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Discrete Probability Distributions: Random Variables and Binomial Distributions

Learn about random variables and probability distributions for discrete random variables, including binomial random variables.

jamesethompson
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Discrete Probability Distributions: Random Variables and Binomial Distributions

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  1. Chapter 5 Discrete Probability Distributions

  2. 5.1 Random Variables • Random Variable • A random variable is a function that assigns a numerical value to each outcome of an experiment. • Example 5.1.1: We flip a coin two times and are interested in the number of heads. • The outcomes are HH, HT, TH and TT. • Let x be the “total number of heads observed”. Then x = 0, 1, or 2. Therefore, the above outcomes can be assigned these numbers as follows. • The variable “total number of heads observed” in this experiment is called a random variable.

  3. 5.1 Random Variables (cont.) • Discrete Random Variable • A random variable that can assume a finite number of possible values is called a discrete random variable. • That is, the possible values of a discrete random variable can be listed or counted. • Example 5.1.2: X is a discrete random variable if X represents the number of people out of 200 who will make an airline reservation and then fail to show up. • Example 5.1.3: You roll two dice, a red die and a blue die. Suppose X is the total of the two dice. X is a discrete random variable because X has finite number of possible values.

  4. 5.1 Random Variables (cont.) • Continuous Random Variable • A random variable that can take any value over some continuous range of values is called a continuous random variable. • Example 5.1.4: Let Y be the amount of rainfall during the month of September. Y is a continuous random variable because the amount of rainfall can be any nonnegative value.

  5. 5.1 Random Variables (cont.) • Probability Distribution • A probability distribution is the list of all possible outcomes of a random variable and their associated probabilities. • Example 5.1.5: Toss three fair coins and let X equal the number of tails observed. • The outcomes are HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. Each outcome has equal probability. • X = 0, 1, 2, and 3. • P(X) is between 0 and 1 (inclusive) • ∑P(X) = 1

  6. 5.2 Probability Distributions for Discrete Random Variables • Three popular methods of describing probabilities associated with a discrete random variable. • List each value of X and its corresponding probability. • Use a histogram to convey the probabilities corresponding to the various values of X. • Use a function that assigns a probability to each value of X.

  7. 3/8 – 2/8 – 1/8 – – Probability 0 1 2 3 x = number of heads 5.2 Probability Distributions for Discrete Random Variables (cont.) • List each value of X and its corresponding probability. Used in Example 5.1.5. • Use a histogram to convey the probabilities corresponding to the various values of X.

  8. 5.2 Probability Distributions for Discrete Random Variables (cont.) • Probability Mass Function (PMF) • A PMF is a function that assigns a probability to each value of X. • P(X = x) =some expression (usually containing x) that produces a probability of observing x = P(x). • P(x) is between 0 and 1 (inclusive) for each x • ∑P(x) = 1 • Example 5.2.1: The function P(X=x) = x/30 for x = 0, 10, and 20 (and zero elsewhere) is a probability mass function because: • P(X=x) = x/30 assigns a probability to each value of x. • ∑P(X=x) = 1

  9. Probability Mass Function (PMF) (cont.) Example 5.2.2: A manager has four employees with 0, 1, 3, and 4 years of job experience. The manager will assign two of the employees at random to a team. Define X to be equal to the total number of years of job experience for the two selected employees. What values can X assume? X = {0+1, 0+3, 0+4 or 1+3, 1+4, 3+4} = {1, 3, 4, 5, 7} What is the probability mass function of X? P(X=i) = 1/6, where i = 1, 3, 5, and 7; and P(X=j) = 2/6, where j = 4. 5.2 Probability Distributions for Discrete Random Variables (cont.)

  10. 5.2 Probability Distributions for Discrete Random Variables (cont.) • Mean of Discrete Random Variables • The mean of a discrete random variable represents the average value of the random variable if you were to observe this variable over an indefinite period of time. • The mean of a discrete random variable is written as µ and µ = ∑xP(x). • Example 5.2.2: Toss three fair coins and let X equal the number of tails observed. Find the mean number of tails observed.

  11. 5.2 Probability Distributions for Discrete Random Variables (cont.) • Variance of Discrete Random Variables • The variance of a discrete random variable, X, is a parameter describing the variation of the corresponding population. • The symbol used is 2. • 2 = ∑(x - µ)2 • P(x) = ∑x2P(x) - µ2 • Example 5.2.3: Toss three fair coins and let X equal the number of tails observed. Find the variance for the number of tails observed.

  12. 5.3 Binomial Random Variable • Binomial Random Variable • A discrete random variable that can assume one of two possible outcomes in each trial of an experiment comprising of n independent trials. • An experiment in which each trial results in one of two mutually exclusive outcomes is called a binomial experiment. • Example 5.3.1: Flip a coin five times. Let X be the number of heads. • A binomial experiment with 5 trials and each trial has two possible outcomes – a head and a tail. Trials (flips) are independent. • X is a binomial random variable.

  13. 5.3 Binomial Random Variable (cont.) • Characteristics of a binomial experiment • The experiment consists of n repetitions, called trials. • Each trial has two mutually exclusive possible outcomes, referred to as success and failure. • The n trials are independent. • The probability for a success for each trial is denoted p; and remains the same for each trial. • The random variable x is the number of successes out of n trials. • Example 5.3.2: Flip a coin five times. Let X be the number of heads. Is it a binomial situation? • n = 5. • Success = head, failure = tail. • The results on one flip do not affect the results on another flip. • p = the probability of flipping a head on a particular flip = ½. • X = the number of heads out of five flips.

