Chapter 11 Section 1

Chapter 11Section 1 Random Variables

6 1 2 3 4 5 Chapter 11 – Section 1 • Learning objectives • Distinguish between discrete and continuous random variables • Identify discrete probability distributions • Construct probability histograms • Compute and interpret the mean of a discrete random variable • Interpret the mean of a discrete random variable as an expected value • Compute the variance and standard deviation of a discrete random variable

Chapter 11 – Section 1 • A randomvariable is a numeric measure of the outcome of a probability experiment • Random variables reflect measurements that can change as the experiment is repeated • Random variables are denoted with capital letters, typically using X (and Y and Z …) • Values are usually written with lower case letters, typically using x (and y and z ...)

Chapter 11 – Section 1 • Examples • Tossing four coins and counting the number of heads • The number could be 0, 1, 2, 3, or 4 • The number could change when we toss another four coins • Examples • Tossing four coins and counting the number of heads • The number could be 0, 1, 2, 3, or 4 • The number could change when we toss another four coins • Measuring the heights of students • The heights could change from student to student

Chapter 11 – Section 1 • A discreterandomvariable is a random variable that has either a finite or a countable number of values • A finite number of values such as {0, 1, 2, 3, and 4} • A countable number of values such as {1, 2, 3, …} • A discreterandomvariable is a random variable that has either a finite or a countable number of values • A finite number of values such as {0, 1, 2, 3, and 4} • A countable number of values such as {1, 2, 3, …} • Discrete random variables are designed to model discrete variables • Discrete random variables are often “counts of …”

Chapter 11 – Section 1 • An example of a discrete random variable • The number of heads in tossing 3 coins (a finite number of possible values) • An example of a discrete random variable • The number of heads in tossing 3 coins (a finite number of possible values) • There are four possible values – 0 heads, 1 head, 2 heads, and 3 heads • An example of a discrete random variable • The number of heads in tossing 3 coins (a finite number of possible values) • There are four possible values – 0 heads, 1 head, 2 heads, and 3 heads • A finite number of possible values – a discrete random variable • An example of a discrete random variable • The number of heads in tossing 3 coins (a finite number of possible values) • There are four possible values – 0 heads, 1 head, 2 heads, and 3 heads • A finite number of possible values – a discrete random variable • This fits our general concept that discrete random variables are often “counts of …”

Chapter 11 – Section 1 • Other examples of discrete random variables • Other examples of discrete random variables • The possible rolls when rolling a pair of dice • A finite number of possible pairs, ranging from (1,1) to (6,6) • Other examples of discrete random variables • The possible rolls when rolling a pair of dice • A finite number of possible pairs, ranging from (1,1) to (6,6) • The number of pages in statistics textbooks • A countable number of possible values • Other examples of discrete random variables • The possible rolls when rolling a pair of dice • A finite number of possible pairs, ranging from (1,1) to (6,6) • The number of pages in statistics textbooks • A countable number of possible values • The number of visitors to the White House in a day • A countable number of possible values

Chapter 11 – Section 1 • A continuousrandomvariable is a random variable that has an infinite, and more than countable, number of values • The values are any number in an interval • A continuousrandomvariable is a random variable that has an infinite, and more than countable, number of values • The values are any number in an interval • Continuous random variables are designed to model continuous variables (see section 1.1) • Continuous random variables are often “measurements of …”

Chapter 11 – Section 1 • An example of a continuous random variable • The possible temperature in Chicago at noon tomorrow, measured in degrees Fahrenheit • An example of a continuous random variable • The possible temperature in Chicago at noon tomorrow, measured in degrees Fahrenheit • The possible values (assuming that we can measure temperature to great accuracy) are in an interval • An example of a continuous random variable • The possible temperature in Chicago at noon tomorrow, measured in degrees Fahrenheit • The possible values (assuming that we can measure temperature to great accuracy) are in an interval • The interval may be something like (–20,110) • An example of a continuous random variable • The possible temperature in Chicago at noon tomorrow, measured in degrees • The possible values (assuming that we can measure temperature to great accuracy) are in an interval • The interval may be something like (–20,110) • This fits our general concept that continuous random variables are often “measurements of …”

Chapter 11 – Section 1 • Other examples of continuous random variables • Other examples of continuous random variables • The height of a college student • A value in an interval between 3 and 8 feet • Other examples of continuous random variables • The height of a college student • A value in an interval between 3 and 8 feet • The length of a country and western song • A value in an interval between 1 and 15 minutes • Other examples of continuous random variables • The height of a college student • A value in an interval between 3 and 8 feet • The length of a country and western song • A value in an interval between 1 and 15 minutes

