Discrete Probability Distribution Tables

Transcription

1 Section 5 A : Discrete Probability Distributions Introduction Discrete Probability Distribution ables A probability distribution table is like the relative frequency tables that we constructed in chapter. hose relative frequency tables had all the possible values of x in the left column and the right column contained the percent (as a decimal) that each outcome occurred out of the. he probability distribution table has the same format. he left column will contain a list of all the the possible values of the random variable x. he right column will contain the probability that each of the possible x values occurs. A Relative Frequency A Probability Distribution able able Of Girls in births Relative Frequency(x) Of Girls in births he left column will contain a list of all the possible outcomes of the variable x. he right column will contain the proportion of the population that each of the possible outcomes occurs. he left column will contain a list of all the possible outcomes of the variable x. he right column will contain the probability that a given value of x will occur. Example 4A Example 4 B. (%) of the women he probability that if I select who have children women from the population of have girls women who have children that they will have girls is. (%).8 (8%) of the women he probability that if I select who have children women from the population of have girls women who have children that they will have girls is.8 (8%) It is easy to see that if % of a population has a given attribute, then there is a % chance that a person selected from that population will have that attribute. Section 5 A Page Eitel

2 he Random Variable A Random Variable is a variable (typically represented by x) whose different numerical values describe all the possible outcomes from a procedure or experiment. A Discrete Random Variable: A random variable is discrete if the possible values for the variable are countable. For the purposes of this chapter we will use whole numbers for the possible values of the Discrete Random Variable. hat is, the possible values for x are,,,, 4,.... Continuous Random Variable: A random variable is continuous if the values are associated with measurements on a continuous scale. Measurements of time, volume, weight, temperature and distance are examples of of such measurements. his unit will only deal with Discrete Random Variables. We will work with continuous random variables in later chapters. Examples of a Discrete Random Variable Example x is the number of heads that occurred when a coin was flipped 4 times in a row. x is a discrete variable because the possible values for x are x =,,, or 4. x is random variable because the value of x will vary with each new set of 4 flips Example x is how many of the 6 patients that used a new drug showed an improvement. x is a discrete variable because the possible values for x are x =,,,, 4, 5 or 6 x is a random variable because the value of x will vary with each new set of 6 patients tested. Example 5 students who attend FLC are asked how many days a week they go to school. x is a discrete variable because the possible values for x are x =,,,, 4, 5 x is random variable because the value of x will vary with each new set of 5 students. Section 5 A Page Eitel

3 here are two types of Discrete Probability Distributions ables One type is based on a Sample and the other is based on a heoretical Model he outcome from flipping a coin 5 times = = 4 A Probability Distribution able A Probability Distribution able based on a Sample based on a heoretical Model he outcomes from he probability if getting a head on a sample of 5 coin flips one flip of the coin is / eads = ails = 4 P() = / and P() = / Of eads in flip of a coin Freq (x) P (x) Of eads in flip of a coin P( ) = 5 = P() = 4 5 =.56.5 he values of will vary he values of are fixed and based on the sample reflect the heoretical population he probabilities for each x value in the Probability able will vary depending on the sample. his table does not reflect the entire population. As the sample size gets larger the values for based on the sample will get closer to the exact population values of he probabilities for each x value in the Probability able of the theoretical model will not vary. hey are not based on a sample. hese probabilities represent the entire population. We do not need to list the entire population to find each. We use formulas to develop the values in the table. We can then use this type of table to answer questions about what probabilities for the variable x can be expected for the entire population. Section 5 A Page Eitel

4 Example Creating a Probability Distribution able based on a theoretical model if the probability of each outcome in the same One way to create a theoretical model of a Probability Distribution able would be to create the entire population sample space based on the theoretical outcomes and then calculate the value of for each x as a proportion of the total sample space. P() = / and P() =./ Procedure: Flip a coin times and record the number of heads. he model below contains the population of all 8 equally likely outcomes. he random variable x ( the number of eads ) can have a value of,, or. he probability of each of the possible values for x is computed as shown below. of eads in flips ow each was calculated for each x.5 P( heads) = number of outcomes with head = 8 =.5.75 P( heads) = number of outcomes with head = 8 = P( heads) = number of outcomes with head = 8 =.75.5 P( heads) = number of outcomes with head = 8 =.5 he third column on the right is not normally shown as part of a probability distribution table. It is included here to help understand how each was calculated. It is also common to state be a decimal number rounded off to a given number of decimal places. For this chapter we will list all P(X) as a decimal number rounded off to decimal places unless stated otherwise. Section 5 A Page 4 Eitel

5 Example Creating a Probability Distribution able based on a theoretical model if the probability of each outcome is NO the same One way to create a theoretical model of a Probability Distribution able would be to create the entire branching diagram and determine the probability for each outcome P(A and B) = P(A) P(B) Procedure: ner and then spin ner again. First / 4 Y Second / 4 / 4 Possible Outcomes Y R Y Y Y R Probabilities P ( Y and Y ) = 4 4 = 6 P ( Y and R ) = 4 4 = 6 / 4 First R / 4 / 4 Second Y R R Y R R P ( R and Y ) = 4 4 = 6 P (R and R ) = 4 4 = 9 6 of times a red is spun in spins P( Reds) = P(Y and Y) = 6 P( Red) = P(R and Y) + P( Y and R) = 6 6 P( Reds) = P(R and R)= 9 6 P( Red) + P( Red) + P( Red ) = = 6 6 = he table above represents all the possible outcomes Since = the table represents the entire population of probabilities for the discrete variable x. Creating a Probability Distribution able based on a theoretical model produces a table that represents the entire population of probabilities for the discrete variable x. Section 5 A Page 5 Eitel

6 Discrete Probability Distribution ables for different samples will not be the same. Procedure: Flip a coin times and record the number of heads. he outcome from a sample of 48 flips he outcome from a sample of flips f heads in flips Freq. (x) n = 48 = frequency of x total sample size n f heads in flips Freq. (x) n = 4 = frequency of x total sample size n 5 P( ) = 6 48 =.4 47 P( ) = 47 =. 7 P() = 7 48 = P()) = 45 =.77 9 P( ) = 9 48 = P( ) = 446 =.7 7 P( ) = 7 48 = P( ) = 55 =.9 he sample of 48 flips did not have values equal to the theoretical model we developed using the population sample space in Example 6. he values of for the table based on samples was closer to the values based on the theoretical model but they were sill not exactly equal to theoretical model. If we conducted this procedure, times we would expect the values of each to get even closer to the theoretical model. he value for P() for large samples would be about.5 he value for P() for large samples would be about.75 he value for P() for large samples would be about.75 he value for P() for large samples would be about.5 You may wonder if there is a way to create a theoretical table that world describe the expected outcomes based on flipping the coins a very large number of times ( n ) without creating the entire population sample space or conducting a large sample. Yes there is. Under some conditions we can construct a Probability Distribution able using probability formulas to find the expected outcomes. he values in a table of this type are considered to be the exact theoretical values for the population probabilities of the variable x. Section 5 A Page 6 Eitel