CH 6 Powerpoint, Study Notes on Discrete Random Variables
Chapter Outline
6.1. Two Types of Random Variables
6.2. Discrete Probability Distributions
6.3. The Binomial Distribution
Random Variable
Definition: A random variable is a variable whose value is a numerical value that is determined by the outcome of an experiment.
A random variable assigns one and only one numerical value to each experimental outcome.
Before an experiment is carried out, its outcome is uncertain.
A random variable can be thought of as representing an uncertain numerical outcome due to its assignment of numbers to experimental outcomes.
Idea of a Random Variable
Example Scenario: In a coffee shop experiment where the number of coffees sold during the day is denoted as ( x ), the outcome (number sold) is uncertain before the day begins, thus ( x ) is a random variable.
Types of Random Variables
Two Main Types:
Discrete Random Variables: Can be counted or listed.
Continuous Random Variables: Not countable.
Discrete Random Variables
A discrete random variable can assume a finite number of possible values or the possible values may take the form of a countable sequence (e.g., 0, 1, 2, …).
Examples of Discrete Random Variables
Number of Customers Entering a Store: Values can be 0, 1, 2, 3, 4, etc. (Countable and Finite).
Number of Hurricanes in a Decade: Values can be counted across a region. (Countable and Finite).
Ratings to a Movie: Possible values {1, 1.5, 2, 2.5, …, 5}.
Production Scenarios:
Inspecting 15 fertilizer types: # productive fertilizer is between 0 and 15.
Answering 35 questions: # correct answers is between 0 and 35.
Quality Inspection of 60 TVs: # defects is between 0 and 60.
Counting cars at a toll between 11:00 & 1:00: # cars is 0, 1, … .
Continuous Random Variables
A continuous random variable may assume any numerical value in one or more intervals on the real number line.
Note: Its possible value is not countable.
Examples of Continuous Random Variables
Temperature of a Cup of Tea: Infinite possibilities due to precision.
Weight of a Person: Infinite possibilities (e.g., 60.7 Kg, 60.67 Kg, etc.).
Height of Adults (Age 18-65): Values can include 165, 173, 163.5, etc.
Lifetime of a Battery: Values in hours could include 178.9, 250.5, etc.
Measuring Time Between Flight Departures: Values measured in minutes or hours.
Discrete Probability Distribution
The value assumed by a random variable depends on the outcome of an experiment, introducing uncertainty.
We can model the random variable by finding or estimating its probability distribution which describes how probabilities are distributed over its values.
The discrete probability distribution is represented in the form of a table, graph, or formula that gives the probability associated with each possible value of the random variable.
Properties of Discrete Probability Distribution
Let ( x ) be a random variable.
( p(x) ) is the discrete probability distribution of the random variable.
For any value ( x ) of the random variable, it must hold that: ( p(x) \geq 0 ).
The sum of all probabilities for the events in the sample space must equal 1.
Example 6.2: Sales of Car Radios at Sound City
Historical Records (over 100 weeks):
3 weeks with no radios sold
20 weeks with 1 radio sold
50 weeks with 2 radios sold
20 weeks with 3 radios sold
5 weeks with 4 radios sold
2 weeks with 5 radios sold
Random Variable (x): Number of TrueSound-XL car radios sold.
Possible Values for x: {0, 1, 2, 3, 4, 5}.
Calculate Probabilities
Probability of No Radio Sold: ( p(0) = \frac{3}{100} = 0.03 ) (3 of 100 weeks).
Continuing this way for all values gives the entire estimated probability distribution.
Graphical Representation
A graph shows the estimated probability distribution of the number of car radios sold during a week.
Example Queries:
Probability (≥2): ( P(X ≥ 2) = p(2) + p(3) + p(4) + p(5) = 0.5 + 0.2 + 0.05 + 0.2 = 0.77 ).
Probability (1 < x ≤ 4): ( P(1 < x ≤ 4) = p(2) + p(3) + p(4) = 0.5 + 0.2 + 0.05 = 0.75 ).
