Examples of Continuous Probability Distributions
[Pages:57]Examples of continuous probability distributions:
The normal and standard normal
The Normal Distribution
f(X)
Changing shifts the
distribution left or right.
Changing increases or decreases the spread.
X
The Normal Distribution: as mathematical function (pdf)
f (x) =
1
-1( x-? )2
e 2
2
Note constants: =3.14159 e=2.71828
This is a bell shaped curve with different centers and spreads depending on ? and
The Normal PDF
It's a probability function, so no matter what the values of ? and , must integrate to 1!
+
1
-1( x-? )2
e 2 dx = 1
- 2
Normal distribution is defined by its mean and standard dev.
E(X)=? =
+
x
1
-1( x-? )2
e 2 dx
- 2
Var(X)=2 =
+
( x 2
1
e
-1( 2
x-?
)2
dx
)
-
?
2
- 2
Standard Deviation(X)=
**The beauty of the normal curve:
No matter what ? and are, the area between ?- and ?+ is about 68%; the area between ?-2 and ?+2 is about 95%; and the area between ?-3 and ?+3 is about 99.7%. Almost all values fall within 3 standard deviations.
68-95-99.7 Rule
68% of the data 95% of the data 99.7% of the data
68-95-99.7 Rule in Math terms...
? +
1
- 1 ( x-? )2
? e 2 dx = .68
? - 2
? + 2
1
- 1 ( x-? )2
? e 2 dx = .95
? - 2 2
? + 3
1
- 1 ( x-? )2
? e 2 dx = .997
? -3 2
................
................
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