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 1-1 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
F x
P X
x
f t dt
−∞
=
≤
=
∫
•
P(a < X < b) = area under the curve
between a and b = F(b) - F(a)
a
b
P(a < X < b)
•
pdf must be nonnegative and the entire
area under the curve is 1.
•
For continuous
random variables only,
(
)
(
)
= (
)
(
)
P a
X
b
P a
X
b
P a
X
b
P a
X
b
≤
≤
=
<
≤
≤
<
=
<
<
Q: Why does that work for continuous
rvs?
A: Because the area of a single point =
0
P(X = a) = 0
(
)
( )
0
a
a
P a
X
a
f x dx
≤
≤
=
=
∫

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