advertisement

Discrete Distributions Chapter 5 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 1 Random Variables Random variable a variable (typically represented by x) that takes a numerical value by chance. For each outcome of a procedure, x takes a certain value, but for different outcomes that value may be different. pg. 205 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 2 Examples: Number of boys in a randomly selected family with three children. Possible values: x=0,1,2,3 The weight of a randomly selected person from a population. Possible values: positive numbers, x>0 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 3 Examples: Genetics: If two pea plants have both green and yellow pod genes, the probability that offspring pea plants has a green pod is .75 That is P(green) = .75 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 4 Discrete and Continuous Random Variables Discrete random variable Either a finite number of values or countable number of values (resulting from a counting process) Continuous random variable Infinitely many values, and those values can be associated with measurements on a continuous scale without gaps or interruptions pg. 206 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 5 Probability Distributions Probability distribution A description that gives the probability for each value of the random variable; often expressed in the format of a table, graph, or formula Pea pod histogram on page 207 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 6 Tables Values: Probabilities: x P(x) 0 1/8 1 3/8 2 3/8 3 1/8 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 7 Graphs The probability histogram is very similar to a relative frequency histogram, but the vertical scale shows probabilities. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 8 Requirements for Probability Distribution ∑ P(x) = 1 where x assumes all possible values. 0 P(x) 1 for every individual value of x. pg. 207 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 9 Mean, Variance and Standard Deviation of a Probability Distribution µ = ∑ [x • P(x)] Mean σ = ∑ [(x – µ) • P(x)] Variance σ = ∑ [x • P(x)] – µ Variance (shortcut) 2 2 2 σ= 2 2 ∑[x 2 • P(x)] – µ 2 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Standard Deviation 10 Using TI-83/84 calculator • Press the STAT button and choose EDIT • Enter the x-values into the list L1 and the P(x) values into the list L2 • Press the STAT button and choose CALC • Choose 1-Var Stats and press ENTER • Type in L1 then , (comma) then L2 on that line, you will see 1-Var Stats L1,L2 • Press ENTER • You will see x-bar=…, it is actually m (mean) and sx=…, it is actually s (st. deviation) Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 11 Finding the Mean, Variance, and Standard Deviation Example 5, page 209 Table 5-1 describes the probability distribution for the number of peas with green pods among five offspring peas obtained from parents both having the green/yellow pair of genes. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 12 Using Excel - Mean • • • • Put the x-values in column 1 Put the P(x)-values in column 2 In column 3 enter “=A1*B1” Copy and paste that cell to the entire column of data • At the bottom of column 3 enter “=sum(c1:cN)” - N is the last row of the data Table 5-3, pg 209 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 13 Using Excel - Mean Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 14 Using Excel – Standard Deviation Continuing • In column 4 enter “=power(A1-$c$M,2)*B1” • Copy and paste that cell to the entire column of data • At the bottom of column 4 enter “=sum(d1:dN)” - N is the last row of the data Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 15 Using Excel – Standard Deviation Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 16 Using Excel to find the mean and standard deviation of a discrete probability distribution More detailed example Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 17 Using Excel to find the mean and standard deviation of a discrete probability distribution (1) Enter the values and their probabilities as separate columns Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 18 Using Excel (continued) (2) Check to make sure the values of P(x) add up to 1: Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 19 Using Excel (continued) (3) In column C, for each value of x, calculate x*P(x): Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 20 Using Excel (continued) (4) The sum of column C will give the value of the mean, μ Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 21 Using Excel (continued) (5) In column D, calculate the values of X2*P(x) Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 22 Using Excel (continued) (6) Calculate the sum of the squares times probability - X2*P(x) Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 23 Using Excel (continued) (7) Use the shortcut formula for the variance: Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 24 Using Excel (continued) (8) Finally, the Standard deviation is just the square root of the variance: Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 25 How to Choose Lottery Numbers • Note the sidebar on page 209 about choosing numbers in a lottery. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 26 Roundoff Rule for 2 µ, σ, and σ Round results by carrying one more decimal place than the number of decimal places used for the random variable x. If the values of x are integers, round µ, σ, and σ2 to one decimal place. Pg 210 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 27 Identifying Unusual Results Range Rule of Thumb According to the range rule of thumb, most values should lie within 2 standard deviations of the mean. We can therefore identify “unusual” values by determining if they lie outside these limits: Maximum usual value = μ + 2σ Minimum usual value = μ – 2σ Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 28 Identifying Unusual Results By Probabilities Using Probabilities to Determine When Results Are Unusual: Unusually high: a particular value x is unusually high if P(x or more) ≤ 0.05. Unusually low: a particular value x is unusually low if P(x or fewer) ≤ 0.05. