Chapter 7 Random Variables 7.1: Discrete and Continuous Random Variables

Random Variables A random variable is a variable whose value is a numerical outcome of a random phenomenon. the basic units of sampling distributions. 2 types: discrete and continuous

Discrete Random Variables A discrete random variable X has a countable number of possible values. The probability distribution of a discrete random variable X lists the values and their probabilities. Value of X: x1 x2 x3 … xk Probability: p1 p2 p3 pk

Probability Distribution The probability pi must satisfy two requirements. Every probability pi is an number between 0 and 1 The sum of the probabilities is 1: p1 + p2 + p3 +…+pk = 1 Find the probability of any event by adding the probabilities pi of the particular values xi that make up the event.

Example: Maturation of male college students In an article in the journal Developmental Psychology (March 1986), a probability distribution for the age X (in years) when male college students began to shave regularly is shown:

X 11 12 13 14 15 P(X) 0.013 0.027 0.067 0.213 X 16 17 18 19 ≥20 P(X) 0.267 0.240 0.093 0.067 0.013 Observations we can make: This is a valid probability distribution. The sum of the probabilities is 1. The random variable of interest is X = the age (in years) when a randomly selected male college student began to shave regularly. The random variable X is discrete. From the histogram or table we can see that The most common age at which a randomly selected male student began shaving is 16. The probability that a randomly selected male college student began shaving at age 16 is 0.267. The probability that a randomly selected male college student began shaving before 15 is P(x<15) = 0.013+0+0.027+0.067=0.107

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Continuous Random Variables A continuous random variable X takes on all values in an interval of numbers. The probability distribution of X is described by a density curve. The probability of any event is the area under the density curve and above the values of X that make up that event. A density curve is a nonnegative function that has area exactly 1 between it and the horizontal axis.

All continuous probability distributions assign probability 0 to every individual outcome.

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Example: Drugs in schools An opinion poll asks a SRS of 1500 American adults what they consider to be the most serious problem facing our schools. Suppose that if we could ask all adults this question, 30% would say “drugs”. What is the probability that the poll result differs from the truth about the population by more than two percentage points? N(.3, 0.0118) We want P-hat< .28 or p-hat >.32 Standardize to find z-scores and the area under the curve. 0.0455 + 0.0455 = 0.0910

Chapter 7 Random Variables 7.2: Means and Variances of Random Variables

Activity 7B Page 481

Mean and expected Value Mean of a probability distribution is denoted by µ, or µx. The mean of the random variable, X is often referred to as the expected value of X.

Mean of a Discrete Random Variable Suppose that X is a discrete random variable whose distribution is To find the mean of X, multiply each possible value by its probability, then add all the products. Value of X: x1 x2 x3 … xk Probability: p1 p2 p3 pk

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Example Using the data from the “Maturation of male college students” example, find and interpret the mean. Mean is 16.186. This is the expected age at which male college students begin to shave regularly.

Variance of a Discrete Random Variable Suppose that X is a discrete random variable whose distribution is and that µ is the mean of X. The variance of X is σx2 = Σ(x1 - µx)2pi. The standard deviation σx of X is the square root of the variance. Value of X: x1 x2 x3 … xk Probability: p1 p2 p3 pk

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Technology Tip To find µx and σx:

Example Using the data from the “Maturation of male college students” example, find the standard deviation.

Use the empirical rule to determine if the “Maturation of male college students” data is normally distributed.

Sampling Distributions The sampling distributions of statistics are just the probability distributions of these random variables.

Law of Large Numbers The average of a randomly selected sample from a large population is likely to be close to the average of the whole population. http://bcs.whfreeman.com/tps3e/default.asp?s=&n=&i=&v=&o=&ns=0&uid=0&rau=0

Law of Large Numbers What is the mean of rolling 3 dice?

Example: Emergency Evacuations A panel of meteorological and civil engineers studying emergency evacuation plans for Florida’s Gulf Coast in the event of a hurricane has estimated that it take between 13 and 18 hours to evacuate people living in low-lying land, with the probabilities shown in the table. Let X = the time it takes a randomly selected person living in low-lying land in Florida to evacuate.

Time to Evacuate Probability (nearest hour) 13 0.04 14 0.25 15 0.40 16 0.18 17 0.10 18 0.03

Is X a discrete random variable or a continuous random variable? Sketch a probability histogram for this data. Find and interpret the mean. Find the standard deviation.

Weather forecasters say that they cannot accurately predict a hurricane landfall more than 14 hours in advance. Find the probability that all residents of low-lying areas are evacuated safely if the Gulf Coast Civil Engineering Department waits until the 14-hour warning before beginning evacuation.

Law of Small Numbers Gambler’s Fallacy is the belief that every segment of a random sequence should reflect the true proportion. This is a myth. There is no law of small numbers!

Rules for Means Rule 1: If X is a random variable and a and b are fixed numbers, then μa+bx = a + bμx Rule 2: If X and Y are random variables, then μX + Y = μX + μY

Rules for Variances Rule 1: If X is a random variable and a and b are fixed numbers, then σ2a+bx = b2σ2x Rule 2: If X and Y are independent random variables, then σ2X + Y = σ2x + σ2Y σ2X - Y = σ2x + σ2Y Rule 2 is also called the addition rule for variances of independent random variables.

Means and Variances Variances add, standard deviations don’t. These rules can extend for more than 2 random variables…just follow the pattern. Page 498 Example 7.14 PRB Extra Example #7- simulation problem

Combining Normal Random Variables Any linear combination of independent Normal random variables is also Normally distributed.

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