Chapter 7: Random Variables and Probability Distributions Advanced Placement Statistics
What we’ll be discussing • Random Variables • Definitions • Discrete and Continuous • Probability Distributions • Properties • Derivations • Mean and Std Deviation • Linear combinations • Binomial, Geometric and Normal
7.1 - Random Variables • A numerical variable whose value depends on the outcome of a chance experiment • Discrete • Set of possible values are isolated points on number line • Continuous • Set of possible values are contained in an interval on number line • We use capital letters to represent random variables/small letters to represent values assigned to random variables • X, Y, Z, W etc • xi yi zi
Problem 7.5 • A point is randomly selected on the surface of a lake that has a maximum depth of 100 ft. Let Y= the depth of the lake at the randomly chosen point. What are the possible values of Y? Is it discrete or continuous?
Problem 7.7 • A box contains four slips of paper marked 1, 2, 3, and 4. Two slips are selected without replacement. List the possible values for each of the following random variables: • X = sum of the two numbers • Y = difference between the first and second number • Z = number of slips selected that show an even number • W = number of slips selected that show a 4
7.2 – Probability Distributions for Discrete Random Variables • The probability distribution of a discrete random variable is a model that describes the long-run behavior of the variable. • A discrete random variable X has a countable number of possible values. The probability distribution of X lists the values and their probabilities. • Common ways to display a probability distribution for a discrete random variable are a table, a probability histogram or a formula.
A Discrete Probability Distribution • The instructor of a large class gives 15% each of A’s and D’s, 30% each of B’s and C’s and 10% F’s. Choose a student at random. The random variable X is defined as the the selected student’s grade. • Possible values of X • Distribution of X • P(grade is a 3 or 4)
Properties of Discrete Probability Distributions For every possible value of X Because P(X) is a probability Because the prob dist lists all possible values of X
Determining Probabilities • Theoretically • Flip a coin 4 times • X : number of heads in four flips • Empirically • A consumer organization evaluates new cars by number of defects. X : the number of defects on a randomly selected car. • Law of Large Numbers • Simulation
Problem • What is the probability distribution of the discrete random variable X that counts the number of heads in four tosses of a coin? Assume the coin is fair and the tosses are independent. • Provide a probability distribution table and a probability distribution histogram.
Feature of the Probability Histogram • The area under the histogram must be equal to 1. • Width of each bar is 1. Height corresponds to probability that the random variable takes that value.
Probability Distribution using a Formula • A contractor is required by a county planning department to submit anywhere from one to five forms in applying for a building permit. The random variable Y : number of forms required. The distribution of this RV can be described by • Is this a valid probability distribution? • What is the probability that at most 3 forms are required? • What is the probability that between two and four forms are required?
Problem • A couple plans to have three children. There are 8 possible arrangements of girls and boys. All 8 arrangements are approx equally likely. • Write down all 8 arrangements. What is the probability of each? • Let X be the number of girls the couple has. What is the probability that X=2? • Find the distribution of X.
Problem 7.9 • Let X be the number of broken eggs in a randomly selected carton of eggs. Suppose the distribution of X is: • What is P(X=4) ? • Interpret P(X=1) • Determine P(X<2), P(X 2)
Sec 7.3 Probability Distributions for Continuous Random Variables • A probability distribution for a continuous random variable X is specified by a mathematical function f(x) and is called a density function.
Properties of a Continuous Probability Function • f(x) 0 • The area under the density curve is equal to 1
Determining Probabilities of Continuous RandomVariables • The probability that X falls in any interval is the area under the density curve within the interval.
