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Section 7.1 Discrete and Continuous Random Variables

Section 7.1 Discrete and Continuous Random Variables. AP Statistics NPHS Miss Rose. Some Review:. In order to be a VALID probability distribution, the probabilities of each event must add up to ____. Every probability is a number between __ and ___.

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Section 7.1 Discrete and Continuous Random Variables

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  1. Section 7.1Discrete and Continuous Random Variables AP Statistics NPHS Miss Rose

  2. Some Review: In order to be a VALID probability distribution, the probabilities of each event must add up to ____. Every probability is a number between __ and ___. How many possible outcomes are there for rolling two dice and finding the sum? AP Statistics, Section 7.1, Part 1

  3. Random Variables • A random variable is a variable whose value is a numerical outcome of a random phenomenon. • For example: Flip three coins and let X represent the number of heads. X is a random variable. • We usually use capital letters to denotes random variables.

  4. Random Variables • A random variable is a variable whose value is a numerical outcome of a random phenomenon. • For example: Flip three coins and let X represent the number of heads. X is a random variable. • The sample space S lists the possible values of the random variable X

  5. Discrete Random Variable • A discrete random variable X has a countable number of possible values. • For example: Flip three coins and let X represent the number of heads. X is a discrete random variable. • We can use a table to show the probability distribution of a discrete random variable.

  6. Discrete Probability Distribution Table

  7. Probability Distribution Table: Number of Heads Flipping 4 Coins

  8. Discrete Probability Distributions • Can also be shown using a histogram AP Statistics, Section 7.1, Part 1

  9. AP Statistics, Section 7.1, Part 1

  10. What is… • The probability of at most 2 heads? • P(X≤3) = P(X=0) + P(X=1) + P(X=2) + P(X=3) • = 0.0625 + 0.25 + 0.375 + 0.25 • = 0.9375 • OR you could use the complement rule • P(X≤3) =1- P(4) AP Statistics, Section 7.1, Part 1

  11. Example: Maturation of 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 begin to shave regularly is shown: • Is this a valid probability distribution? How do you know? • What is the random variable of interest? • X= the age in years when male college student begin shaving • Is the random variable discrete? • Yes AP Statistics, Section 7.1, Part 1

  12. Example: Maturation of 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 begin to shave regularly is shown: • What is the most common age at which a randomly selected male college student began shaving? • What is the probability that a randomly selected male college student began shaving at age 16? • P(X=16) = 0.267 • What is the probability that a randomly selected male college student was at least 13 before he started shaving? • P(X≥13) = 1 – P(X=11) = 1 – 0.013 = 0.987 AP Statistics, Section 7.1, Part 1

  13. Continuous Random Variable • A continuous random variableX takes all values in an interval of numbers.

  14. Distribution of Continuous Random Variable

  15. Distribution of Continuous Random Variable • 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. • The probability that X = a particular value is 0

  16. Distribution of a Continuous Random Variable • There are two ways to find the given probability. • (1) Find the area less than 0.5 and greater than 0.8, then add them • (2) Find the area between 0.5 and 0.8 and subtract that answer from 1.

  17. AP Statistics, Section 7.1, Part 1

  18. Density Curves What density curve have you already learned about? The Normal distribution!

  19. Normal distributions as probability distributions • Suppose X has N(μ,σ) then we can use our tools to calculate probabilities. • One tool we will need is our formula for standardizing variables: Z = X – μ σ

  20. Do you remember?--p.141 The annual rate of return on stock indexes is approximately Normal. Since 1945, the Standard & Poors index has had a mean yearly return of 12%, with a standard deviation of 16.5%. In what proportion of years does the index gain 25% or more? AP Statistics, Section 7.1, Part 1

  21. Do you remember? --p.143 The annual rate of return on stock indexes is approximately Normal. Since 1945, the Standard & Poors index has had a mean yearly return of 12%, with a standard deviation of 16.5%. In what proportion of years does the index gain between 15% and 22%? AP Statistics, Section 7.1, Part 1

  22. Cheating in School • A sample survey puts this question to an SRS of 400 undergraduates: “You witness two students cheating on a quiz. Do you do to the professor?” Suppose if we could ask all undergraduates, 12% would answer “Yes” • We will learn in Chapter 9 that the proportion p=0.12 is a population parameterand that the proportion ô of the sample who answer “yes” is a statistic used to estimate p. • We will see in Chapter 9 that ô is a random variable that has approximately the N(0.12, 0.016) distribution. • The mean 0.12 of the distribution is the same as the population parameter. The standard deviation is controlled mainly by the sample size.

