Cumulative probability of normal distribution

[Suisse]

Hi,
I've been trying to figure this one out but could really do with some help.
I'm looking to derive the cumulative probability from normal distribution when the mean and standard deviation are known variables. I know i can use the NORMDIST function in excel (which looks like NORMDIST(x,mean,stdev,1[to indicate cumulative]) but i'd like to be able to derive this manually.

E.g. i have a part which fails at a mean of 30,000 hours with a StDev of 5000, if x = 42500 hours what is the cumulative probability?

Thanks

 

[Dr.Peterson]

What do you mean by "derive", and what do you mean by "this"?

If you want an algebraic formula, it can't be done. That's why we have tables and built-in functions.

See http://mathworld.wolfram.com/NormalDistribution.html, which says,

Neither Phi(z) nor erf can be expressed in terms of finite additions, subtractions, multiplications, and root extractions, and so both must be either computed numerically or otherwise approximated.​

 

[HallsofIvy]

The normal distribution function, for mean 40000 and standard deviation 5000, is \(\displaystyle \frac{1}{\sqrt{2\pi}} e^{\frac{-(x- 40000)^2}{5000}}\). The cumulative probability to x= 42500 is \(\displaystyle \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{42500} e^{\frac{-(x- 40000)^2}{5000}}dx\). You could, I suppose, do that using paper and pencil to do a numerical integration- but I wouldn't want to do it!