CHAPTER 3 PROBABILITY DISTRIBUTIONS - The Hong Kong Polytechnic ...

CHAPTER 3

PROBABILITY DISTRIBUTIONS

Page

Contents

3.1

Introduction to Probability Distributions

51

3.2

The Normal Distribution

56

3.3

The Binomial Distribution

60

3.4

The Poisson Distribution

64

Exercise

Objectives:

68

After working through this chapter, you should be able to:

(i)

understand basic concepts of probability distributions, such as

random variables and mathematical expectations;

(ii)

show how the Normal probability density function may be used to

represent certain types of continuous phenomena;

(iii)

demonstrate how certain types of discrete data can be represented by

particular kinds of mathematical models, for instance, the Binomial

and Poisson probability distributions.

50

Chapter 3: Probability Distributions

3.1

Introduction to Probability Distributions

3.1.1

Random Variables

A random variable (R.V.) is a variable that takes on different numerical values

determined by the outcome of a random experiment.

Example 1

An experiment of tossing a coin 4 times.

Notation :

Capital letter, X - Random variable

Lowercase, x - a possible value of X

A random variable is discrete if it can take on only a limited number of values.

A random variable is continuous if it can take any value in an interval.

The probability distribution of a random variable is a representation of the

probabilities for all the possible outcomes. This representation might be algebraic,

graphical or tabular.

A table or a formula listing all possible values that a discrete variable can take on,

together with the associated probability is called a discrete probability distribution.

Example 2

The probability distribution of the number of heads when a coin is tossed 4 times.

x

Pr(X = x)

0

1

2

3

4

1

16

4

16

6

16

4

16

1

16

51

Chapter 3: Probability Distributions

? 4?

? ?

? x?

,

Pr(X = x) =

16

i.e.

x = 0, 1, 2, 3, 4

In graphic form :

1.

2.

Total area of rectangle = 1

Pr(X = 1) = shaded area

Example 3

An experiment of tossing two fair dice.

Let random variable X be the sum of two dice.

The probability distribution of X

Sum, x

P(X = x)

2

3

4

5

6

7

8

9

10

11

12

1

36

2

36

3

36

4

36

5

36

6

36

5

36

4

36

3

36

2

36

1

36

The probability function, f(x), of a discrete random variable X expresses the

probability that X takes the value x, as a function of x. That is

=

f ( x ) Pr

=

( X x)

where the function is evaluated at all possible values of x.

Properties of probability function Pr ( X = x ) :1.

Pr ( X= x ) ¡Ý 0 for any value x.

2.

The individual probabilities sum to 1; that is

¡Æ Pr ( X= x=)

1.

x

Example 4

Find the probability function of the number of boys on a committee of 3 selected at

random from 4 boys and 3 girls.

52

Chapter 3: Probability Distributions

Continuous Probability Distribution

1.

2.

3.

3.1.2

The total area under this curve bounded by the x axis is equal to one.

The area under the curve between lines x = a and x = b gives the probability

that X lies between a and b, which can be denoted by Pr(a ¡Ü X ¡Ü b).

We call f(x) a "probability density function", i.e. p.d.f.

Mathematical Expectations

Expectations for Discrete Random variables

The expected value is the mean of a random variable.

Example 5

A review of textbooks in a segment of the business area found that 81% of all pages

of text were error-free, 17% of all pages contained one error, while the remaining 2%

contained two errors. Find the expected number of errors per page.

Let random variable X be the number of errors in a page.

x

0

1

2

Pr ( X = x )

0.81

0.17

0.02

53

Chapter 3: Probability Distributions

Expected number of errors per page

= 0¡Á 0.81 + 1¡Á 0.17 + 2¡Á 0.02

= 0.21

The expected value, E [ X ] , of a discrete random variable X is defined as

E [ X=

] or ? X

x Pr ( X x )

¡Æ=

x

Definition :

Let X be a random variable. The expectation of the squared discrepancy about the

2

mean, E ?( X ? ? X ) ? , is called the variance, denoted ¦Ò X2 , and given by

?

?

2

=

¦Ò X2 E ?( X ? ? X ) ?

Var ( X ) or

?

?

=

x)

¡Æ ( x ? ? X ) Pr ( X =

2

x

=

¡Æx

2

Pr ( X= x ) ? ? X2

x

Properties of a random variable

Let X be a random variable with mean ? X and variance ¦Ò X2 and a, b are constants.

1.

E [ aX + b ]= a ? X + b

2.

Var ( aX + b ) =

a 2¦Ò X2

Sums and Differences of random variables

Let X and Y be a pair of random variables with means ? X and ?Y and variances ¦Ò X2

and ¦Ò Y2 , and a, b are constants.

1.

E [ aX + bY ] = a ? X + b?Y

2.

E [ aX ? bY ] = a ? X ? b?Y

3.

If X and Y are independent random variables, then

Var ( aX + bY ) = a 2¦Ò X2 + b 2¦Ò Y2

Var ( aX ? bY ) = a 2¦Ò X2 + b 2¦Ò Y2

54

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