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Chapter 6
Continuous Random
Variables and Probability
Distributions
©
Continuous Random Variables
A random variable is continuous if
it can take any value in an interval.
Cumulative Distribution
Function
The cumulative distribution function, F(x), for a
continuous random variable X expresses the
probability that X does not exceed the value of x,
as a function of x
F ( x)  P( X  x)
Shaded Area is the Probability That
X is Between a and b
0
a
b
x
Probability Density Function for a
Uniform 0 to 1 Random Variable
f(x)
1
0
1
x
Areas Under Continuous Probability
Density Functions

1.
2.
Let X be a continuous random variable with the
probability density function f(x) and cumulative
distribution F(x). Then the following properties
hold:
The total area under the curve f(x) = 1.
The area under the curve f(x) to the left of x0 is
F(x0), where x0 is any value that the random
variable can take.
Properties of the Probability Density
Function
f(x)
Comments
1
0
Total area under
the uniform
probability density
function is 1.
0
x0
1
x
Properties of the Probability Density
Function
Comments
f(x)
Area under the uniform
probability density
function to the left of
x0 is F(x0), which is
equal to x0 for this
uniform distribution
because f(x)=1.
1
0
0
x0
1
x
Reasons for Using the Normal
Distribution
1. The normal distribution closely approximates
the probability distributions of a wide range of
random variables.
2. Distributions of sample means approach a
normal distribution given a “large” sample size.
3. Computations of probabilities are direct and
elegant.
4. The normal probability distribution has led to
good business decisions for a number of
applications.
Probability Density Function for
a Normal Distribution
0.4
0.3
0.2
0.1
0.0

x
Probability Density Function of
the Normal Distribution
The probability density function for a normally
distributed random variable X is
f ( x) 
1
2
2
e
 ( x   ) 2 / 2 2
for -   x  
Where  and 2 are any number such that - <  < 
and - < 2 <  and where e and  are physical
constants, e = 2.71828. . . and  = 3.14159. . .
Properties of the Normal
Distribution
Suppose that the random variable X follows a normal distribution
with parameters  and 2. Then the following properties hold:
i.
The mean of the random variable is ,
E( X )  
ii.
The variance of the random variable is 2,
iii.
The shape of the probability density function is a symmetric
bell-shaped curve centered on the mean .
By knowing the mean and variance we can define the normal
distribution by using the notation
iii.
E[( X   X ) 2 ]   2
X ~ N ( , )
2
Effects of  on the Probability Density
Function of a Normal Random Variable
0.4
0.3
Mean = 6
Mean = 5
0.2
0.1
0.0
1.5
2.5
3.5
4.5
5.5
6.5
7.5
8.5
x
Effects of 2 on the Probability Density
Function of a Normal Random Variable
0.4
Variance = 0.0625
0.3
0.2
Variance = 1
0.1
0.0
1.5
2.5
3.5
4.5
5.5
6.5
7.5
8.5
x
Cumulative Distribution Function
of the Normal Distribution
Suppose that X is a normal random variable with
mean  and variance 2 ; that is X~N(, 2). Then
the cumulative distribution function is
F ( x0 )  P( X  x0 )
This is the area under the normal probability
density function to the left of x0,. As for any proper
density function, the total area under the curve is 1;
that is F() = 1.
Shaded Area is the Probability that X
does not Exceed x0 for a Normal
Random Variable
f(x)
x0
x
Range Probabilities for Normal
Random Variables
Let X be a normal random variable with cumulative
distribution function F(x), and let a and b be two
possible values of X, with a < b. Then
P(a  X  b)  F (b)  F (a)
The probability is the area under the corresponding
probability density function between a and b.
Range Probabilities for Normal
Random Variables
f(x)
a

b
x
The Standard Normal
Distribution
Let Z be a normal random variable with mean 0 and
variance 1; that is
Z ~ N (0,1)
We say that Z follows the standard normal distribution.
Denote the cumulative distribution function as F(z), and
a and b as two numbers with a < b, then
P(a  Z  b)  F (b)  F (a)
Standard Normal Distribution with
Probability for z = 1.25
0.8944
z
1.25
Finding Range Probabilities for
Normally Distributed Random Variables
Let X be a normally distributed random variable with mean 
and variance 2. Then the random variable Z = (X - )/ has a
standard normal distribution: Z ~ N(0, 1)
It follows that if a and b are any numbers with a < b, then
b 
a
P ( a  X  b)  P
Z

 
 
b 
a 
 F
  F

  
  
where Z is the standard normal random variable and F(z)
denotes its cumulative distribution function.
Computing Normal Probabilities
A very large group of students obtains test scores that are
normally distributed with mean 60 and standard deviation
15. What proportion of the students obtained scores
between 85 and 95?
95  60 
 85  60
P(85  X  95)  P
Z

15 
 15
 P(1.67  Z  2.33)
 F (2.33)  F (1.67)
 0.9901  0.9525  0.0376
That is, 3.76% of the students obtained scores in the range 85 to 95.