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Chapter 6. Continuous Random Variables and Probability Distributions


Continuous Random Variables and Probability Density Functions

ď&#x201A;§ Continuous Random Variables A random variable X is said to be continuous if its set of possible values is an entire interval of numbers â&#x20AC;&#x201C; that is , if for some A<B, any number x between A and B is possible

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Continuous Random Variables and Probability Density Functions

 Example If a chemical compound is randomly selected and its PH X is determined, then X is a continuous rv because any PH value between 0 and 14 is possible. If more is know about the compound selected for analysis, then the set of possible values might be a subinterval of [0, 14], such as 5.5 ≤ x ≤ 6.5, but X would still be continuous. 0

5.5

6.5

14

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Continuous Random Variables and Probability Density Functions

ď&#x201A;§ Probability Distribution for Continuous Variables Suppose the variable X of interest is the depth of a lake at a randomly chosen point on the surface. Let M be the maximum depth, so that any number in the interval [0,M] is a possible value of X.

0

4.1(a)

M

Measured by meter

0

M 4.1(b) Measured by centimeter

Discrete Cases

0

M 4.1(c) A limit of a sequence of discrete histogram Continuous Case

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Continuous Random Variables and Probability Density Functions

 Probability Distribution Let X be a continuous rv. Then a probability distribution or probability density function (pdf) of X is f(x) such that for any two numbers a and b with a ≤ b

P ( a  X  b) 

b

a

f ( x ) dx

The probability that X takes on a value in the interval [a,b] is the area under the graph of the density function as follows. f(x ) a

b 5

x


Continuous Random Variables and Probability Density Functions

 A legitimate pdf should satisfy 1. f ( x ) ³ 0 for all x 2.

¥

f ( x)dx  area under the entire graph of f ( x) 1 f(x ) a

b

6

x


Continuous Random Variables and Probability Density Functions

 pmf (Discrete) vs. pdf (Continuous) p(x) 0

f(x) 0

M

P(X=c) = p(c)

M

P(X=c) = f(c) ?

P( X  c) 

7

c

c

f ( x) dx  0


Continuous Random Variables and Probability Density Functions

 Proposition If X is a continuous rv, then for any number c, P(X=c)=0. Furthermore, for any two numbers a and b with a<b,

P(a≤X ≤b) = P(a<X ≤b) = P(a ≤X<b) = P(a <X<b) Impossible event :the event contain no simple element P(A)=0  A is an impossible event ?

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Continuous Random Variables and Probability Density Functions

 Example (Cont’) ì0.15e -0.15( x -0.5) x ³ 0.5 f ( x)  í 0 otherwise î

1. f(x) ≥ 0; ¥ 2. to show ¥ f ( x) ³ 0dx  1, we use the result

¥

¥

¥

f ( x )dx   0.15e

0.15( x 5)

0.5

 0.15e

0.075

dx  0.15e

0.075

1  (0.15)(0.5) × e 1 0.15

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¥

0.5

¥

a

e

 kx

e 0.15 x dx

1  ka dx  e k


Continuous Random Variables and Probability Density Functions

 Example (Cont’) f(x)

P( X  5)

0.15

0.5

2

4

5

5

¥

0.5

6

P( X  5)   f ( x)dx   .15e

0.15( x 5)

5

 0.15e

0.075

8

10

dx  0.15e

æ 1 0.15 x ö ×ç e ÷  0.491  P( X < 5) è 0.15 ø 0.5 10

x 0.075

5

0.5

e 0.15 x dx


Cumulative Distribution Functions and Expected Values

 Cumulative Distribution Function The cumulative distribution function F(x) for a continuous rv X is defined for every number x by x

F ( x)  P( X  x)   f ( y )dy ¥

For each x, F(x) is the area under the density curve to the left of x as follows F(x) f(x)

1 F(8)

F(8)

0.5 5

8

10

x

5 11

8

10

x


Cumulative Distribution Functions and Expected Values

 Example Let X, the thickness of a certain metal sheet, have a uniform distribution on [A, B]. The density function is shown as follows. f(x)

f(x)

1 BA

1 BA

A

B

x

A x B For x < A, F(x) = 0, since there is no area under the graph of the density function to the left of such an x. For x ≥ B, F(x) = 1, since all the area is accumulated to the left of such an x. 12


Cumulative Distribution Functions and Expected Values

 Using F(x) to compute probabilities Let X be a continuous rv with pdf f(x) and cdf F(x). Then for any number a P( X  a )  1  F (a )

and for any two numbers a and b with a<b P(a  X  b)  F (b)  F (a) f(x)

-

= a

b

b 13

a


Cumulative Distribution Functions and Expected Values

 Example Suppose the pdf of the magnitude X of a dynamic load on a bridge is given by ì1 3   x, 0  x  2 f ( x)  í 8 8 î0, otherwise

For any number x between 0 and 2, x

F ( x) 

¥

thus

x

1 3 x 3 f ( y )dy   (  x)dy   x 2 8 8 8 16 0

ì0, x < 0 x x 1 3 x 3  F ( x)  í  f ( y )dy   (  x)dy   x 2 , x Î [0, 2] 8 8 8 16 0  ¥ î1, x  2 14


Cumulative Distribution Functions and Expected Values

 Example (Cont’) P(1  X  1.5)  F (1.5)  F (1)  0.297

P( X  1)  1  F ( X  1)  0.688 f(x)

