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I am trying to calculate the following probability

P(X>a) a is the lower bound of integration

where X is following normal distribution with parammter $\mu$ and $\sigma$. I tried the integartion of probability density function of normal distribution but I did something wrong I guess

$\int_{a}^{}$ f(x) dx

Thank you in advance

Angelıque
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  • It's very unclear what you are asking... Are you looking for $a$? There will be multiple values that work because you have an inequality ( $\geq 0.5$). The normal density function does not have an elementary antiderivative, so I would recommend using a $z$-score table instead. – kccu Apr 08 '19 at 22:24
  • Thank you beasically I am asking the integration of normal distribution whith the bounds of [a,infinity) . Thank you so much (Note, a is already known) – Angelıque Apr 08 '19 at 22:27
  • Again, you cannot calculate this by hand using an integral. You need to use a $z$-score and a table. – kccu Apr 08 '19 at 22:29

1 Answers1

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If I properly understand, what you want is to compute, for given $(a,\mu,\sigma)$ $$P(a)=\frac{1}{\sqrt{2 \pi \sigma ^2}}\int_a^\infty e^{-\frac{(x-\mu )^2}{2 \sigma ^2}}\,dx$$ Using $\sigma >0$, you have $$P(a)=\frac{1}{2} \text{erfc}\left(\frac{a-\mu }{\sigma\sqrt{2} }\right)=\frac 12\left( 1-\text{erf}\left(\frac{a-\mu }{\sigma\sqrt{2} }\right)\right)$$ This involves non elementary functions but you can get quite good approximations using for example $$\mathrm{erf}\!\left(t\right)\sim \sqrt{1-\exp\Big(-\frac 4 {\pi}\,\frac{1+\alpha \,t^2}{1+\beta\, t^2}\,t^2 \Big)}$$ where $$\alpha=\frac{10-\pi ^2}{5 (\pi -3) \pi }\qquad \text{and} \qquad \beta=\frac{120-60 \pi +7 \pi ^2}{15 (\pi -3) \pi }$$

Have a look here.

Edit

In a quite recent paper is given the very nice approximation $$\frac{1}{2} \left(1+\text{erf}\left(\frac{t}{\sqrt{2}}\right)\right)\sim 2^{-22^{1-41^{t/10}}}$$

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Claude Leibovici
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  • Thank you so much Claude, just I have a small question, in the link you sent how did you compute the a as in the follwing, should not it be "ln" instead of "log" I am talkin about a=$\frac{-2 log [4(1-F(a)F(a)]}{k}$, If you can help me , I will be very happy , thank you in advence – Angelıque Apr 09 '19 at 14:28
  • @Angelıque. For me, there is only one logarithm : as for many people, by default, $\log(x)$ stands for $\kn(x)$. Cheers :-) – Claude Leibovici Apr 09 '19 at 17:06
  • Thank you so much Claude, it is clear now. Just one last question, is it possible to approximate erf (a/$\root_{2}$) without taking the square of erf function. If I can approximate erf (a/$\root_{2}$) directly, It will be more useful fo me. – Angelıque Apr 09 '19 at 19:33
  • @Angelıque. SInce $x=\frac a {\sqrt 2}$ then $a=x{\sqrt 2}$ – Claude Leibovici Apr 10 '19 at 02:24