Probability
QMT200
CHAPTER 3: PROBABILITY DISTRIBUTION
3.1
RANDOM VARIABLES AND PROBABILITY DISTRIBUTION
Random variables is a quantity resulting from an experiment that, by chance, can assume different values. Examples of random variables are the number of defective light bulbs produced during the week and the heights of the students is a class. Two types of random variables are discrete random variables and continuous random variable.
3.2
DISCRETE RANDOM VARIABLE
A random variable is called a discrete random variable if its set of posibble outcomes is countable. Probability distribution is a listing of all the outcomes of an experiment and the probability associated with each outcome. For example, the probability distribution of rolling a die once is as below: Outcome, x Probability, P(x) 1 1 6 2 1 6 3 1 6 4 1 6 5 1 6 6 1 6
The probability distribution for P(x) for a discrete random variable must satisfy two properties: 1. The values for the probabilities must be from 0 to 1; 0 ≤ ( ) ≤ 1 2. The sum for P(x) must be equal to 1; ∑ ( ) = 1
QMT200
3.2.1 FINDING MEAN AND VARIANCE Mean of X is also referred to as its “expected value”.
= ( ) Where: = ∑[ ( )]
( )=
= (
) − [ ( )]
(
)=
[
( )] = ( )
Example 1 An experiment consists of tossing two coins simultaneously. Write down the sample space. If X is the number of tails observed, determine the probability distribution of X.
QMT200
Example 2 The following table shows the probability of the number of long-distance telephone calls made in a month by residents of a sample of urban households. X 0 1 2 3 4 5 6 7 8 9 10 P(X=x) 0.02 0.05 0.08 0.11 0.14 0.22 0.28 0.04 0.03 0.02 0.01
a) Find the mean number of calls per household. b) Calculate the variance c) Find (1 < ≤ 6)
QMT200
3.2.2 MEAN AND VARIANCE OF LINEAR COMBINATIONS OF RANDOM VARIABLES Rules of Mean For any constant a and b, a) b) c) d) ( )= ( )= ( ) ( ± )= ( )± ( ) ( ± )= ( )± ( ) Rules of Variance For any
CHAPTER 3: PROBABILITY DISTRIBUTION
3.1
RANDOM VARIABLES AND PROBABILITY DISTRIBUTION
Random variables is a quantity resulting from an experiment that, by chance, can assume different values. Examples of random variables are the number of defective light bulbs produced during the week and the heights of the students is a class. Two types of random variables are discrete random variables and continuous random variable.
3.2
DISCRETE RANDOM VARIABLE
A random variable is called a discrete random variable if its set of posibble outcomes is countable. Probability distribution is a listing of all the outcomes of an experiment and the probability associated with each outcome. For example, the probability distribution of rolling a die once is as below: Outcome, x Probability, P(x) 1 1 6 2 1 6 3 1 6 4 1 6 5 1 6 6 1 6
The probability distribution for P(x) for a discrete random variable must satisfy two properties: 1. The values for the probabilities must be from 0 to 1; 0 ≤ ( ) ≤ 1 2. The sum for P(x) must be equal to 1; ∑ ( ) = 1
QMT200
3.2.1 FINDING MEAN AND VARIANCE Mean of X is also referred to as its “expected value”.
= ( ) Where: = ∑[ ( )]
( )=
= (
) − [ ( )]
(
)=
[
( )] = ( )
Example 1 An experiment consists of tossing two coins simultaneously. Write down the sample space. If X is the number of tails observed, determine the probability distribution of X.
QMT200
Example 2 The following table shows the probability of the number of long-distance telephone calls made in a month by residents of a sample of urban households. X 0 1 2 3 4 5 6 7 8 9 10 P(X=x) 0.02 0.05 0.08 0.11 0.14 0.22 0.28 0.04 0.03 0.02 0.01
a) Find the mean number of calls per household. b) Calculate the variance c) Find (1 < ≤ 6)
QMT200
3.2.2 MEAN AND VARIANCE OF LINEAR COMBINATIONS OF RANDOM VARIABLES Rules of Mean For any constant a and b, a) b) c) d) ( )= ( )= ( ) ( ± )= ( )± ( ) ( ± )= ( )± ( ) Rules of Variance For any