Demonstrate that, for a standard normal random variable:  $$1-\Phi(x) \approx \frac{\varphi(x)}x$$  for large values of $x$.
27128
I know that an easy to handle formula for the CDF of a normal distribution is somewhat missing, due to the complicated error function in it.  However, I wonder if there is a a nice formula for $N(c_{-} \leq x &lt; c_{+}| \mu, \sigma^2)$. Or what the "state of the art" approximation for this problem might be.
110317
I am wondering what is the probability density function for the normal cdf $\Phi (aX+b)$, where $\phi$ is the usual standard normal cumulative distribution function  I want to calculate $\mathbb{E}[\Phi(aX+b)]$ but i am stuck on how to get the distribution. thank you =]  note: X is normally distributed
41108
I'm trying to prove this conclusion but have some problems with one of the steps.  Assume $X_1,\ldots,X_n,\ldots$ is a sequence of Gaussian random variables, converging almost surely to $X$, prove that $X$ is Gaussian.  We use characteristics function here. Since $|\phi_{X_n}(t)|\leq 1$, by dominated convergent theorem, we have for any $t$  $$ \lim_{n\rightarrow\infty}e^{it\mu_n-t^2\sigma_n^2/2}=\lim_{n\rightarrow \infty}\phi_{X_n}(t) = \lim_{n\rightarrow \infty}\mathbb{E}\left[e^{itX_n}\right] = \mathbb{E}\left[e^{itX}\right] = \phi_X(t) $$  this is the step that I cannot figure out: $e^{it\mu_n-t^2\sigma_n^2/2}$ converges for any $t$ if and only if $\mu_n$ and $\sigma_n$ converges.   Let $\mu=\lim_n \mu_n$, and $\sigma=\lim_n\sigma_n$, then $\phi_X(t)=e^{it\mu-t^2\sigma^2/2}$, which proves that $X$ is a Gaussian random variable.  Why can we get that $\mu_n$ and $\sigma_n$ converge? This looks intuitive for me, but I cannot make a rigorous prove.
9536
Given  $X \sim \mathcal{N}(\mu,\,\sigma^{2})$, what is the covariance between $X$ and $e^X$?
49152
such as find  MLE of $P(X&gt;12)$ when $X_i$ ~ $N(\mu,\sigma^2) $
87354
Let  $X\sim \mathcal{N}(6,1)$ and $Y\sim\mathcal{N}(7,1)$ be two independent normal variables. Find $Pr(X&gt;Y)$. the answer is $0.2389$ but I do not know how to do it.
48979
Let $X_1, \ldots, X_n$ be i.i.d. continuous uniform random variables on $\left[\vartheta-\frac{1}{2},\vartheta+\frac{1}{2}\right]$ for some $\vartheta \in \mathbb R$.    I am trying to solve $\mathbb E[(T-\vartheta)^2]$ where $T(X) = \frac{1}{2}\left(\min_{1 \le i \le n} X_i + \max_{1 \le i \le n} X_i\right)$ or simplify it such that it becomes clear that it is independent of $\vartheta$.   I have tried to approach this by a variety of conversion but always end up with a term that involes $\mathbb E\left[\left(\min_{1 \le i \le n} X_i\right)\left(\max_{1 \le i \le n} X_i\right)\right]$.  I can't find a way to actually calculate it. Do you have a hint or another way of approaching the task?
115076
Please help me solve this problem:  Let $X$ and $Y$ be independent standard normal random variables, let $Z$ have an arbitrary density function, and form $Q = \frac{X + YZ}{\sqrt{1+ Z^2}}$. Prove that $Q$ also has a standard normal density function.   Thanh you
1061
How can I prove that a random variable taking values in $[0,1]$ has variance no larger than $\frac{1}{4}$? If it matters, discrete and continuous proofs are both welcome.
19140
Let $X$ and $Y$ be real-valued random variables defined on the same space.  Let's use $\phi_X$ to denote the characteristic function of $X$.  If $\phi_{X+Y}=\phi_X\phi_Y$ then must $X$ and $Y$ be independent?
19657
Suppose I have  a bivariate normal $(x,y)\sim \mathcal N(0,\Sigma)$ .  Is there an easy formula for the expectation $\mathrm E(x\mid y&gt;0)$ ?
122536
Suppose $(\Omega,\mathcal F,P)$ is a probability space and that $\mathcal G$ is a sub-sigma-algebra of $\mathcal F$. If $X$ is an integrable, non-negative random variable with the same distribution as $E[X\mid \mathcal G]$, how does one show that $X=E[X\mid \mathcal G]$ a.s.?
52420
Suppose $X_1, X_2, X_3, \ldots, X_n$ be independent and identically distributed random variables having an exponential distribution with mean $\frac{1}{\lambda}$.  If $S_n = X_1 + X_2 + \ldots + X_n$ and $N = \inf\{n \geq 1: S_n &gt; 1\}$, then what would be the value of $\mathrm{Var}(N)$ ?
3733
What is the distribution of the square of a non-standard normal random variable (i.e., the mean is not equal to 0 and the variance is not equal to 1)?
40488
Show that for a non-negative random variable $X$,   $$\mathbb E(X)=\int_0^\infty \mathbb P(X\ge x) \, dx.$$   I started with  $$\mathbb E(X)=\int_0^\infty x \, dF(x)=\int_0^\infty \int_0^x dt\,dF(x).$$ This is my try.
48180
Given $x_1, \ldots, x_N$, independent and all distributed as a   Gaussian with mean $\mu$ and variance $\sigma^2$. Then, the average   $$\bar{x} = \frac{1}{N}\sum_{i=1}^Nx_i$$ is distributed as a Gaussian   with mean $\mu$ and variance $\frac{\sigma^2}{N}.$   This is a very well-known result. Anyway, I'm looking around to find a proof for this and I'm not having luck.
118661
Show that $\phi(n)=\frac{n}{2}$ if and only if $n=2^k$ for some positive integer k. I think I have it figured and would like to see if I am on the right track. Thank you.