectation $\mathrm E(x\mid y&gt;0)$ ?
22712
... and why ?  Assuming $X_1$,$X_2$ are independent random-variables with mean $\mu_1,\mu_2$ and variance $\sigma^2_1,\sigma^2_2$ respectively. My basic statistics book tells me that the distribution of the $X_1-X_2$ has the following properties:    $E(X_1-X_2)=\mu_1-\mu_2$ $Var(X_1-X_2)=\sigma^2_1 +\sigma^2_2$   Now let's say $X_1$, $X_2$  are t-distributions with $n_1-1$, $n_2-2$ degrees of freedom. What is the distribution of $X_1-X_2$ ?   This question has been edited: The original question was "What are the degrees of freedom of the difference of two t-distributions ?". mpiktas has already pointed out that this makes no sense since $X_1-X_2$ is not t-distributed, no matter how approximately normal $X_1,X_2$ (i.e. high df) may be.
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?
9536
Given  $X \sim \mathcal{N}(\mu,\,\sigma^{2})$, what is the covariance between $X$ and $e^X$?
123155
Let $X_n$ be iid random variables with $P(X_n=(-1)^kk)=\dfrac{C}{k^2\log k}$ for all $k\geq 2$. Here $C$ is just a constant so that the sum of probabilities is $1$. Find constant $\mu$ such that $S_n/n\to \mu$ in probability where $S_n=\sum_{k=1}^nX_k$.   Here $E(X_n)=\infty$. I'm a bit lost after this. In general how does one find constants like this?
232
Given two independent random variables $X\sim \mathrm{Gamma}(\alpha_X,\beta_X)$ and $Y\sim \mathrm{Gamma}(\alpha_Y,\beta_Y)$, what is the distribution of the difference, i.e. $D=X-Y$?  If the result is not well-known, how would I go about deriving the result?
109955
Suppose $X, Y$ are independent random variables with $X\sim NB(r,p)$ and $Y\sim NB(s,p)$. Then $$X + Y \sim NB(r+s,p)$$  How do I go about proving this? I'm not sure where to begin, I'd be glad for any hint.
105717
Let $X_1$ and $X_2$ be independent Normal random variables with mean $\mu_1$ and $\mu_2$, and variances $\sigma_1$ and $\sigma_2$. Let $Y = X_2-X_1 + c$, where $c$ is a constant. For notational simplicity, let $Y = X' + c$, where $X' = X_2-X_1$ The moment generating function of a Normal random variable $X_n\sim N(μ_n,σ_n^2)$ is $$M_{X_n } (t)=\text{exp}\Big(μ_n t+\frac{σ_n^2 t^2}{2}\Big)$$ Then, $$M_{X'} (t)=\text{exp}\Big[t(\mu_1+\mu_2)+\frac{t^2}{2}(\sigma^2_1+\sigma^2_2)\Big]$$ Thus, by the uniqueness property of moment generating function, $X'$ is said to follow a Normal distribution with mean $(\mu_1+\mu_2)$ and variance $(\sigma^2_1+\sigma^2_2)$. How do I proceed from here to derive the following:  First, I want to establish that $Y$ is also a Normal random variable using the moment generating function. The constant $c$ bugs me, but I cannot ignore it. As per my understanding, $Y\sim N\big(c+\mu_1+\mu_2,\sigma^2_1+\sigma^2_2\big)$ (and I could be completely wrong). But I don't understand how to apply the moment generating function to prove that.  Secondly, I want to derive $P(Y \leq a)$, where $a$ is a constant. If I am not mistaken, $P(Y \leq a)$ can be written as $$P(Y \leq a)=\Phi\Bigg(\frac{a-X_2-X_1 + c}{\sqrt{2(\sigma^2_1+\sigma^2_2)}}\Bigg).$$ But how do I arrive at that.   Any help is high solicited and appreciated.
9232
Given   $$ f_{XY}(x,y) = \frac{1}{2\pi \sqrt{1-\rho^2}} \exp \left( -\frac{x^2 +y^2 - 2\rho xy}{2(1-\rho^2)} \right) $$  $Y = Z\sqrt{1-\rho^2} + \rho X$  And   $$ f_{XZ}(x,z) = \frac{1}{2\pi } \exp \left( -\frac{x^2 +z^2}{2} \right) $$  Show that $P(X&gt;0,Y&gt;0)= \frac{1}{4}+\frac{1}{2\pi}(\arcsin \rho) $  I'm supposed to use the fact that X and Z are independent standard normal random variables, but I don't quite understand how. Any help would be greatly appreciated.
109702
I'm trying to get more familiar with $E(XY) = E(X)E(Y)$, and all examples that I have come across seems to satisfy this expression. My question then is, if there exist an example of two random variables $X$ and $Y$ for which $E(XY) \neq E(X)E(Y)$?  I am looking for an example using a joint probability distribution table.  Thank you.
120850
I would like to find the conditional bivariate normal distribution. There are two dependent normal variables with the same distribution and the correlation coefficient $\rho$: $X,Y \sim N(\mu, \sigma^2)$. I would like to get $P(X|Y&gt;M)$. I found the conditional expectation of $X$ given that $Y$ is bigger than $M$: $E(X|Y&gt;M)= \mu + \rho \sigma \frac{\phi(\frac{M-\mu}{\sigma})}{1-\Phi(\frac{M-\mu}{\sigma})}$. But what is the conditional variance of $var(X|Y&gt;M)$? Is it $(1-\rho^2)\sigma^2 $, as it would be in the case of $var(X|Y=M)$, where variance does not depends on $M$? And is the conditional distribution $N(E(X|Y&gt;M),var(X|Y&gt;M))$?
97787
$X$ and $Y$ are independent and identically distribued (i.i.d.), $X+Y$ and $X-Y$ are independent, $\mathbb{E}(X)=0$ and $\mathbb{E}(X^2)=1$. Show that $X\sim N(0,1)$.  We should use characteristic functions to prove this. Any ideas?
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?
23827
Let $X$ and $Y$ be independent $Exp(1)$-distributed random variables. Find the conditional distribution of $X$ given that $X + Y = c$ ($c$ is a positive constant).  this is my idea:  $$f_{X \mid X+Y}(x\mid c) = \frac{f_{X,X+Y}(x,c)}{f_{X+Y}(c) } $$  if this is the right definition then i might calculate the righthand side to obtain the distribution but Iam unsure
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 $\mathbb{E}[XY] \geq \mathbb{E}[X] \mathbb{E}[Y]$.
32232
If $X_i$, $i =1,2$ are independent and have normal distribution with mean $0$ and variance $\sigma_i ^2$. Show that $X_1 + X_2$ has a normal distribution with mean $0$ and variance $\sigma_1^2 + \sigma_2^2$.  Any idea is appreciated.
52511
There are two Random Variables X and Y. Both of them are Normally distributed with 0 mean and variance a and b respectively. X and Y are independent of each other.  What is the probability of   What should be the easiest way to do this?