  23. Continuous Random Variable • P-hat (proportion of the sample who answered drugs) is a random variable that has approximately the N(0.12, 0.016) distribution. • What is the probability that the poll result differs from the truth about the population by more than two percentage points? (p=0.12) • P(p-hat<0.10 or p-hat>0.14)

  24. = 1 – (0.10 – 0.12 ≤ p-hat-0.12 ≤ 0.14-0.12) • 0.016 0.016 0.016 • = 1 – (-1.25 ≤ Z ≤ 1.25) • = 1 – (0.8944 – 0.1056) • = 1 – (0.7888) • = 0.2112 • P(p-hat<0.10 or p-hat>0.14) • Let’s start with the complement rule • P(p-hat<0.10 or p-hat>0.14) = 1 – (0.10≤p-hat≤0.14)

  25. Check Point Discrete (a) vs. Continuous (b) You flip a coin and count the number of tails you get in 30 trials. You have each student in Stats run a mile and calculate the time in seconds. You count the number of cookies an SRS of 30 students can eat in one minute. AP Statistics, Section 7.1, Part 1

  26. Check Point P-hat (proportion of the sample who answered drugs) is a random variable that has approximately the N(0.12, 0.016) distribution. What is the probability that the poll result is greater than 13%? What is the probability that the poll result is less than 10%? AP Statistics, Section 7.1, Part 1

  27. Random Variables: MEAN • The Michigan Daily Game you pick a 3 digit number and win $500 if your number matches the number drawn. • There are 1000 three-digit numbers, so you have a probability of 1/1000 of winning • Taking X to be the amount of money your ticket pays you, the probability distribution is: Payoff X: $0 $500 Probability: 0.999 0.001 • We want to know your average payoff if you were to buy many tickets. • Why can’t we just find the average of the two outcomes (0+500/2 = $250?

  28. Random Variables: Mean • So…what is the average winnings? (Expected long-run payoff) Payoff X: $0 $500 Probability: 0.999 0.001

  29. Random Variables: Mean The mean of a probability distribution

  30. Random Variables: Example • The Michigan Daily Game you pick a 3 digit number and win $500 if your number matches the number drawn. • You have to pay $1 to play • What is the average PROFIT? • Mean = Expected Value Payoff X: -$1 $499 Probability: 0.999 0.001

  31. Random Variables: Variance (the average of the squared deviation from the mean) The standard deviationσ of X is the square root of the variance

  32. Random Variables: Example • The Michigan Daily Game you pick a 3 digit number and win $500 if your number matches the number drawn. • The probability of winning is .001 • What is the variance and standard deviation of X?

  33. Technology • When you work with a larger data set, it may be a good idea to use your calculator to calculate the standard deviation and mean. • Enter the X values into List1 and the probabilities into List 2. Then 1-Var Stats L1, L2 will give you μx (as x-bar) and σx (to find the variance, you will have to square σx) • EX: find μx and σ2x for the data in example 7.7 (p.485)

  34. Exercise… • Pretend to “simulate” tossing a fair coin 10 times and write down the results. AP Statistics, Section 7.2, Part 1

  35. Law of Large Numbers • Draw independent observations at random from any population with finite mean μ. • Decide how accurately you would like to estimate μ. • As the number of observations drawn increases, the mean x-bar of the observed values eventually approaches the mean μ of the population as closely as you specified and then stays that close.

  36. Example • The distribution of the heights of all young women is close to the normal distribution with mean 64.5 inches and standard deviation 2.5 inches. • What happens if you make larger and larger samples…

  37. Law of Small Numbers • Most people incorrectly believe in the law of small numbers. • “Runs” of numbers, etc.

  38. Assignment: Exercises: 7.3, 7.7, 7.9, 7.12-7.15, 7.24, 7.27, 7.32-7.34

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