CHAPTER 6 Random Variables Discrete and Continuous Random Variables

Discrete and Continuous Random Variables COMPUTE probabilities using the probability distribution of a discrete random variable. CALCULATE and INTERPRET the mean (expected value) of a discrete random variable. CALCULATE and INTERPRET the standard deviation of a discrete random variable. COMPUTE probabilities using the probability distribution of certain continuous random variables.

Random Variables and Probability Distributions A probability model describes the possible outcomes of a chance process and the likelihood that those outcomes will occur. Consider tossing a fair coin 3 times. Define X = the number of heads obtained X = 0: TTT X = 1: HTT THT TTH X = 2: HHT HTH THH X = 3: HHH Value 1 2 3 Probability 1/8 3/8 A random variable takes numerical values that describe the outcomes of some chance process. The probability distribution of a random variable gives its possible values and their probabilities. We could estimate this probability distribution using the simulation methods from ch 5. Interpret P(X > 1) = 7/8 If we repeatedly flip 3 coins and record the number of heads, we will get 1 or more heads 87.5% of the time. Reminder: “distribution” is the same since Ch 1: the possible values a variable can take and how often it takes those values

Discrete Random Variables There are two main types of random variables: discrete and continuous. If we can find a way to list all possible outcomes for a random variable and assign probabilities to each one, we have a discrete random variable. Discrete Random Variables And Their Probability Distributions A discrete random variable X takes a fixed set of possible values with gaps between. The probability distribution of a discrete random variable X lists the values xi and their probabilities pi: Value: x1 x2 x3 … Probability: p1 p2 p3 … The probabilities pi must satisfy two requirements: Every probability pi is a number between 0 and 1. The sum of the probabilities is 1. To find the probability of any event, add the probabilities pi of the particular values xi that make up the event. Set of possible values can be infinite, ex. If X = number of rolls of a fair die needed to get a 6, (no upper limit for values of X) “gaps” illustration: shoe size (discrete) vs foot length (continuous)

Example - How many languages? Imagine selecting a U.S. high school student at random. Define the random variable X = number of languages spoken by the randomly selected student. The table below gives the probability distribution of X, based on a sample of students from the U.S. Census at School database. Problem: (a) Show that the probability distribution for X is legitimate. (b) Make a histogram of the probability distribution. Describe what you see. (c) What is the probability that a randomly selected student speaks at least 3 languages? Languages: 1 2 3 4 5 Probability: 0.630 0.295 0.065 0.008 0.002 In (b) describe what you see, just as you would describe any other distribution SOCS—outliers are far less common so we will spend less time addressing them for probability distributions

Example - How many languages? Show that the probability distribution for X is legitimate. All the probabilities are between 0 and 1 and they add to 1, so this is a legitimate probability distribution. Languages: 1 2 3 4 5 Probability: 0.630 0.295 0.065 0.008 0.002 (b) Make a histogram of the probability distribution. Describe what you see. Shape: The histogram is strongly skewed to the right. Center: The median is 1, but the mean will be slightly higher due to the skewness. Spread: The number of languages varies from 1 to 5, but nearly all of the students speak just one or two languages. (c) What is the probability that a randomly selected student speaks at least 3 languages? P(X > 3) = P(X = 3) + P(X = 4) + P(X = 5) = 0.065 + 0.008 + 0.002 = 0.075. There is a 0.075 probability of randomly selecting a student who speaks three or more languages. How would the probability change if P(X > 3)?

Example - Roulette On an American roulette wheel, there are 38 slots numbered 1 through 36, plus 0 and 00. Half of the slots from 1 to 36 are red; the other half are black. Both the 0 and 00 slots are green. Suppose that a player places a simple $1 bet on red. If the ball lands in a red slot, the player gets the original dollar back, plus an additional dollar for winning the bet. If the ball lands in a different-colored slot, the player loses the dollar bet to the casino. Let’s define the random variable X = net gain from a single $1 bet on red. Determine the probability distribution. Value −$1 $1 Probability 20 38 18 38

Example - Roulette What is the player’s average gain? In the long run, the player loses $1 in 20 of every 38 games and gains $1 in 18 of every 38 games. Imagine a hypothetical 38 bets. The player’s average gain is 𝜇 𝑥 = −1+ −1 +…+ −1 +1+1+…+1 38 𝜇 𝑥 = 20 −1 +18(1) 38 𝜇 𝑥 = 20(−1) 38 + 18 1 38 𝜇 𝑥 =(−1) 20 38 +(1) 18 38 𝜇 𝑥 =−$0.05 Value −$1 $1 Probability 20 38 18 38 This means the player loses (and the casino gains) an average of five cents per $1 bet in many, many plays of the game.

Mean (Expected Value) of a Discrete Random Variable When analyzing discrete random variables, we follow the same strategy we used with quantitative data – describe the shape, center, and spread, and identify any outliers. The mean of any discrete random variable is an average of the possible outcomes, with each outcome weighted by its probability. Suppose that X is a discrete random variable whose probability distribution is Value: x1 x2 x3 … Probability: p1 p2 p3 … To find the mean (expected value) of X, multiply each possible value by its probability, then add all the products: This formula is on the formula sheet

Mean (Expected Value) of a Discrete Random Variable A baby’s Apgar score is the sum of the ratings on each of five scales, which gives a whole-number value from 0 to 10. Let X = Apgar score of a randomly selected newborn Compute the mean of the random variable X. Interpret this value in context. We see that 1 in every 1000 babies would have an Apgar score of 0; 6 in every 1000 babies would have an Apgar score of 1; and so on. So the mean (expected value) of X is: The mean Apgar score of a randomly selected newborn is 8.128. This is the average Apgar score of many, many randomly chosen babies.

Example - How many languages? Recall the random variable X = number of languages spoken by the randomly selected student. Compute the mean of the random variable X and interpret this value in context. 𝜇 𝑋 =𝐸 𝑋 = 1 0.630 + 2 0.295 + 3 0.065 + 4 0.008 + 5 0.002 =1.457 If we were to randomly select many, many U.S. high school students, the average number of languages spoken would be about 1.457. Note that the expected value of a random variable does not necessarily equal one of the possible values of the variable. Do not round to an integer. Languages: 1 2 3 4 5 Probability: 0.630 0.295 0.065 0.008 0.002

Discrete and Continuous Random Variables COMPUTE probabilities using the probability distribution of a discrete random variable. CALCULATE and INTERPRET the mean (expected value) of a discrete random variable. CALCULATE and INTERPRET the standard deviation of a discrete random variable. COMPUTE probabilities using the probability distribution of certain continuous random variables. Read p. 345-352 ccc 1, 3, 5, 7, 9, 11, 13