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Chapter 6
Continuous Random
Variables and Probability
Distributions
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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)
Cumulative Distribution
Function
F(x)
1
0
1
Cumulative Distribution Function for a Random variable Over 0 to 1
Cumulative Distribution
Function
Let X be a continuous random variable
with a cumulative distribution function
F(x), and let a and b be two possible
values of X, with a < b. The probability
that X lies between a and b is
P(a  X  b)  F (b)  F (a)
Probability Density Function

1.
2.
3.
4.
Let X be a continuous random variable, and let x be any
number lying in the range of values this random variable
can take. The probability density function, f(x), of the
random variable is a function with the following properties:
f(x) > 0 for all values of x
The area under the probability density function f(x) over all
values of the random variable X is equal to 1.0
Suppose this density function is graphed. Let a and b be
two possible values of the random variable X, with a<b.
Then the probability that X lies between a and b is the area
under the density function between these points.
The cumulative density function F(x0) is the area under the
probability density function f(x) up to x0
f ( x0 ) 
x0
 f ( x)dx
xm
where xm is the minimum value of the random variable 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
Rationale for Expectations of
Continuous Random Variables
Suppose that a random experiment leads
to an outcome that can be represented by
a continuous random variable. If N
independent replications of this
experiment are carried out, then the
expected value of the random variable is
the average of the values taken, as the
number of replications becomes infinitely
large. The expected value of a random
variable is denoted by E(X).
Rationale for Expectations of
Continuous Random Variables
Similarly, if g(x) is any function of the
random variable, X, then the expected value
of this function is the average value taken by
the function over repeated independent
trials, as the number of trials becomes
infinitely large. This expectation is denoted
E[g(X)]. By using calculus we can define
expected values for continuous random
variables similarly to that used for discrete
random variables.
E[ g ( x)]   g ( x) f ( x)dx
x
Mean, Variance, and Standard
Deviation
i.
Let X be a continuous random variable. There are two important
expected values that are used routinely to define continuous
probability distributions.
The mean of X, denoted by X, is defined as the expected value
of X.
 X  E(X )
ii.
The variance of X, denoted by X2, is defined as the expectation
of the squared deviation, (X - X)2, of a random variable from its
mean
2
2
X
X
  E[( X   ) ]
Or an alternative expression can be derived
2
2
2
X
X
The standard deviation of X, X, is the square root of the
variance.
  E( X )  
iii.
Linear Functions of Variables
Let X be a continuous random variable with
mean X and variance X2, and let a and b any
constant fixed numbers. Define the random
variable W as
W  a  bX
Then the mean and variance of W are
W  E (a  bX )  a  b X
and
  Var (a  bX )  b 
2
W
2
and the standard deviation of W is
W  b  X
2
X
Linear Functions of Variable
An important special case of the previous
results is the standardized random
variable
Z
X  X
X
which has a mean 0 and variance 1.
Reasons for Using the Normal
Distribution
1.
2.
3.
4.
The normal distribution closely
approximates the probability distributions
of a wide range of random variables.
Distributions of sample means approach a
normal distribution given a “large” sample
size.
Computations of probabilities are direct
and elegant.
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.
iii.
iii.
The variance of the random variable is 2,
E[( X   X ) 2 ]   2
The shape of the probability density function is a
symmetric bell-shaped curve centered on the mean 
as shown in Figure 6.8.
By knowing the mean and variance we can define the
normal distribution by using the notation
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 illustrated in Figure 6.10. 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.
Approximating Binomial Probabilities
Using the Normal Distribution
Let X be the number of successes from n
independent Bernoulli trials, each with probability
of success . The number of successes, X, is a
Binomial random variable and if n(1 - ) > 9 a
good approximation is
 a  n

b

n


P ( a  X  b)  P
Z
 n (1   )

n

(
1


)


Or if 5 < n(1 - ) < 9 we can use the continuity
correction factor to obtain
 a  0.5  n
b  0.5  n

P ( a  X  b)  P
Z
 n (1   )
n (1   )

where Z is a standard normal variable.




The Exponential Distribution
The exponential random variable T (t>0) has a
probability density function
f (t )  e
 t
for t  0
Where  is the mean number of occurrences per
unit time, t is the number of time units until the
next occurrence, and e = 2.71828. . . Then T is
said to follow an exponential probability distribution.
The cumulative distribution function is
F (t )  1  e
 t
for t  0
The distribution has mean 1/ and variance 1/2
Probability Density Function for an
Exponential Distribution with  = 0.2
f(x)
Lambda = 0.2
0.2
0.1
0.0
0
10
20
x
Joint Cumulative Distribution Functions
Let X1, X2, . . .Xk be continuous random variables
i.
Their joint cumulative distribution function,
F(x1, x2, . . .xk) defines the probability that
simultaneously X1 is less than x1, X2 is less than x2,
and so on; that is
F ( x1 , x2 ,, xk )  P( X 1  x1  X 2  x2   X k  xk )
ii.
The cumulative distribution functions F(x1), F(x2), . .
.,F(xk) of the individual random variables are called
their marginal distribution functions. For any i,
F(xi) is the probability that the random variable Xi does
not exceed the specific value xi.
F ( x1 , x2 ,, xk )  F ( x1 ) F ( x2 ) F ( xk )
iii.
The random variables are independent if and only if
Covariance
Let X and Y be a pair of continuous random
variables, with respective means x and y.
The expected value of (x - x)(Y - y) is called
the covariance between X and Y. That is
Cov( X , Y )  E[( X   x )(Y   y )]
An alternative but equivalent expression can
be derived as
Cov( X , Y )  E ( XY )   x  y
If the random variables X and Y are
independent, then the covariance between
them is 0. However, the converse is not true.
Correlation
Let X and Y be jointly distributed random
variables. The correlation between X and Y is
  Corr ( X , Y ) 
Cov( X , Y )
 X Y
Sums of Random Variables
Let X1, X2, . . .Xk be k random variables with means 1, 2,.
. . k and variances 12, 22,. . ., k2. The following
properties hold:
i.
The mean of their sum is the sum of their means;
that is
E ( X 1  X 2    X k )  1  2    k
ii.
If the covariance between every pair of these random
variables is 0, then the variance of their sum is the
sum of their variances; that is
Var ( X 1  X 2    X k )   12   22     k2
However, if the covariances between pairs of random
variables are not 0, the variance of their sum is
K 1
K
Var ( X 1  X 2    X k )   12   22     k2  2  Cov( X i , X j )
i 1 j i 1
Differences Between a Pair of
Random Variables
Let X and Y be a pair of random variables with means X
and Y and variances X2 and Y2. The following
properties hold:
i.
The mean of their difference is the difference of their
means; that is
E( X  Y )   X  Y
ii.
If the covariance between X and Y is 0, then the
variance of their difference is
Var ( X  Y )   X2   Y2
iii.
If the covariance between X and Y is not 0, then the
variance of their difference is
Var( X  Y )   X2   Y2  2Cov( X , Y )
Linear Combinations of
Random Variables
The linear combination of two random variables, X
and Y, is
W  aX  bY
Where a and b are constant numbers.
The mean for W is,
W  E[W ]  E[aX  bY ]  a X  bY
The variance for W is,
 W2  a 2 X2  b 2 Y2  2abCov( X , Y )
Or using the correlation,
 W2  a 2 X2  b 2 Y2  2abCorr ( X , Y ) X  Y
If both X and Y are joint normally distributed
random variables then the resulting random
variable, W, is also normally distributed with mean
and variance derived above.