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Stat 48 Lecture May 9
th
Normal Distribution
Dr Linda Penas
Spring 2007
1
Continuous Distributions
(p. 106)
DEFN
: A random variable is said to
be continuous
if its values can be put
into a 11 correspondence to the real
numbers.
(i.e., measured on a continuous scale)
Continuous Prob. Distributions
DEFN
: probability density function
(pdf):
for a continuous random variable X
is a
curve such that the area under the curve
between two points a and b is equal to
the probability that the random variable
X falls between a and b.
Continuous Prob. Distributions
Cumulative Distribution Function (cdf
)
of a continuous random variable X is
given by
()
(
)
x
Fx PX x
ftd
t
−∞
=≤
=
∫
•
P(a < X < b) = area under the curve
between a and b = F(b)  F(a)
ab
P(a < X < b)
•
pdf must be nonnegative and the entire
area under the curve is 1.
•
For continuous
random variables only,
=
(
)
(
)
P
aXb P
aXb
P
P
≤≤=
<≤
≤<
=<
<
Q: Why does that work for continuous
rvs?
A: Because the area of a single point =
0
P(X = a) = 0
(
)
0
a
a
Pa X a
f xdx
≤
≤=
=
∫
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View Full Document Stat 48 Lecture May 9
th
Normal Distribution
Dr Linda Penas
Spring 2007
2
NORMAL DISTRIBUTION
The
normal
or
Gaussian
distribution is the cornerstone
of most methods of estimation
and hypothesis.
NORMAL DISTRIBUTION
Can be used to model things such as
birth
weight,
blood
pressures,
lifelengths of components, etc
Even random variables which are not
normally distributed may often be
approximated
by
the
normal
distribution.
Defn
:
A continuous random variable X is
said to have a normal distribution
with
mean
μ
and variance,
σ
2
, if
X has pdf
given by:
()
2
2
2
2
1
(; , )
e
x
p
,
,
2
2
,
0
x
nx
x
μ
μσ
σ
πσ
⎧⎫
−
⎪⎪
=−−
∞
<
<
∞
⎨⎬
⎩⎭
−∞ <
< ∞
>
We write
2
~
( ,
)
XN
A distribution with pdf, f(x), is
said to be symmetric
about c if
you can draw a vertical line
passing through c, cut the curve
in half using that vertical line,
flip one side over and the two
sides coincide for all x.
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This note was uploaded on 04/10/2008 for the course STAT 48 taught by Professor Penas during the Spring '08 term at UC Riverside.
 Spring '08
 Penas
 Normal Distribution

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