Chapter 11 – Section 1 • The probabilitydistribution of a discrete random variable X relates the values of X with their corresponding probabilities • The probabilitydistribution of a discrete random variable X relates the values of X with their corresponding probabilities • A distribution could be • In the form of a table • In the form of a graph • In the form of a mathematical formula

Chapter 11 – Section 1 • If X is a discrete random variable and x is a possible value for X, then we write P(x) as the probability that X is equal to x • If X is a discrete random variable and x is a possible value for X, then we write P(x) as the probability that X is equal to x • Examples • In tossing one coin, if X is the number of heads, then P(0) = 0.5 and P(1) = 0.5 • In rolling one die, if X is the number rolled, thenP(1) = 1/6

Chapter 11 – Section 1 • Properties of P(x) • Since P(x) form a probability distribution, they must satisfy the rules of probability • 0 ≤ P(x) ≤ 1 • ΣP(x) = 1 • In the second rule, the Σ sign means to add up the P(x)’s for all the possible x’s

Chapter 11 – Section 1 • An example of a discrete probability distribution • All of the P(x) values are positive and they add up to 1

Chapter 11 – Section 1 • An example that is not a probability distribution • Two things are wrong • An example that is not a probability distribution • Two things are wrong • P(5) is negative • An example that is not a probability distribution • Two things are wrong • P(5) is negative • The P(x)’s do not add up to 1

Chapter 11 – Section 1 • A probabilityhistogram is a histogram where • The horizontal axis corresponds to the possible values of X (i.e. the x’s) • The vertical axis corresponds to the probabilities for those values (i.e. the P(x)’s) • A probability histogram is very similar to a relative frequency histogram

Chapter 11 – Section 1 • An example of a probability histogram • The histogram is drawn so that the height of the bar is the probability of that value

Chapter 11 – Section 1 • The meanofaprobabilitydistribution can be thought of in this way: • There are various possible values of a discrete random variable • The meanofaprobabilitydistribution can be thought of in this way: • There are various possible values of a discrete random variable • The values that have the higher probabilities are the ones that occur more often • The meanofaprobabilitydistribution can be thought of in this way: • There are various possible values of a discrete random variable • The values that have the higher probabilities are the ones that occur more often • The values that occur more often should have a larger role in calculating the mean • The meanofaprobabilitydistribution can be thought of in this way: • There are various possible values of a discrete random variable • The values that have the higher probabilities are the ones that occur more often • The values that occur more often should have a larger role in calculating the mean • The mean is the weighted average of the values, weighted by the probabilities

Chapter 11 – Section 1 • The mean of a discrete random variable is μX = Σ [ x • P(x) ] • The mean of a discrete random variable is μX = Σ [ x • P(x) ] • In this formula • x are the possible values of X • P(x) is the probability that x occurs • Σ means to add up these terms for all the possible values x

Multiply Multiply Multiply again Multiply again Multiply again Multiply again Multiply again Multiply again Chapter 11 – Section 1 • Example of a calculation for the mean • Example of a calculation for the mean • Example of a calculation for the mean • Example of a calculation for the mean • Add: 0.2 + 1.2 + 0.5 + 0.6 = 2.5 • The mean of this discrete random variable is 2.5

Chapter 11 – Section 1 • The calculation for this problem written out μX = Σ [ x • P(x) ] = [1• 0.2] + [2• 0.6] + [5• 0.1] + [6• 0.1] = 0.2 + 1.2 + 0.5 + 0.6 = 2.5 • The mean of this discrete random variable is 2.5

Chapter 11 – Section 1 • The mean can also be thought of this way (as in the Law of Large Numbers) • The mean can also be thought of this way (as in the Law of Large Numbers) • If we repeat the experiment many times • The mean can also be thought of this way (as in the Law of Large Numbers) • If we repeat the experiment many times • If we record the result each time • The mean can also be thought of this way (as in the Law of Large Numbers) • If we repeat the experiment many times • If we record the result each time • If we calculate the mean of the results (this is just a mean of a group of numbers) • The mean can also be thought of this way (as in the Law of Large Numbers) • If we repeat the experiment many times • If we record the result each time • If we calculate the mean of the results (this is just a mean of a group of numbers) • Then this mean of the results gets closer and closer to the mean of the random variable

