Discrete Random Variables and Their Probability Distributions

Transcription

1 Chapter 5 Discrete Random Variables and Their Probability Distributions Mean and Standard Deviation of a Discrete Random Variable Computing the mean and standard deviation of a discrete random variable is slightly different than computing the mean and standard deviation of a set of data values. In a set of data values, each number in the data set weighs equally in the computation. However, in a discrete random variable, the possible data values are given along with the likelihood of each value occurring on any given single trial. As was the case for a set of data values, the TI-84 calculator can be used to calculate the mean and standard deviation of a discrete random variable by either manually using the formulas or by using a built-in function. We will begin with manually using the formulas. Example: Number of Breakdowns Per Week Example 5-3 gives the probability distribution of the number of breakdowns per week for a machine based on past data. Enter the number of breakdowns into a list named X, and the probability into a list named PROBX. Mean of a Discrete Random Variable The formula to calculate the mean of a discrete probability distribution is. Move the cursor to highlight the name of the empty list next to PROBX.

2 ( ) and select X. ( ) and select PROBX. µ = 1.8 breakdowns per week. Now find the sum of this list of values. From the home screen, ( ). Select 5: sum( from the MATH menu. Standard Deviation of a Discrete Random Variable The formula to calculate the standard deviation of a discrete probability distribution is Move the cursor to highlight the name of the empty list next to PROBX. ( ) and select X. ( ) and select PROBX. Now find the square root of the difference of the sum of this list of values and the square of the mean. From the home screen, ( ). ( ). Select 5: sum( from the MATH menu. and then press. σ = 1.03 breakdowns per week. 2

3 Using TI-84 Plus Built-In Functions For Discrete Probability Distributions The TI-84 Plus function 1-Var Stats will also calculate the numerical descriptive statistics for a Discrete Probability Distribution. We will use the same probability distribution as above, which we stored in Lists X and PROBX. Select 1: 1-Var Stats from the CALC menu. At the List: prompt, press ( ) and select list X. At the FreqList: prompt, press ( ) and select list PROBX. to highlight Calculate and then press. The screen will display the descriptive statistics, which includes the population mean and standard deviation. Binomial Distribution There are many situations in statistics where you need to generate probabilities from distributions where the numbers are not equally likely to occur. One of the most commonly used distributions used in statistics is the Binomial distribution. Compute Binomial Probabilities The command for computing a probability of x successes for a Binomial distribution is binompdf(. Example: DVD s Suppose that 5% of all DVD players manufactured by an electronics company are defective. Three DVD players are selected at random. What is the probability that exactly one of them is defective? That is, what is P(x = 1)? 3

4 ( ). Select A: binompdf(from the DISTR menu. At the trials: prompt, type 3. At the p: prompt, type At the x value: prompt, type 1. The calculator will translate the information you enter into the command binompdf(n, p, x), where x is the number successes out of n trials, each with probability p of success. The correct probability of 1 success in 3 trials, when each trial has a 0.05 probability of success, is Compute Cumulative Binomial Probabilities The command for the probability for a cumulative number of successes from 0 to x for a discrete Binomial distribution with n number of trials and p probability of success on any given single trial is binomcdf(. In other words, this function calculates P(number of successes x). Using the same Binomial distribution of 3 DVD players as above, what is the probability that zero or one of them is defective? That is, find P(x 1). ( ). Select B: binomcdf( from the DISTR menu. At the trials: prompt, type 3. At the p: prompt, type At the x value: prompt, type 1. The result is or 99.3% chance that one or less of them is defective. Randomly Generating Number of Successes From a Binomial Distribution The TI-84 Plus has a built-in function to generate random numbers from a specific Binomial distribution. The random number represents an x value, the number of successes. 4

5 Select 7:randBin( from the PRB menu. The screen shot above shows the results when the Binomial experiment is repeated 5 times. Each time there were 3 trials and the numbers of successes were 1, 2, 1, 0, and 1 respectively. The syntax for the randbin( function is randbin(n, p, r). This will generate r random numbers representing x the number of successes from a binomial distribution with n number of trials and p probability of success on a given trial. Note: if r = 1, you may omit it. Hypergeometric Distribution Compute Hypergeometric Probabilities The TI-84 does not have a built-in function to calculate hypergeometric probabilities. To compute these probabilities, you will have to use the combinations function described previously to evaluate the hypergeometric probability formula. Example: Auto parts An auto parts manufacturer produced 25 parts, 5 of which were defective. If the manufacturer shipped 4 parts to a dealer, what is the probability that exactly 3 of them were good parts? To use the hypergeometric formula, notice that the population size N is 25, the number of successes r is 20, the number of failures N r is 5, the number of trials n is 4, the number of successes in four trials x is 3, and the number of failures in four trials x is 1. and select 3: ncr from the PRB menu. 5

6 and select 3: ncr from the PRB menu. and select 3: ncr from the PRB menu. The probability that the dealer receives exactly 3 good parts in the shipment of 4 parts is Compute Poisson Probabilities Poisson Distribution The command for computing the probability of x occurrences within a given interval for a discrete Poisson distribution with a mean number of occurrences λ is poissonpdf(λ, x). In other words, this function calculates P(number of successes = x). Example: Telemarketing Suppose that a household receives, on average, 9.5 telemarketing calls per week. Find the probability that the household receives 6 calls this week. ( ). Select C: poissonpdf( from the DISTR menu. At the λ: prompt, type 9.5. At the x value: prompt, type 6. The result is % chance that the household receives 6 calls this week. Find the probability that the household receives 10 calls this week. There is 12.4% chance that the household receives 10 calls this week. 6

7 Compute Cumulative Poisson Probabilities The command for computing the probability of at most x occurrences (cumulative) within a given interval for a discrete Poisson distribution with a mean number of occurrences λ is poissoncdf(λ, x). In other words, this function calculates P(number of successes x). Using the same Poisson distribution of Telemarketing calls as above, what is the probability that the household receives at most 6 calls this week? That is, find P(x 6). ( ). Select C: poissonpdf( from the DISTR menu. At the λ: prompt, type 9.5. At the x value: prompt, type 6. The probability that the household receives 6 or fewer calls this week is , or 16.5%. 7