Expected Value of Discrete Random Variable
If the experiment is repeated an indefinitely large number of times, the mean (expected value) of all observed values can be calculated.
The expected value ( E(X) ) is calculated as
( E(X) = \sum (x imes p(x)) ) over all possible values of x.
Example Calculation for Expected Value
Average Number of Radios Sold Per Week:
( E(X) = 0 imes 0.03 + 1 imes 0.2 + 2 imes 0.5 + 3 imes 0.2 + 4 imes 0.05 + 5 imes 0.02 = 2.1 )
Hence, the average number of radios sold per week is ( 2.1 ).
Example 6.4: Life Insurance Case
Insurance Details:
Policy amount: $20,000
Annual premium: $300
Probability of death: 0.001.
Random Variable ( x ): Profit made by the insurance company during a year.
Expected Profit Calculation
To find expected profit:
( E(X) = 300(0.999) + (-19700)(0.001) )
Result: ( E(X) = 280 ).
This indicates a company average profit of $280 per policy per year.
Variance of a Discrete Random Variable
Definition: Variance measures the average of the squared deviations of values from the mean.
Formula: Variance of a discrete random variable is defined as:
( Var(X) = \sum (x - \mu)^2 imes p(x) ) where ( \mu ) is the expected mean.
Standard Deviation of a Discrete Random Variable
Definition: The standard deviation is the square root of the variance and represents the spread of values from their expected value.
Example 6.2: Selling Car Radios - Variance and Interpretation
Variance Calculation:
( Var(X) = 0.89 )Standard Deviation Calculation:
( \sigma = \sqrt{0.89} = 0.9434 ).Interpretation: If another satellite radio has a mean of 2.1 radio sales and a standard deviation of 1.2254, the ClearTone-400 shows greater variability in sales compared to TrueSound-XL.
Interval of a Discrete Random Variable
Concept: Similar to population measurements, we can find the probability that a random variable lies within a set number of standard deviations from its mean.
Example using ( \mu = 2.1 ) and ( \sigma = 0.9434 ). To find the interval [0.2132, 3.9868] covering ±2 standard deviations leads to:
Identifying ( x = 1, 2, 3 ) as falling within the interval.
Probabilities for those values yield: ( P(X=1)+P(X=2)+P(X=3) = 0.2 + 0.5 + 0.2 = 0.9 ).
Thus, there is a 90% chance that sales will be within two standard deviations of the mean.
Binomial Distribution
Definition: The binomial distribution is a vital discrete probability distribution characterized by:
Experiment consists of ( n ) identical trials.
Each trial results in either “success” or “failure.”
Probability of success ( p ) is constant across trials.
Probability of failure ( q = 1 - p ).
The trials are independent.
Example 6.6: Purchase at a Discount Store
Situation: 40% of customers make a purchase. What is the probability that 2 of the next 3 customers will buy something?
This scenario fits a binomial distribution.
Probability Calculation:
Sample space outcomes for making purchases are characterized by two possible outcomes (success S / failure F).
With ( P(S) = 0.4 ), we derive ( P(F) = 0.6 ).
Total outcomes can be illustrated through a tree diagram and calculated for probabilities of particular outcomes like ( P(SSF), \ P(SFS), \ P(FSS) ).
Combined probability of 2 purchases in 3 trials:
( P(X=2) = 3 \times (0.4)^2 \times (0.6) = 0.288 ).
Thus, there's a 28.8% chance that two out of three customers will make a purchase.
Further Examples on Binomial Distribution
Probability calculations for varying numbers of customers making purchases and the associated implications.
Illustrations with factorial arrangements and connections to the binomial table for broader insights into trends and expectations.
Mean, Variance, and Standard Deviation of a Binomial Random Variable
Formulas:
Mean: ( \mu = np )
Variance: ( \sigma^2 = npq )
Standard Deviation: ( \sigma = \sqrt{npq} )
Where ( n ) represents the number of trials and ( p ) is the probability of success.