Dear All,
I need to write an R function which computes values of Probabilities for
the (standard) normal distribution, ¦µ(z) for z > 0 by summing this power
series. (We should accumulate terms until the sum doesn't change). I also
need to make the function work for vector of values z.
The initial fomular is
¦µ(z) = ( 1/ sqrt(2¦Ð) )* ¡Ò e^(-t^2/2)dt (¡Ò is from -¡Þ, z)
= 1/ 2 + ( 1/ sqrt(2¦Ð) )* ¡Ò e^(-t^2/2)dt (¡Ò is from 0 to z)
I can substituting x = -t^2/2 into the series expansion for the exponential
function
e^x = ¡Æ x^n/n! (¡Æ is from n=0 to ¡Þ)
I can obtain the series
e^(-t^2/2) = ¡Æ (-1)^k*t^2k / 2^k*k! (¡Æ is from n=0 to ¡Þ)
This series can be integrated term by term to obtain the formula
¦µ(z) = 1/ 2 + ( 1/ sqrt(2¦Ð) ) * ¡Æ (-1)^k*z^(2k+1) / (2^k*k! *(2k+1))
(¡Æ is from n=0 to ¡Þ)
I know how to write the R function for exponential function e^x
expf = function (x)
{
x=ifelse((neg=(x<0)),-x,x)
n=0;term=1
oldsum=0; newsum=1
while(any(newsum != oldsum)) {
oldsum=newsum
n=n+1
term = term*x/n
newsum = newsum+term}
ifelse(neg, 1/newsum, newsum)
}
I know it will be similar to the above coding, but I don¡¯t know exactly how
should we modify the above coding in order to get Probabilities for the
(standard) normal distribution, ¦µ(z) for z > 0.
Can anybody advise me on this??
Thanks a lot.
Rene.
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