shown here is more accurate than that included in the SLE application. For the upper-tail distribution
with x = 2, mean = 0 and standard deviation = 2,the SLE returns a result of .022750062014, which is in
error by about 8E-8.
It is occasionally necessary to find the inverse of distribution function, that is, to find the value of the
random variable x, given a probability, a mean and a standard deviation. There is no closed-form
solution to this problem. One possiblity is to use the built-in nSolve() function with cnd(). For example,
suppose you know that the distribution has a mean of 7, a standard deviation of 1. What is the value of
the random variable x for which the probability is 0.2? Use nSolve() like this:
nSolve(cnd(x,7,1)=0.2,x)
which returns x = 6.158379 in about 19 seconds on a HW2 92+.. You can (and should) check this
solution with
cnd(6.158379,7,1) which returns 0.2, as hoped.
The execution time can be considerably reduced by providing nSolve() with a good initial guess for the
value of x. If the initial guess is good enough, we don't need to use nSolve() at all, and the result is
returned very quickly.
For the standard CND, with mean = 0 and standard deviation = 1, Odeh and Evans give this estimating
function for x
p
for a given probability p:
and and 10
-20
< p < 0.5 [5]
x
p
= f
(
t
)
= t+
p
4
t
4
+p
3
t
3
+p
2
t
2
+p
1
t+p
0
q
4
t
4
+q
3
t
3
+q
2
t
2
+q
1
t+q
0
t =−2ln
(
p
)
and
p
0
= -0.32223 24310 88 q
0
= 0.09934 84626 060
p
1
= -1.0 q
1
= 0.58858 15704 95
p
2
= -0.34224 20885 47 q
2
= 0.53110 34623 66
p
3
= -0.02042 31202 45 q
3
= 0.10353 77528 50
p
4
= -0.45364 22101 48 E-4 q
4
= 0.38560 70063 4 E-2
For the general case in with mean m and standard deviation s, we have
[6]
x
p
= s $ f
(
t
)
+ m
using the identity in equation [2]. Finally, we use the symmetry of the normal distribution about the
mean to account for p 0.5, like this:
m
[7]
x
p
=
[
sgn
(
p − 0.5
)][
s $ f
(
t
)]
+ m
Here, sgn(x) is a 'sign' function such that
[8]
sgn
(
x
)
=
−1forx< 0
1forxm 0
We cannot use the built-in sign() function, because it returns the expression 'sign(0)' for x=0, and we
want 1. This is fixed with a when() function:
6 - 68
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