Linked Questions

Possible Duplicate: Proof of upper-tail inequality for standard normal distribution Proof that $x \Phi(x) + \Phi'(x) \geq 0$ $\forall x$, where $\Phi$ is the normal CDF Let $X$ be a normal $N(0,...

How to formally prove the following inequality - $$\int_t^{\infty} e^{-x^2/2}\,dx > e^{-t^2/2}\left(\frac{1}{t} - \frac{1}{t^3}\right)$$

By comparison with the integral of $ \frac xae^{-x^{2}}$ Show that $\int_a^\infty e^{-x^{2}}dx≤ \frac 1{2a}e^{-a^{2}} $ Given that $a>0$.

I tried taking exponentials and using Markov's Inequality, but this only gave me an upper bound of $\sqrt{2\pi}$. I'm not sure how to begin to approach this question - can anyone give a hint?

I want to establish the following inequality for $x>0$: $$\phi(x) \left( \frac{1}{x} - \frac{1}{x^3}\right)\leq 1- \Phi(x) \leq \phi(x) \frac{1}{x}$$ with $\phi(x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{...

How do you show that $\lim_{x\to \infty} 1-\Phi(x) \sim \phi(x)/x$? In the previous, I'm using $\Phi$ to refer to the standard normal CDF and $\phi$ to refer to the standard normal pdf. Thanks!!

As title. Can anyone supply a simple proof that $$x \Phi(x) + \Phi'(x) \geq 0 \quad \forall x\in\mathbb{R}$$ where $\Phi$ is the standard normal CDF, i.e. $$\Phi(x) = \int_{-\infty}^x \frac{1}{\...

My advisor told me to look up the proof of the following standard estimate so that we can adapt it to the case where we are dealing with something similar but including the addition of a polynomial ...

While perusing old unanswered puzzle questions, I came across this one. I found it quite interesting, but a bit vague, so I've decided to recast it. A party is to be held at a restaurant. The ...

Suppose X is a 0 mean Gaussian random variable with variance 1. I'm trying to find a lower bound on $P(X>\lambda)$. Specifically I'd like to derive a lower bound of the form $c e^{-C\lambda^2}$ ...

Let $S_n=\dfrac{B_n - np}{\sqrt{n\cdot p\cdot (1-p)}}$ be a random variable which has the standardized binomial distribution. From Chebyshev's inequality I know that $$P(|S_n| \ge x) \le \frac{1}{x^2}$...

I am interested in upper tail bounds (or bounds on deviation from the mean) for t-distribution with n degrees of freedom (http://en.wikipedia.org/wiki/Student's_t-distribution) A bound that is of the ...

Suppose random variables $X$ and $Y$ are i.i.d. Normal$(0,1)$. Consider the following events, where $\varepsilon>0, c>0$: $$\begin{align*} Q&=\{(x,y)\in\Bbb R^2: x>c, y>c\}\\ C&=\{(...

Find out the area in percentage under standard normal distribution curve of random variable $Z$ within limits from $-3$ to $3$. my try: probability density function of standard normal distribution is ...

According to this post, I found for $X \sim N(0,1)$, $x > 0$ the result that \begin{align} \frac{1}{\sqrt{2\pi}}\big(\frac{1}{x}-\frac{1}{x^3}\big)e^{-\frac{x^2}{2}} \leq P(X>x) \leq \frac{1}{\...