Linked Questions

3
votes
1answer
7k views

bounds on normal distribution [duplicate]

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,...
1
vote
2answers
163 views

Integral with inequality: $\int_a^\infty e^{-x^{2}}dx≤ \frac 1{2a}e^{-a^{2}} $ [duplicate]

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 $a>0$.
1
vote
1answer
129 views

limiting behavior of standard normal survivor function [duplicate]

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!!
2
votes
1answer
70 views

Double inequality involving gaussian and erf functions [duplicate]

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{...
15
votes
5answers
2k views

Proof that $x \Phi(x) + \Phi'(x) \geq 0$ $\forall x$, where $\Phi$ is the normal CDF

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}{\...
8
votes
2answers
4k views

Proof of an estimate for the tail of a normal distribution

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 ...
15
votes
4answers
442 views

Selecting a menu

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 ...
3
votes
2answers
2k views

Lower bound on $P(X>\lambda)$ where $X$ is Gaussian.

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 $c e^{-C\lambda^2}$ ...
0
votes
1answer
657 views

Upper-tail inequality for t-distribution

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 ...
3
votes
2answers
536 views

Approximation of the binomial distribution

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}$...
1
vote
1answer
744 views

Upper bound inequality cumulative normal distribution

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}{\...
3
votes
1answer
183 views

Integral inequality of exponent

How to formally prove the following inequality - $$\int_t^{\infty} e^{-x^2/2}\,dx > e^{-t^2/2}\left(\frac{1}{t} - \frac{1}{t^3}\right)$$
0
votes
1answer
184 views

Inequalities involving the Gaussian integral

For $x>0$, how to prove $$\frac{e^{-\frac{x^2}{2}}}{x}-\int_x^\infty e^{-\frac{t^2}{2}}dt\leq \frac{e^{-\frac{x^2}{2}}}{x^3}$$ I did the following let $f(x)=\frac{e^{-\frac{x^2}{2}}}{x^3}-\frac{e^{...
7
votes
1answer
182 views

Truncated taylor series inequality

I came across the following fact in a paper and am having trouble understanding why it is true: Consider the error made when truncating the expansion for $e^a$ at the $K$th term. By choosing $K = O(\...
2
votes
2answers
136 views

how to find area under normal distribution curve

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 ...

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