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Chapter 4

Chapter 4. Discrete Random Variables. Two Types of Random Variables. Random Variable: variable that assumes numerical values associated with random outcomes of an experiment only one numerical value is assigned to each sample point Two types of Random Variable: Discrete Continuous.

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Chapter 4

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  1. Chapter 4 Discrete Random Variables

  2. Two Types of Random Variables • Random Variable: • variable that assumes numerical values associated with random outcomes of an experiment • only one numerical value is assigned to each sample point • Two types of Random Variable: • Discrete • Continuous

  3. Two Types of Random Variables • Discrete Random Variable: • Random variable that has a finite, or countable number of distinct possible values • Example: number of people born in July • Continuous Random Variable: • Random variable that has an infinite number of distinct possible values • Example: average age of people born in July

  4. Probability Distributions for Discrete Random Variables • The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable. • We can describe a discrete probability distribution with a table, graph, or equation.

  5. Probability Distributions for Discrete Random Variables • Two Basic Properties of a Discrete Probability Distribution: where the summation of p(x) is over all possible values of x. for all values of x 1. 2.

  6. Probability Distributions for Discrete Random Variables: Example 1 • Experiment: tossing 2 coins simultaneously • Random variable X represents the number of heads observed

  7. Probability Distributions for Discrete Random Variables: Example 1 (continued) • Probability Distribution of Discrete Random Variable X

  8. Expected Values of Discrete Random Variables • The mean, or expected value of a discrete random variable is:

  9. Expected Values of Discrete Random Variables • The variance of a discrete random variable is:  2 = E[(x - )2]=(x - )2p(x)=x 2 p(x) -  2 and standard deviation is

  10. Probability Distributions for Discrete Random Variables: Example • A dust mite allergen level that exceeds 2 micrograms per gram (mg/g) of dust has been associated with the development of allergies. Consider a random sample of four homes and let x be the number of homes with a dust mite level that exceeds 2 mg/g. The probability distribution for x is shown in the following table:

  11. Probability Distributions for Discrete Random Variables: Example (continued) • Is it a valid probability distribution? • Find the probability that three or four of the homes in the sample have a dust mite level that exceeds 2 mg/g. • Find the probability that fewer than two homes in the sample have a dust mite level that exceeds 2 mg/g. • Find E(x). Give a meaningful interpretation of the result. • Find .

  12. Expected Values of Discrete Random Variables • Probability Rules for a Discrete Random Variable

  13. The Binomial Random Variable • Characteristics of a Binomial Random variable: • An experiment consist of n identical trials. • There are two possible outcomes on each trial, denoted as S (success) and F (failure). • Probability of success (p) is constant from trial to trial. Probability of failure (q)=1-p. • Trials are independent. • The binomial random variable is the number of S’s in n trials.

  14. The Binomial Random Variable • Heart association claims that only 10% of US adults over 30 can pass the President’s Physical Fitness commission’s minimum requirements. • Select 4 adults at random, administer the test. What is the probability that none of the adults passes the test? • S = pass, F = Fail, p(S)=.10, q=1-p=1-.10=.90

  15. The Binomial Random Variable • Use multiplicative rule to calculate probabilities of the possible outcomes: P(SSSS) = .1*.1*.1*.1=.14=.0001 P(FSSS) = .9*.1*.1*.1=.9*.13=.0009 ….. P(FFFF) = .9*.9*.9*.9=.94=.6561

  16. The Binomial Random Variable • What is the probability that 3 of the 4 adults pass the test? P(3 of the 4 adults pass the test) = 4(.1)3(.9)=4(.09) = .0036 • What is the probability that 3 of the 4 adults fail the test? P(3 of the 4 adults fail the test) = 4(.9)3(.1)=4(.0729) = .2916 • Do you see a pattern?

  17. The Binomial Random Variable • Formula for the Binomial Probability Distribution p(x): where p = probability of success on single trial q = 1-p n = number of trials x = number of successes in n trials

  18. The Binomial Random Variable • Mean: • Variance: • Standard deviation

  19. Binomial Tables • Binomial tables are cumulative tables, entries represent cumulative binomial probabilities • Make use of additive and complementary properties to calculate probabilities of individual x’s, or x being greater than a particular value.

  20. Binomial Tables • If x < 2, and p =.1, n =10, then P(x<2) =.930 • If x = 2, and p =.1, n =10, then P(x=2) = P(x<2) - P(x<1)=.930-.736 = .194 • If x >2, and p = .1, n =10, then P(x>2) = 1- P(x<2) =1-.930 = .322

  21. The Binomial Random Variable:Example • A national poll conducted by The New-York Times (May 7, 2000) revealed that 80% of Americans believe that after you die, some part of you leaves on, either in a next life on earth or in heaven. Consider a random sample of 10 Americans.

  22. The Binomial Random Variable:Example (continued) • What is a success in terms of the problem? • What is the probability of the success? • What is the probability of observing three Americans who believe in life after death in a sample of 10? • What is the probability of observing more than four Americans who believe in life after death in a sample of 10?

  23. The Binomial Random Variable:Example (continued) • What is the probability of observing at most 7 Americans who believe in life after death in a sample of 10? • What is the probability of observing at least 1 American who believe in life after death in a sample of 10? • In a random sample of 10 Americans, how many would you expect to believe in life after death?

  24. Poisson Probability Distribution A discrete random variable following this distribution is often useful in estimating the number of occurrences over a specified interval of time or space. It is a discrete random variable that may assume an infinite sequence of values (x = 0, 1, 2, . . . ).

  25. Poisson Probability Distribution Examples: the number of knotholes in 14 linear feet of pine board the number of vehicles arriving at a toll booth in one hour

  26. Poisson Probability Distribution Properties of a Poisson Experiment: The probability of an occurrence is the same for any two intervals of equal length. The occurrence or nonoccurrence in any interval is independent of the occurrence or nonoccurrence in any other interval.

  27. Poisson Probability Distribution The Poisson probability distribution is given by the formula where x is the number of occurrences in an interval, is the mean number of occurrences in an interval, e is the base of the natural logarithm (e = 2.7182)

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