Transcript Chapter 6

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
(Figure 6.1)
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
(Figure 6.3)
0
a
b
x
Probability Density Function for a
Uniform 0 to 1 Random Variable
(Figure 6.4)
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
(Figure 6.6 (a))
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
(Figure 6.6 (b))
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
(continued)
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
Let X be a continuous random variable. There are two important expected
values that are used routinely to define continuous probability
distributions.
i.
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
  E[( X   X ) ]
2
X
2
Or an alternative expression can be derived
  E( X )  
2
X
iii.
2
2
X
The standard deviation of X, X, is the square root of the variance.
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
 W2  Var (a  bX )  b 2 X2
and the standard deviation of W is
W  b  X
Linear Functions of Variable
(continued)
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. 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
(Figure 6.8)
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  as shown in Figure
6.8.
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
(Figure 6.9 (a))
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
(Figure 6.9 (b))
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
(Figure 6.10)
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
(Figure 6.12)
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
(Figure 6.13)
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
(Example 6.6)
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
(Figure 6.27)
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.
iii.
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.
The random variables are independent if and only if
F ( x1 , x2 ,, xk )  F ( x1 ) F ( x2 ) F ( xk )
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
ii.
E ( X 1  X 2    X k )  1  2    k
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,
  a   b   2abCorr ( X , Y ) X  Y
2
W
2
2
X
2
2
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.
Key Words
 Approximating Binomial
Probabilities Using the
Normal Distribution
 Area Under Continuous
Probability Density
Functions
 Correlation
 Covariance
 Cumulative Distribution
Function
 Cumulative Distribution
Function of the Normal
Distribution
 Differences Between Pairs
of Random Variable
 Expectations of Continuous
Random Variables
 Exponential Distribution
 Finding Range Probabilities
for Normal Random
Variables
 Joint Cumulative
Distribution Function
 Linear Combinations of
Random variables
 Linear Functions of
Random Variables
Key Words
(continued)
 Mean of a Continuous
Random Variable
 Probability Density
Function
 Probability Density
Function of the Normal
Distribution
 Properties of a Normal
Distribution
 Range Probabilities for
Normal Random
Variables
 Range Probabilities
Using a Cumulative
Distribution Function
 Standard Deviation:
Continuous Random
Variable
 Standard Normal
Distribution
 Sums of Random
Variables
 Uniform Distribution
 Variance