Chapter 11 – Section 1 • The expectedvalue of a random variable is another term for its mean • The expectedvalue of a random variable is another term for its mean • The term “expected value” illustrates the long term nature of the experiments – as we perform more and more experiments, the mean of the results of those experiments gets closer to the “expected value” of the random variable

Chapter 11 – Section 1 • The variance of a discrete random variable is computed similarly as for the mean • The variance of a discrete random variable is computed similarly as for the mean • The mean is the weighted sum of the values μX = Σ [ x • P(x) ] • The variance of a discrete random variable is computed similarly as for the mean • The mean is the weighted sum of the values μX = Σ [ x • P(x) ] • The variance is the weighted sum of the squared differences from the mean σX2 = Σ [ (x – μX)2 • P(x) ] • The variance of a discrete random variable is computed similarly as for the mean • The mean is the weighted sum of the values μX = Σ [ x • P(x) ] • The variance is the weighted sum of the squared differences from the mean σX2 = Σ [ (x – μX)2 • P(x) ] • The standard deviation, as we’ve seen before, is the square root of the variance … σX = √ σX2

Chapter 11 – Section 1 • The variance formula σX2 = Σ [ (x – μX)2 • P(x) ] can involve calculations with many decimals or fractions • An equivalent formula is σX2 = [ Σx2 • P(x) ] – μX2 • This formula is often easier to compute

Chapter 11 – Section 1 • For variables and samples, we had the concept of a population variance (for the entire population) and a sample variance (for a sample from that population) • These probability distributions model the complete population • These are population variance formulas • There is no analogy for sample variance here

Chapter 11Section 2 The Binomial Probability Distribution

1 2 3 4 Chapter 11 – Section 2 • Learning objectives • Determine whether a probability experiment is a binomial experiment • Compute probabilities of binomial experiments • Compute the mean and standard deviation of a binomial random variable • Construct binomial probability histograms

Chapter 11 – Section 2 • A binomial experiment has the following structure • A binomial experiment has the following structure • The first test is performed … the result is either a success or a failure • A binomial experiment has the following structure • The first test is performed … the result is either a success or a failure • The second test is performed … the result is either a success or a failure. This result is independent of the first and the chance of success is the same • A binomial experiment has the following structure • The first test is performed … the result is either a success or a failure • The second test is performed … the result is either a success or a failure. This result is independent of the first and the chance of success is the same • A third test is performed … the result is either a success or a failure. The result is independent of the first two and the chance of success is the same

Chapter 11 – Section 2 • Example • A card is drawn from a deck. A “success” is for that card to be a heart … a “failure” is for any other suit • Example • A card is drawn from a deck. A “success” is for that card to be a heart … a “failure” is for any other suit • The card is then put back into the deck • Example • A card is drawn from a deck. A “success” is for that card to be a heart … a “failure” is for any other suit • The card is then put back into the deck • A second card is drawn from the deck with the same definition of success. • Example • A card is drawn from a deck. A “success” is for that card to be a heart … a “failure” is for any other suit • The card is then put back into the deck • A second card is drawn from the deck with the same definition of success. • The second card is put back into the deck • Example • A card is drawn from a deck. A “success” is for that card to be a heart … a “failure” is for any other suit • The card is then put back into the deck • A second card is drawn from the deck with the same definition of success. • The second card is put back into the deck • We continue for 10 cards

Chapter 11 – Section 2 • A binomialexperiment is an experiment with the following characteristics • A binomialexperiment is an experiment with the following characteristics • The experiment is performed a fixed number of times, each time called a trial • A binomialexperiment is an experiment with the following characteristics • The experiment is performed a fixed number of times, each time called a trial • The trials are independent • A binomialexperiment is an experiment with the following characteristics • The experiment is performed a fixed number of times, each time called a trial • The trials are independent • Each trial has two possible outcomes, usually called a success and a failure • A binomialexperiment is an experiment with the following characteristics • The experiment is performed a fixed number of times, each time called a trial • The trials are independent • Each trial has two possible outcomes, usually called a success and a failure • The probability of success is the same for every trial

Chapter 11 – Section 2 • Notation used for binomial distributions • The number of trials is represented by n • The probability of a success is represented by p • The total number of successes in n trials is represented by X • Because there cannot be a negative number of successes, and because there cannot be more than n successes (out of n attempts) 0 ≤ X ≤ n