1

7/8

F(x)

1/8 0

2 15

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Cumulative Distribution Functions and Expected Values

 Obtaining f(x) form F(x) If X is a continuous rv with pdf f(x) and cdf F(x), then at every x at which the derivative F’(x) exists, F’(x)=f(x)

f ( x)  F ( x) F ( x)  f ( x)

x

F ( x)  P( X  x)   f ( y )dy ¥

x

f ( x)  F '( x)  (  f ( y )dy ) ' ¥

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Cumulative Distribution Functions and Expected Values

 Example When X has a uniform distribution, F(x) is differentiable except at x=A and x=B, where the graph of F(x) has sharp corners. Since F(x)=0 for x<A and F(x)=1 for x>B, F’(x)=0=f(x) for such x. For A<x<B

d x A 1 F '( x)  ( )  f ( x) dx B  A B  A

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Cumulative Distribution Functions and Expected Values

 Example The distribution of the amount of gravel (in tons) sold by a particular construction supply company in a given week is a continuous rv X with pdf ì3  (1  x 2 ) 0  x  1 f ( x)  í 2  otherwise î0

The cdf of sales for any x between 0 and 1 is x

F ( x) 

 0

3 3 3 3 y 3 x (1  y 2 )dy  ( y  ) | yy  0x  ( x  ) 2 2 3 2 3

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4.2 Cumulative Distribution Functions and Expected Values

 The median

~, The median of a continuous distribution, denoted by  ~ th  is the 50 percentile, so satisfies 0.5=F( ), that is, half ~ the area under the density curve is to the left of  and ~ half is to the right of  Symmetric Distribution

~

~ 19

~


4.2 Cumulative Distribution Functions and Expected Values

 Expected/Mean Value The expected/mean value of a continuous rv X with pdf f(x) is ¥

 X  E ( X )   xf ( x)dx ¥

 X  E ( X )  å x × p( x) xÎD

Discrete Case

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4.2 Cumulative Distribution Functions and Expected Values

 Example 4.9 (Ex. 4.8 Cont’) The pdf of weekly gravel sales X was ì3  (1  x 2 ) 0  x  1 f ( x)  í 2  otherwise î0

So ¥

1

3 3 x2 x 4 x 1 3 2 E ( x)   xf ( x) dx   x (1  x )dx  (  ) |x 0  2 2 2 4 8 ¥ 0

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4.2 Cumulative Distribution Functions and Expected Values

 Expected value of a function If X is a continuous rv with pdf f(x) and h(X) is any function of X, then ¥

E [ h( X ) ]  h ( X )   h( x ) f ( x ) dx ¥

h ( X )  E (h( X ))  å h( x) × p ( x) xÎD

Discrete Case

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4.2 Cumulative Distribution Functions and Expected Values

 Example 4.10 Two species are competing in a region for control of a limited amount of a certain resource. Let X = the proportion of the resource controlled by species 1 and suppose X has pdf ì1 0  x  1 f ( x)  í î0 otherwise which is a uniform distribution on [0,1]. Then the species that controls the majority of this resource controls the amount ì1  X if 0  x < 12 h( X )  max( X ,1  X )  í 1 î X if 2  X  1 The expected amount controlled by the species having majority control is then E [ h( X )]   max( x, x  1) × f ( x )dx   max( x,1  x) × 1dx ¥

1

¥

0

1

1

  (1  x) × 1dx  1 x × 1dx  0

2

2

23

3 4


Cumulative Distribution Functions and Expected Values

 The Variance The variance of a continuous random variable X with pdf f(x) and mean value μ is ¥

s X  V ( X )   ( x   ) 2 f ( x)dx  E[( X   ) 2 ] 2

¥

The standard deviation (SD) of X is

sX  V (X )

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4.2 Cumulative Distribution Functions and Expected Values

 Proposition E (aX  b)  aE ( X )  b V ( X )  E ( X 2 )  [ E ( X )]2 The Same Properties as Discrete Cases

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4.5 Other Continuous Distributions

 The Lognormal Distribution A nonnegative rv X is said to have a lognormal distribution if the rv Y = ln(X) has a normal distribution . The resulting pdf of a lognormal rv when ln(X) is normally distributed with parameters μ and σ is

ì 1  ( ln ( x )   ) 2 ( 2s 2 ) e  f ( x;  , s )  í 2 sx  0 î

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x³0 x<0


4.5 Other Continuous Distributions

 Mean and Variance E( X )  e

 s 2 2

;V ( X )  e

2  s 2

(

s2

× e

)

1

 The cdf of Lognormal Distribution F ( x;  , s )  P ( X  x )  P[ ln ( X )  ln ( x ) ]

ln ( x )   ö æ æ ln ( x )   ö  Pç Z  ÷  ç ÷ s s è ø è ø 35


4.5 Other Continuous Distributions

 The Beta Distribution A random variable X is said to have a beta distribution with parameters α, β, A, and B if the pdf of X is a 1 b 1 ì 1 G(a  b ) æ x  A ö æ B  x ö × , A x B  ç ÷ ç ÷ f ( x;a , b , A, B )  í B  A G ( a ) ×G ( b ) è B  A ø è B  A ø  0 otherwise î

The case A = 0, B =1 gives the standard beta distribution. And the mean and variance are B  A ) ab ( a 2   A  ( B  A) × ; s  2 a b ( a  b ) ( a  b  1) 2

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Lec5 continuous rv  
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