Download Math 4030 – 3a Continuous Random Variables

MATH 4030 – 4B
CONTINUOUS RANDOM
VARIABLES
 Density Function
 PDF and CDF
 Mean and Variance
 Normal Distribution
Difficulties:
• Probability at a value
• Addition rule
• Axiom of probability
• Zero probability event vs. empty event
Approach (Sec. 5.1):
• Probability density function (pdf) f(x)
• Area for probability  cumulative
distribution function (cdf) F(x)
• Integral for area, fundamental theorem
of calculus
Properties of f(x) and F(x):
1) f ( x)  0

2)
 f ( x)dx  1.
-
x
3) F ( x )  P( X  x) xdx f1 x(t )dt1 64  1  65 .
4
4

1
2
3


-  3  1
3
3
b
4) P(a  x  b)  P(a  x  b)   f ( x )dx  F (b)  F (a )
a
Mean and Variance:


 xf ( x )dx
-
2 

2
2
(
x


)
f
(
x
)
dx




2

-
Normal N(, 2) (Sec. 5.2):
1
f ( x) 
e
2

Mean   xf ( x )dx  

x   2

2 2
-4
-3
-2
-1
0
1
,    x  .

Variance 
2
2


x


f
(
x
)
dx





Standard Normal N(0, 1):
1
e
2
f ( x) 
x
F ( x) 


 x2
2
1
e
2
,    x  .
t 2
2
dt ,    x  .
2
3
4
5
z notation:
A z-value that the probability for Z to be
greater than this value is exactly .
Or the cut point of the standard normal curve
that makes the area of the right tail exactly .
From General Normal to Standard Normal
If X ~ N(µ, 2), then the transformation
Z = (X - µ) /  results Z ~ N(0, 1).
6
Normal Approximation to
Binomial (Sec. 5.3)
Bi (n, p )  N (np, np(1  p ))
Or if X has binomial distribution with parameters n
and p, then
X  np
Z
~ N (0,1)
np(1  p)
• For large n, np>15, and n(1-p)>15.
• Correction for continuity