1 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY S TATISTICS Chapter 4 Probability Distributions

2 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Chapter 4 Probability Distributions 4-1 Overview 4-2 Random Variables 4-3 Binomial Probability Distributions 4-4 Mean, Variance, Standard Deviation for the Binomial Distribution

3 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 4-1 Overview This chapter will deal with the construction of probability distributions by combining the methods of Chapter 2 with the those of Chapter 3. Probability Distributions will describe what will probably happen instead of what actually did happen.

4 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Figure 4-1 Combining Descriptive Statistics Methods and Probabilities to Form a Theoretical Model of Behavior

5 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 4-2 Random Variables

6 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Definitions  Random Variable a variable (typically represented by x ) that has a single numerical value, determined by chance, for each outcome of a procedure  Probability Distribution a graph, table, or formula that gives the probability for each value of the random variable

7 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Probability Distribution Number of Girls Among Fourteen Newborn Babies x P(x)P(x) Table 4-1

8 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Figure 4-3 Probability Histogram

9 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Definitions  Discrete random variable has either a finite number of values or countable number of values, where ‘countable’ refers to the fact that there might be infinitely many values, but they result from a counting process.  Continuous random variable has infinitely many values, and those values can be associated with measurements on a continuous scale with no gaps or interruptions.

10 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Requirements for Probability Distribution The sum of all the probabilities of the distribution equals 1. Σ P( x ) = 1 where x assumes all possible values All probabilities of the distribution must fall between 0 and 1 inclusive. 0  P ( x )  1 for every value of x

11 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Mean µ =  [x P (x)] Variance  2 =  [ (x - µ) 2 P (x )] Standard Deviation  =  2 Mean, Variance and Standard Deviation of a Probability Distribution

12 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Roundoff Rule for µ,  2, and  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,  2, and  to one decimal place.

13 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Example A 4 point pop quiz was given and the scores ranged from 0 to 4, with the corresponding probabilities: 0.05, 0.2, 0.25, 0.3, 0.2 Write the probability distribution in a table. Verify whether a probability distribution is given. Compute the mean, variance and standard deviation of the probability distribution.

14 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Solution: Write the probability distribution in a table. X (Quiz Score)P(x)

15 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Solution: Verify whether a probability distribution is given Yes it is a probability distribution because the sum of P(x) for all values of x is 1 and P(x) for all values of x is between 0 and 1, inclusive. X (Quiz Scores)P(x) Sum1.0

16 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Solution: Compute the mean Enter x values into L 1, Enter P(x) into L 2. Compute the products L 3 = L 1 * L 2. Find the sum(L 3 ) which is the Mean: μ = 2.4 L1L2L3=L1*L2 xP(x)x * P(x) Sum1.02.4

17 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Solution: Compute the variance and standard deviation. Having already found the mean of 2.4 Compute L 4 = (L ) 2 * L 2. Find the sum(L 4 ) which is the  2 = 1.34  1.3 Find the  which is the square root of  2  = = 1.16  1.2 Mean: μ = 2.4 Variance: σ 2 = 1.3 Std Dev σ = 1.2 L1L2L3 = L1*L2L4=(L1-2.4) 2 *L2 xP(x)x * P(x)(x- μ) 2 *P(x) (0-2.4) 2 *0.05 = (1-2.4) 2 *0.2 = (2-2.4) 2 *0.25 = (3-2.4) 2 *0.3 = (4-2.4) 2 *0.2 = Sum

18 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Definition Expected Value The average(mean) value of outcomes, if the trials could continue indefinetly. E =  [ x P( x )]

19 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman The Daily Number allows you to place a bet that the three-digit number of your choice. It cost $1 to place a bet in order to win $500. What is the expected value of gain or loss?

20 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman E =  [x P(x)] Event Win Lose x $499 - $1

21 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman E =  [x P(x)] Event Win Lose x $499 - $1 P(x)

22 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman E =  [x P(x)] Event Win Lose x $499 - $1 P(x) x P(x)

23 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman E =  [x P(x)] Event Win Lose x $499 - $1 P(x) x P(x) E = -$.50

24 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Expected Value This means that in the long run, for each $1 bet, we can expect to lose an average of $.50. In actuality a player either loses $1 or wins $499, there will never be a loss of $.50. If an expected value is $0, that means that the game is fair, and favors no side of the bet.

25 Chapter 4. Section 4-1 and 4-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Alternate Solution: Excluding the cost of playing. Event Win Lose x $500 0 P(x) x P(x) E = $.50, Which means the average win is $.50. If the game cost $.50 to play E = 0 and it would be a “fair” game and not favor either side of the bet. If the game cost $.75 to play E = -.25 and it would not be a “fair” game and would favor the house.