Chapter 11 – Section 2 • In our card drawing example • Each trial is the experiment of drawing one card • The experiment is performed 10 times, so n = 10 • In our card drawing example • Each trial is the experiment of drawing one card • The experiment is performed 10 times, so n = 10 • The trials are independent because the drawn card is put back into the deck • In our card drawing example • Each trial is the experiment of drawing one card • The experiment is performed 10 times, so n = 10 • The trials are independent because the drawn card is put back into the deck • Each trial has two possible outcomes, a “success” of drawing a heart and a “failure” of drawing anything else • The probability of success is 0.25, the same for every trial, so p = 0.25 • In our card drawing example • Each trial is the experiment of drawing one card • The experiment is performed 10 times, so n = 10 • The trials are independent because the drawn card is put back into the deck • Each trial has two possible outcomes, a “success” of drawing a heart and a “failure” of drawing anything else • The probability of success is 0.25, the same for every trial, so p = 0.25 • X, the number of successes, is between 0 and 10

Chapter 11 – Section 2 • The word “success” does not mean that this is a good outcome or that we want this to be the outcome • A “success” in our card drawing experiment is to draw a heart • If we are counting hearts, then this is the outcome that we are measuring • There is no good or bad meaning to “success”

1 2 3 4 Chapter 11 – Section 2 • Learning objectives • Determine whether a probability experiment is a binomial experiment • Compute probabilities of binomial experiments • Compute the mean and standard deviation of a binomial random variable • Construct binomial probability histograms

Chapter 11 – Section 2 • We would like to calculate the probabilities of X, i.e. P(0), P(1), P(2), …, P(n) • Do a simpler example first • For n = 3 trials • With p = .4 probability of success • Calculate P(2), the probability of 2 successes

Chapter 11 – Section 2 • For 3 trials, the possible ways of getting exactly 2 successes are • S S F • S F S • F S S • For 3 trials, the possible ways of getting exactly 2 successes are • S S F • S F S • F S S • The probabilities for each (using the multiplication rule) are • 0.4 • 0.4 • 0.6 = 0.096 • 0.4 • 0.6 • 0.4 = 0.096 • 0.6 • 0.4 • 0.4 = 0.096

Chapter 11 – Section 2 • The total probability is P(2) = 0.096 + 0.096 + 0.096 = 0.288 • But there is a pattern • Each way had the same probability … the probability of 2 success (0.4 times 0.4) times the probability of 1 failure (0.6) • The probability for each case is 0.42 • 0.61

Chapter 11 – Section 2 • There are 3 cases • S S F could represent choosing a combination of 2 out of 3 … choosing the first and the second • S F S could represent choosing a second combination of 2 out of 3 … choosing the first and the third • F S S could represent choosing a third combination of 2 out of 3 • There are 3 cases • S S F could represent choosing a combination of 2 out of 3 … choosing the first and the second • S F S could represent choosing a second combination of 2 out of 3 … choosing the first and the third • F S S could represent choosing a third combination of 2 out of 3 • These are the 3 = 3C2 ways to choose 2 out of 3

Chapter 11 – Section 2 • Thus the total probability P(2) = .096 + .096 + .096 = .288 can also be written as P(2) = 3C2• .42 • .61 • Thus the total probability P(2) = .096 + .096 + .096 = .288 can also be written as P(2) = 3C2• .42 • .61 • In other words, the probability is • The number of ways of choosing 2 out of 3, times • The probability of 2 successes, times • The probability of 1 failure

Chapter 11 – Section 2 • The general formula for the binomial probabilities is just this • The general formula for the binomial probabilities is just this • For P(x), the probability of x successes, the probability is • The number of ways of choosing x out of n, times • The probability of x successes, times • The probability of n-x failures • The general formula for the binomial probabilities is just this • For P(x), the probability of x successes, the probability is • The number of ways of choosing x out of n, times • The probability of x successes, times • The probability of n-x failures • This formula is P(x) = nCx px (1 – p)n-x

Chapter 11 – Section 2 • Example • A student guesses at random on a multiple choice quiz • There are n = 10 questions in total • There are 5 choices per question so that the probability of success p = 1/5 = .2 • What is the probability that the student gets 6 questions correct?

Chapter 11 – Section 2 • Example continued • This is a binomial experiment • There are a finite number n = 10 of trials • Each trial has two outcomes (a correct guess and an incorrect guess) • The probability of success is independent from trial to trial (every one is a random guess) • The probability of success p = .2 is the same for each trial

Chapter 11 Section 1