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where μ is the mean, and σ
2
is the variance of the distribution. To calculate the
value of f(
μ,σ
2
,x) for the normal distribution, use function NDIST with the
following arguments: the mean,
μ, the variance, σ
2
, and, the value x , i.e.,
NDIST(
μ,σ
2
,x). For example, check that for a normal distribution, f(1.0,0.5,2.0)
= 0.20755374.
Normal distribution cdf
The calculator has a function UTPN that calculates the upper-tail normal
distribution, i.e., UTPN(x) = P(X>x) = 1 - P(X<x). To obtain the value of the
upper-tail normal distribution UTPN we need to enter the following values: the
mean,
μ; the variance, σ
2
; and, the value x, e.g., UTPN((μ,σ
2
,x)
For example, check that for a normal distribution, with
μ = 1.0, σ
2
= 0.5,
UTPN(0.75) = 0.638163. Use UTPN(1.0,0.5,0.75) = 0.638163.
Different probability calculations for normal distributions [X is N(
μ,σ
2
)] can be
defined using the function UTPN, as follows:
Θ P(X<a) = 1 - UTPN(
μ, σ
2
,a)
Θ P(a<X<b) = P(X<b) - P(X<a) = 1 - UTPN(
μ, σ
2
,b) - (1 - UTPN(μ, σ
2
,a)) =
UTPN(
μ, σ
2
,a) - UTPN(μ, σ
2
,b)
Θ P(X>c) = UTPN(
μ, σ
2
,c)
Examples: Using
μ = 1.5, and σ
2
= 0.5, find:
P(X<1.0) = 1 - P(X>1.0) = 1 - UTPN(1.5, 0.5, 1.0) = 0.239750.
P(X>2.0) = UTPN(1.5, 0.5, 2.0) = 0.239750.
P(1.0<X<2.0) = F(1.0) - F(2.0) = UTPN(1.5,0.5,1.0) - UTPN(1.5,0.5,2.0) =
0.7602499 - 0.2397500 = 0.524998.
The Student-t distribution
The Student-t, or simply, the t-, distribution has one parameter ν, known as the
degrees of freedom of the distribution. The probability distribution function (pdf)
is given by