  14. 5.3 Binomial Random Variable (cont.) • Binomial Distribution • The probability distribution of a binomial random variable is called a binomial distribution. • Probability Mass Function (PMF) of a binomial random variable is: • Example 5.3.3: A lawyer estimates that 40% of the cases in which she represented the defendant were won. If the lawyer is presently representing 10 defendants, what is the probability that 5 of the cases will be won? • n = 10; x = 5; p = 0.4

  15. 5.3 Binomial Random Variable (cont.) • Using Binomial Table A.1 to Determine Probabilities • The binomial PMFs have been tabulated in Table A.1 for various values of n and p. • If n = 4 and p = 0.3 and you wish to find the P(2) locate n = 4 and x = 2. • Go across to p = 0.3 and you will find the corresponding probability (after inserting the decimal in front of the number). This probability is 0.265.

  16. .3 – .2 – .1 – .3 – .2 – .1 – Probability Probability x x 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 (n = 10, p = .5) (n = 10, p = .8) .3 – .2 – .1 – .3 – .2 – .1 – (a) (b) Probability Probability x x 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 20 (n = 10, p = .2) (n = 20, p = .2) (c) (d) 5.3 Binomial Random Variable (cont.) • Shape of the Binomial Distribution • Approximately bell-shaped (symmetric) if p is near ½ or if n is large. • Skewed left for p > ½ and small n. • Skewed right for p < ½ and small n.

  17. 5.3 Binomial Random Variable (cont.) • Cumulative Binomial Probabilities • Finding P(X ≤ k), that is, finding the total probability of all successes up to and including k. • P(X ≤ k) = P(X = 0) + P(X = 1) + P(X = 2) + ….. + P(X = k). • P(X < k) = P(X = 0) + P(X = 1) + P(X = 2) + ….. + P(X = k - 1). • P(X > k) = 1 - P(X ≤ k). • P(X ≥ k) = 1 - P(X < k). • Using Binomial Table A.2 • If n = 4 and p = 0.3 and you wish to find the P(x ≤ 2) locate n = 4 and x = 2. • Go across to p = 0.3 and you will find the corresponding probability (after inserting the decimal in front of the number). This probability is 0.916.

  18. 5.3 Binomial Random Variable (cont.) • Example 5.3.4: Suppose that 60% of all the employees of ABC Company favor unionization. A poll of 20 employees is taken to determine the number who favor unionization. • Find the probability that at least 10 employees favor unionization. • p = 0.6, n = 20, P(x ≤ 10) = 0.245 [Using Table A.2] • Find the probability that more than 12 employees favor unionization. • P(x > 12) = 1 – P(x ≤ 12) = 1 – 0.584 (From Table A.2) = 0.416

  19. 5.3 Binomial Random Variable (cont.) • Mean and Variance of a Binomial Random Variable • µ = np • 2 = np(1 - p) • Example 5.3.5: Suppose that 60% of all the employees of ABC Company favor unionization. A poll of 20 employees is taken to determine the number who favor unionization. Find the mean and standard deviation of those who favor unionization. • p = 0.6, n = 20, µ = np = 20(0.6) = 12. • 2 = np(1 - p) = 20(0.6)(1 – 0.6) = 4.8,  = sqrt(4.8) = 2.19.

  20. 5.3 Binomial Random Variable (cont.) • Finding Probabilities with Excel • P(X ≤ 1) = P(X = 0) + P(X = 1). Assume n = 100 and p = 0.05. • P(X = 0) is shown below.

  21. 5.3 Binomial Random Variable (cont.) • Finding Probabilities with Excel • P(X = 1) is shown below.

  22. 5.3 Binomial Random Variable (cont.) • Finding Probabilities with Excel • P(X ≤ 1) is shown below.

  23. 5.4 Poisson Distribution • The Poisson distribution is useful for counting the number of times a particular event occurs over a specified period of time or over a specified area. • For example, arrivals of customers at a service facility follow Poisson Distribution. • Conditions for the Poisson Distribution: • The number of occurrences in one measurement unit are independent of the number of occurrences in any other nonoverlapping measurement unit. • The expected number of occurrences in any given measurement unit are proportional to the size of the measurement unit. • Events can not occur at exactly the same point in the measurement unit.

  24. 5.4 Poisson Distribution (cont.) • PMF of Poisson Distribution: • Mean and Variance of a Poisson Random Variable • Mean of X = ∑xP(x) = µ = expected number of occurrences. • Variance of X = 2 = ∑x2P(x) - µ2 = µ = expected number of occurrences. • Example 5.4.1: A large bakery determined that the expected number of delivery truck breakdowns per day is 1.5. Assume that the number of breakdowns is independent from day to day. • What is the probability that there will be exactly two breakdowns tomorrow? • µ = 1.5, P(X = 2) = 0.251. • What is the probability that there will be exactly two breakdowns during next two days? • µ = 2(1.5) = 3, P(X = 2) = 0.224.

  25. 5.4 Poisson Distribution (cont.) • Finding Poisson Probabilities with Statistical Software

  26. 5.4 Poisson Distribution (cont.) • Finding Poisson Probabilities with Statistical Software

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