Example • X: amount of time (min) taken by a clerk to process a form • Suppose that X has a probability distribution with density function • Find
More about Continuous Random Variable Probabilities As opposed to discrete Random Variables Cumulative areas To find these cumulative areas we use integral calculus or tables (Table 2 in the Appendix)
Sec 7.4 Mean and Standard Deviation of a Random Variable • Mean value of a random variable X Where the distribution is centered Also called the Expected Value • Standard Deviation of a random variable X Variability in the probability distribution
Mean ValueA Measure of Center • Calculation of the mean from data • Sample mean – average of actual observations/data • Histogram • Calculation of the mean of a random variable • Only do calculations for discrete random variables • Given values of the random variable and associated probabilities • The mean of a random variable is the long run average you would expect – also called an Expected Value • A weighted average of possible outcomes
Example from Text • Consider an experiment consisting of random selection of an autos licensed in a certain state. • X: # of low beam headlights that need adjustment (possible values 0, 1, 2)
Calculate Mean of Random Variable Probability Distribution Expected Value
Standard DeviationA Measure of Spread • The analogous relationship between and also applies to s and • That is, sample standard deviation of data and the standard deviation of a discrete random variable
Mean and Standard Deviation when X is Continuous • Computations for and for continuous random variables are determined using integral calculus • Both terms give you the same information about the distribution of the random variable – measure of center and spread about the center
Mean and Variance of a RV that is a Linear Function of another RV • Sometimes we may know the mean and variance of a random variable but we may be interested in the behavior of some function of that random variable
For example • Consider randomly selecting customers of a gas company • X: number of gallons provided to customer • We’re looking at 2 pricing models • 1. $2 per gallon • 2. $50 + $1.80 per gallon • Y: amount billed
Continued • For model 1: Y=2X • For model 2: Y=50 + 1.8X • Find
Mean and Variance of a RV that is a Linear Function of another RV • If X is a random variable with mean and variance • Y is a random variable defined as • Y is a linear function of X
To complete our example • For model 1: Y=2X
continuing • For model 2: Y=50 + 1.8X
Rules for Linear Combination of Random Variables • For random variables X and Y
In summary • Rules for Means of Random Variables • Rules for Variances of Random Variables
An example • Gains Communications sells communications devices. The have military and civilian divisions. The military division has the following distribution: • The civilian division has the following distribution:
continued • Find the mean of the random variables X and Y • If Gains Communications makes a $2000 profit on each military unit sold and $3500 profit on each civilian unit sold, what is the mean military profit? Civilian profit? • Define Z as the total profit. Find the expected profit from all sales. • If the standard deviation for military profits is $3M and the standard deviation for civilian profits is $1.2M, what is the standard deviation for total profits?
Summary • What is a Random Variable? • A variable whose value is based on the outcome of a chance experiment • Continuous and discrete random variables • Probability distribution of random variables • Long run behavior of a RV • Table, Histogram, Formula, Properties • Density Curve, Properties • Mean and SD of a random variable • Center and spread of distribution • Rules for Linear Functions and Linear Combinations of Random Variables
Commonly Encountered Distributions • Discrete • Binomial • Geometric • Continuous • Normal • Normal approximation to the binomial
Binomial Distribution • Suppose we are interested in observing the birth of the next 25 babies in a particular hospital. We’re interested in the following probabilities: • Probability that at least 15 are female • Probability that between 10 and 15 are female • Probability that exactly 5 are female • Probability that no more than 6 are female
Characteristics of a Binomial Experiment • (1) A fixed number of observations called trials • We’re interested in the next 25 births • (2) Each trial results in one of only 2 mutually exclusive events • Male or female • We call the event we are interested in a success • (3) The probability that a trial results in a success is the same trial to trial • Probability of a female is .5 – • (4) The outcomes of the trials are independent • The binomial random variable X is the number of successes observed when the experiment is performed
Ways to Determine Probabilities • Binomial Formula • Binomial Tables • TI-83 • Simulation
Binomial Formula • Let’s consider a simpler problem to learn about the binomial formula • Let’s consider the birth of the next 4 babies • Does this meet the 4 properties? • X: number of females born • n: number of trials is 4 • Baby born is either a male or female – mutually exclusive • Trials are independent • Probability of success is .5
The Formula • Let’s consider case X=1 • How many possible outcomes are there in this experiment? • How many where X=1? What are they? • What is ? • Sum of the 4 mutually exclusive events
The Formula continued • Where does the 4 come from? • Number of ways of arranging 1 success among 4 trials • Binomial Coefficient -- nCk
The Binomial Distribution • n: number of independent trials in a binomial experiment • : probability of success in any trial For k = 0, 1, 2, …, n We call n and the parameters of the distribution – B (n, )
Using the Binomial Formula • Determine • Create a probability distribution table
Binomial Probability Table • Appendix X in back of textbook • Enter first with n • Enter column with appropriate • Enter row with appropriate value for X = k • Limitations:
Use the Binomial Table • Suppose we are interested in observing the birth of the next 25 babies in a particular hospital. We’re interested in the following probabilities: • Probability that at least 15 are female • Probability that between 10 and 15 are female • Probability that exactly 5 are female • Probability that no more than 6 are female
TI-83 Calculator • Let’s do the same problem with the calculator • 2nd DISTR • Key down to binompdf or binomcdf • Binompdf (n, p, k) • Binomcdf (n, p, k) • Cumulative Distribution Function • Sum of the probabilities up to the value of X • Determines the probability of obtaining at most X successes
Mean and Variance of the Binomial Distribution • Mean • Variance