Lecture Note 11
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MAKING DECISIONS UNDER UNCERTAINTY
Most times, one would have to make a decision before observing the outcome. For example, invest in a business today and attain profit/loss tomorrow; plan electricity generation capacity today in anticipation of next year’s demand which is uncertain; toss a coin, the result is uncertain and could be “Head” or “Tail”.
- How can one make better decisions in such an uncertain environment? This is done with the help of probabilities.
PROBABILITIES
A probability gives a measure of uncertainty. That is, the probability of an event is a measure of its likelihood or chance of occurrence.
Statistical Experiment
A Statistical Experiment can be described as any situation, specifically set up or occurring naturally. All Statistical Experiments have three things in common;
1. The experiment can have more than one possible outcomes 2. Each possible outcome can be specified in advance
3. The outcome of an experiment depends on chance, e.g. tossing a dice (possible outcomes≡{1,2,3,4,5,6})
4. After the experiment is performed, only one of the possible outcomes will occur irrespective of the prior probabilities or chances.
Outcomes of an Experiment
An outcome set of an experiment is the specification of all possible distinct results (i.e. outcomes) of the experiment when it is performed.
E.g. Toss a coin;
- Outcome set = {Head, Tail}
After tossing however, only “Head” or “Tail” and not both will be realized. Tossing a dice
- Outcome set = {1,2,3,4,5,6} Playing the lottery
- Outcome set = {Win or Lose} Playing soccer
Statistical Event
An event is any subset of a given outcome set of an experiment that is of interest. Thus, after performing the experiment, the event may or may not occur.
Example: Suppose the experiment is tossing a dice. Outcome set = {1,2,3,4,5,6}
What are the chances that 2 or 3 might show up? Event set = {2, 3}
Mutually Exclusive Events
Two events of the same experiment are mutually exclusive if their respective event sets do not overlap. In other words, when event 1 occurs, event 2 cannot occur.
Independent Events
Two events, A and B, are independent if the outcome of one of the event in no way affects the outcome of the occurrence (or not) of the other.
Measuring/Estimating Probabilities
Empirical (Relative Frequency) Probabilities: Given an event E, denote the probability of E by P(E), then, ( )
Examples
1. The economy of Ghana has always been Excellent, Good, O.K., or Bad. The following table is a breakdown of the number of times the economy was either Excellent, or Good, or O.K., or Bad in the last 50 years.
What is the probability that next year the economy will be Excellent, Good, O.K. or Bad? Solution
( )
Economy Number of times
Excellent 5
Good 25
O.K. 10
Bad 10
( )
( )
( )
Probability = 0 means the event will never occur
Probability = 1 means the event will surely/always occur
Probability between 0 and 1 measures the likelihood of chances that the event will occur *Probability can never be outside the range 0 to 1
Probability can also be determined using expert opinion.
2. Below is a table of ages of students attending a summer camp. If a student is randomly picked from a darkened room containing all the students, what are the chances that the student picked is aged 25 – 29?
Age of student Frequency/Number Present
10 – 14 4
15 – 19 8
20 – 24 5
25 – 29 2
30 - 34 1
( )
Rules of Probabilities
Addition Rule
Multiplication Rule (In particular independent)
Complement Rule
Conditional Probability
Bayes Rule
In all, given an outcome set x = {x1, x2, x3, …, xn} with respective probabilities P = {P1, P2, P3, …, Pn}, the
following must hold for the probabilities;
∑
Given an event E with probability P(E), P(Ē) = 1 – P(E)
Random Variable
A Random Variable, usually written Xi, is a variable whose possible values are a numerical outcome of a
random phenomenon.
1. Discrete Random Variables
- May take on only a countable number of distinct values such as 1, 2, 3, 4.
- If a random variable can take only a finite number of distinct values, then, it must be discrete. 2. Continuous Random Variables
- A variable which can take on infinite number of possible values.
- Continuous Random Variables are usually measurements like height, weight, pressure, etc.
Probability Distribution
A Probability Distribution of a Random Variable describes the chances of the occurrence of the possible outcomes of the random variable. That is, it lists for each outcome, the probability of occurrence
Example:
1. Tossing a dice
Outcome = {1, 2, 3, 4, 5, 6} Probability Distribution Probability ={ }
2. I m p I m p l i c a t i o n : Implication:
Demand for next year is uncertain among these five events. A demand of 2500 is highly probable and a demand of 3000 is least probable.
X1 X2 X3 X4 X5
Electricity Demand Next Year
(MW) 1800 2200 2500 2800 3000
Probability (X) 0.2 0.1 0.35 0.3 0.05
X1 X2 X3 X4 X5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
1800 2200 2500 2800 3000
P(X)
Demand
Probability
Expectation (Mean) of a Random Variable
Given a random variable x = {x1, x2, x3… xn} with respective probabilities P = {P1, P2, P3… Pn}, the
expectation/mean of the Random Variable x is given as:
( ̅) ∑
Example:
Using the illustration above (Electricity Demand),
( ̅) ( ) ( ) ( ) ( ) ( )
Variance of a Random Variable
Given a Random Variable x = {x1, x2, x3… xn} with respective probabilities P = {P1, P2, P3… Pn}, the variance
of X is given as
( ) ∑ ( ̅ ) ( ̅
) ( ̅ ) ( ̅ ) ( ̅ )
where X is the expectation of the Random Variable(X) √ ( )
NB: A smaller standard deviation implies lesser uncertainty A larger standard deviation implies greater uncertainty
Example:
Using the illustration above (Electricity Demand),
( ) ( ) ( ) ( ) ( ) ( )
( ) √
( ) ( ) ( )
Example
The economy of Ghana could be Excellent, Good, O.K. or Bad next year with respective probabilities 0.2, 0.5, 0.2 and 0.1. The table below gives the respective returns of an investment next year.
Economy Return
Excellent 25%
Good 15%
O.K. 10%
Bad -10%
The economy of the United States next year will also be excellent, good, O.K. or Bad with respective probabilities and returns of 0.5, 0.3, 0.15 and 0.05; 10%, 3%, 1% and 0.5%.
1. Find the expected returns of GH¢100,000 investment in both Ghana’s economy and the United States’ economy.
2. Quantify the risk associated with the investment. 3. Which country is more economically viable to invest?
Solution 1&2 For Ghana
( ̅)
∑
( ̅)
( ) ( ) ( ) ( ) ( )
∑ ( ̅ ) ( ̅
) ( ̅ ) ( ̅ ) ( ̅
) ( )
( ) ( ) ( ) ( ( ))
√
For US
( ̅) ∑
( ̅)
( ) ( ) ( ) ( ) ( )
∑ ( ̅ ) ( ̅
) ( ̅ ) ( ̅ ) ( ̅
) ( )
( ) ( ) ( ) ( ) √
