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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 $N(0,1)$ randon variable. Show that $\mathbb{P}(X>t)\le\frac{1}{\sqrt{2\pi}t}e^{-\frac{t^2}{2}}$, for $t>0$.

Using markov inequality shows that $P(X>t)\le \frac{\mathbb{E}(X)}{t}$ but I dont know how to bound the expected value

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    In fact, this question has [been added to the faq](http://meta.math.stackexchange.com/questions/1868/list-of-generalizations-of-common-questions).2011-10-22

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The following argument is a standard one that can be used far more generally.

$P(X>t)=\int_t^\infty \frac{1}{\sqrt{2\pi}}e^{-x^2/2}\,dx.$ We are integrating from $x=t$ to infinity. So over the region of integration, $x \ge t$, and therefore $\frac{x}{t}\ge 1$. It follows that $\int_t^\infty \frac{1}{\sqrt{2\pi}}e^{-x^2/2}\,dx< \frac{1}{\sqrt{2\pi}t}\int_t^\infty xe^{-x^2/2}\,dx.$

Now we can integrate explicitly. An antiderivative of $xe^{-x^2/2}$ is $-e^{-x^2/2}$, and therefore $\int_t^\infty xe^{-x^2/2}\,dx=e^{-t^2/2},$
which yields the desired result.

Comment: "General" inequalities, such as those of Markov (and therefore Chebyshev) are often much too weak to prove reasonably sharp bounds for specific distributions. In this case, if we apply the Markov Inequality directly to $X$, we conclude that $P(|X|>t) \le \frac{1}{t}E(|X|)$. The expectation of the absolute value of the standard normal is an easily computed constant. So the bound we get from applying the Markov Inequality directly to $X$ is of the form $\frac{K}{t}$. For large $t$, this is enormously larger than the truth, since it entirely misses the rapidly decaying $e^{-t^2/2}$ term.

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    @Dilip Sarwate: Repetition, or essential repetition, is very frequent on StackExchange. But often it is easier to answer a question than to search for close relatives. In this case, the question in fact started out different, because of the OP's inadvertent omission of the $t$ in the denominator.2011-10-22