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Math/Stat 370 Chapter 3 Part 2
Section 3-5 Continuous RVs
The Normal RV
Definition
a normal (Gaussian) rv X has pdf
e forall x
f(x)
(x )( 2 )
2 2
where - < < + and > 0. We write X~N(,
2 ).
N(,
2 ) has been used to model IQ, SAT scores, heights.
Mean and Variance : E(X)= and V(X)=
2
Remarks
f(x) is symmetric about .
68-95-99.7 Rule (Empirical Rule)
- 68% of all values fall between .
- 95% of all values fall between 2 .
- 99.7% of all values fall between 3 .
Z~N(=0,
2 =1) is called a standard normal rv. We write Z~N(0,1).
e forall z
f(z)
z 2
2
Probabilities for N(,
2 ) are computed using N(0,1).
Computing N(0,1) Probabilities
Let Z~N(0,1). Values of
z t 2 e dt
(z ) P(Z z )
2
are given in Table I on
pages 460-461.
Example 3i. Z~N(0,1)
i. P(Z 1.63) ii. P(Z > -0.59)
iii. P(-0.29 < Z < 2.11)
Computing N( , ) Probabilities
Proposition
If ~N( 0 , 1 ).
X -
X ~N(, ), then Z
2
Example 3j. X~N(3,2)
i. P(X < 1.5) ii. P(1.5 < X 3.2)
iii. Find the 95
th percentile of X.
Example 3k. X=yield strength (ksi) of steel ~ N(=42, =3.75)
i. What proportion of yield strengths falls between 30 and 50 ksi?
ii. What yield strength separates the weakest 30% from the others?
Definition
gamma function
0
r 1 x (r ) x e dx where r > 0
Properties of ( )
(k)=(k-1)! for any positive integer k
Definition
a gamma rv X has pdf
0 o.w.
x e x 0
(r )
f(x)
r 1 x
r
where r, > 0. We write X~Gam(r,).
Note:
The density is right skewed (see plots on page 79).
Gam(r,) models time between engine malfunctions, lengths of time to complete
maintenance checkout for an auto or aircraft engine, rainfall.
Mean and Variance E(X)=r/, V(X)=r/
2
The Weibull RV
Definition
a Weibull rv X has pdf
0 o.w.
e x 0
x
f(x )
x
1
where >0 is the scale parameter and β>0 is the shape parameter. We write
X~Wei(,β).
Wei(,β) models lifetime data and describes a wide variety of failure mechanisms.
cdf of Wei( ,β))
Mean and Variance
2
2 2
,V(X) 1
E( X) 1
Example 3n. X=maximum flood level ~ Wei(=1.5, β=0.6)
a. What is the probability that flood level will be
i. more than 0.5?
ii. will not exceed 0.8?
b. Compute the mean and variance of X.
