HP F2226A - 48GII Graphic Calculator User Manual page 567

•
Exponential, F(x) = 1 - exp(-x/β)
•
Weibull, F(x) = 1-exp(-αx
(Before continuing, make sure to purge variables α and β). To find the inverse
cdf's for these two distributions we need just solve for x from these expressions,
i.e.,
Exponential:
For the Gamma and Beta distributions the expressions to solve will be more
complicated due to the presence of integrals, i.e.,
•
Gamma, p
∫
p =
•
Beta,
A numerical solution with the numerical solver will not be feasible because of
the integral sign involved in the expression. However, a graphical solution is
possible. Details on how to find the root of a graph are presented in Chapter
12. To ensure numerical results, change the CAS setting to Approx. The
function to plot for the Gamma distribution is
Y(X) = ∫(0,X,z^(α-1)*exp(-z/β)/(β^α*GAMMA(α)),z)-p
For the Beta distribution, the function to plot is
Y(X) =
∫(0,X,z^(α-1)*(1-z)^(β-1)*GAMMA(α+β)/(GAMMA(α)*GAMMA(β)),z)-p
To produce the plot, it is necessary to store values of α, β, and p, before
attempting the plot. For example, for α = 2, β = 3, and p = 0.3, the plot of
Y(X) for the Gamma distribution is shown below. (Please notice that, because
β
)
Weibull:
1
x
∫
α − 1
exp(
z
α
β ( α )
0
Γ α + β
x ( )
α − 1
⋅ z ⋅
1 (
Γ α Γ ⋅ β
0
( ) ( )
z
)
dz
β
β − 1
− z dz
)
Page 17-14