Download p(x)

LECTURE UNIT 4.3
Normal Random Variables and
Normal Probability
Distributions
Understanding Normal
Distributions is Essential for
the Successful Completion of
this Course
Recall: Probability
Distributions p(x) for a
Discrete Random Variable


p(x) = Pr(X=x)
Two properties
1. 0  p(x)  1 for all values of x
2.  all x p(x) = 1
Graph of p(x); x binomial n=10
p=.5; p(0)+p(1)+ … +p(10)=1
The sum of all the
areas is 1
Think of p(x) as the area
of rectangle above x
p(5)=.246 is the area
of the rectangle above 5
Recall: Continuous r. v. x

A continuous random variable can assume
any value in an interval of the real line (test:
no nearest neighbor to a particular value)
Discrete rv: prob dist function
Cont. rv: density function
Discrete random
variable
p(x): probability
distribution function
for a discrete random
variable x

Continuous random
variable
f(x): probability density
function of a
continuous random
variable x

Binomial rv n=100 p=.5
The graph of f(x) is a smooth
curve
f(x)
Graphs of probability density
functions f(x)


Probability density functions come in many
shapes
The shape depends on the probability
distribution of the continuous random
variable that the density function represents
Graphs of probability density functions f(x)
1.2
1
f(x)
0.8
0.6
0.4
f(x)
f(x)
0.2
0
0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3 3.3 3.6 3.9 4.2 4.5 4.8 5.1 5.4 5.7 6 6.3 6.6 6.9 7.2 7.5 7.8 8.1 8.4 8.7 9 9.3 9.6 9.9
Probabilities:
area under
graph of f(x)
P(a < X < b)
f(x)
a
b
X
P(a < X < b) = area under the density curve
b
between a and b.
P(a  X  b) =  f(x)dx
P(X=a) = 0
a
P(a < x < b) = P(a < x < b)
Properties of a probability
density function f(x)


0  p(x)  1
 p(x)=1
Think of p(x) as the area
of rectangle above x
The sum of all
the areas is 1


f(x)0 for all x
the total area under the
graph of f(x) = 1
Total area
under curve
=1
f(x)
x
Important difference


1. 0  p(x)  1 for all
values of x
2. all x p(x) = 1
values of p(x) for a
discrete rv X are
probabilities: p(x) =
Pr(X=x);


1. f(x)0 for all x
2. the total area under
the graph of f(x) = 1
values of f(x) are not
probabilities - it is
areas under the graph
of f(x) that are
probabilities
Next: normal random variables