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 29 Uniform Probability Distribution A uniform probability distribution has the following property: P(x) = c The value c is a constant, so every event is just a likely as every other event. If there are n events P(x) = 1 n Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 30 Binomial Probability Distribution A binomial probability distribution results from a procedure that meets all the following requirements: 1. The procedure has a fixed number of trials. 2. The trials must be independent. (The outcome of any individual trial doesn’t affect the probabilities in the other trials.) 3. Each trial must have all outcomes classified into two categories (commonly referred to as success and failure). 4. The probability of a success remains the same in all trials. pp 218-219 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 31 Notation for Binomial Probability Distributions S and F (success and failure) denote the two possible categories of all outcomes; p and q denote the probabilities of S and F, respectively: P(S) = p (p = probability of success) P(F) = 1 – p = q (q = probability of failure) Note that “success” may not be desirable. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 32 Notation (continued) n denotes the fixed number of trials. x denotes a specific number of successes in n trials, so x can be any whole number between 0 and n, inclusive. p denotes the probability of success in one of the n trials. q denotes the probability of failure in one of the n trials. P(x) denotes the probability of getting exactly x successes among the n trials. pg 219 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 33 Methods for Finding Probabilities We will now discuss three methods for finding the probabilities corresponding to the random variable x in a binomial distribution. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 34 Method 1: Using the Binomial Probability Formula P(x) = n-x x n! •p •q (n – x )!x! for x = 0, 1, 2, . . ., n where n = number of trials x = number of successes among n trials p = probability of success in any one trial q = probability of failure in any one trial (q = 1 – p) pg 221 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 35 Rationale for the Binomial Probability Formula P(x) = n! n-x x •p •q (n – x )!x! The number of outcomes with exactly x successes among n trials Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 36 Binomial Probability Formula P(x) = n! • (n – x )!x! Number of outcomes with exactly x successes among n trials Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. px • n-x q The probability of x successes among n trials for any one particular order 37 Method 2: Using TI-83/84 • Press 2nd VARS to get the DISTR menu • Scroll down to binomialpdf( and press ENTER • Type in three values: n, p, x (separated by commas) and close the parenthesis • You see a line like binomialpdf(10,.3,6) • Press ENTER and read the probability of the value x (successes) in n trials Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 38 Alternative use of TI-83/84 • Press 2nd VARS to get the DISTR menu • Scroll down to binomialcdf( and press ENTER • Type in three values: n, p, x (separated by commas) and close the parenthesis • You see a line like binomialcdf(10,.3,6) • Press ENTER and read the combined probability of all values from 0 to x (i.e., probability that there are at most x successes) Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 39 Method 3: Excel • Use an Excel canned function Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 40 Using Excel (2007 or later) for Binomial Distributions (1) With the cell you want highlighted, From the home tab, click the arrow beside the ∑ button, and select “more functions” Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 41 Using Excel (2007 or later) for Binomial Distributions (2) In the pop-up window, scroll down and select the category “statistical” Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 42 Using Excel (2007 or later) for Binomial Distributions (3) Under “select a function” scroll down to BINOMDIST and click “ok” Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 43 Using Excel (2007 or later) for Binomial Distributions (4) Fill in the values required Number_s # of successes Trials # of trials Probability_s probability of success Cumulative TRUE if you want the probability of x or fewer successes, FALSE if you just want the probability of exactly x successes. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 44 Using Excel (2007 or later) for Binomial Distributions (5) Now look back at the cell you had highlighted: From now on, you can just enter this short form if you find it quicker. Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 45 Example An “unfair coin” has a 0.55 probability of getting heads and is tossed 10 times • What is the probability of getting exactly 5 heads? • What is the probability of getting at least 4 heads? Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 46 Example Probability of heads: p = 0.55 Number of tosses: n = 10 “Exactly 5 heads” → P(5) P(5) = 0.234 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 47 Example Probability of heads: p = 0.55 Number of tosses: n = 10 “at most 4 heads” → ∑P(4) ∑P(4) = 0.262 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 48 Example An “unfair coin” has a 0.55 probability of getting heads and is tossed 10 times p = 0.55 n = 10 • What is the probability of getting exactly 5 heads? P(5) = 0.234 • What is the probability of getting at least 4 heads? ∑P(4) = 0.262 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 49 Binomial Distribution: Formulas Mean µ = n•p Variance σ2 = n • p • q Std. Dev. σ = n•p•q Where n = number of fixed trials p = probability of success in one of the n trials q = probability of failure in one of the n trials Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 50 Interpretation of Results It is especially important to interpret results. The range rule of thumb suggests that values are unusual if they lie outside of these limits: Maximum usual values = µ + 2 Minimum usual values = µ – 2 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 51