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Random Variables OBJECTIVE Construct a probability distribution. Find measures of center and spread for a probability distribution. RELEVANCE To find the likelihood of all possible outcomes of a probability distribution and to describe the distribution. Definition…… Random Variable – a random variable, x, represents a numerical value associated with each outcome of a probability experiment; values are determined by chance (Mutually exclusive) Example…… You have 4 True or False questions and you observe the number correct. Random Variable (x) # correct Possible Values of x 0, 1, 2, 3, 4 Example…… Count the number of siblings in your family. Random Variable (x) # of siblings Possible Values 0, 1, 2, 3, ….. Example…… Toss 5 coins and observe the number of heads. Random Variable (x) # of heads Possible Values 0, 1, 2, 3, 4, 5 2 Types of Random Variables…… Discrete – can be counted Ex: # of joggers Continuous – can be measured Ex: Height, Weight, Temp, Time, Distance Probability Distributions – Discrete …… Probability Distribution…… Consists of the values a random variable can assume and the corresponding probabilities of the variables. The probabilities used are theoretical. Example…… Construct a probability distribution of tossing 2 coins and getting heads. Remember: x = the number of heads possible p(x) = the probability of getting those heads Answer…… The sample space for tossing 2 coins consists of TT HT TH HH The probability distribution of getting heads… x 0 1 2 P(x) 1/4 2/4 1/4 Now Graph It…… 2 4 P (x ) 1 4 0 1 x 2 Example…… Construct a probability distribution of tossing 3 coins and getting heads. Remember: x = the # of heads possible P(x) = the probability of getting those heads Answer…… The sample space of tossing 3 coins consists of HHH HHT HTH HTT THH THT TTH TTT The probability distribution…. x 0 1 2 3 P(x) 1/8 3/8 3/8 1/8 Graph It…… 3 8 P (x ) 2 8 1 8 0 1 2 x 3 Example…… Construct a probability distribution for rolling a die. The probability distribution…. x 1 2 3 4 5 6 P(x) 1/6 1/6 1/6 1/6 1/6 1/6 2 Requirements for a Probability Distribution…… 1. 0 P( x) 1 Probability has to fall between 0 and 1 for individual probabilities 2. P( x) 1 All the P(x)’s must add up to 1. Is it a probability distribution? Example 1…… x 0 5 10 15 20 P(x) 1/5 1/5 1/5 1/5 1/5 YES Example 2…… x 1 2 3 4 P(x) 1/4 1/8 1/16 9/16 4/16 2/16 YES Example 3….. x 2 3 7 NO P( x) 1 P(x) 0.5 0.3 0.4 Example 4….. x 0 2 P(x) -1.0 1.5 2 6 0.3 0.2 NO Individual P(x) does not fall between 0 and 1 Complete the Chart…… 1 – 0.72 = 0.28 x 1 2 3 4 5 P(x) 0.2 0.35 0.1 0.07 ? Probability Functions Probability Function…… A rule that assigns probabilities to the values of the random #’s. Example…… The following function, is a probability distribution for x = 1, 2, 3, 4. Write the probability distribution. x P( x) 10 Answer…… x 1 2 3 4 P(x) 1/10 2/10 3/10 4/10 You can set a formula in the lists using your calculator by putting x’s in L1 and setting L2 as L1/10……or you can just plug it in by hand. Would the function x P( x) 10 Be a probability distribution for x= 2, 3, 4, 5? Answer…… Here is the distribution…… The sum of the P(x) values adds up to a number greater than 1….therefore it is NOT a probability distribution. x 2 3 4 5 P(x) 2/10 3/10 4/10 5/10 Example…… Write the probability distribution for the function y x 3 if x = 0, 1, 2 x 0 1 2 P(x) 0 1/3 2/3 Mean, Standard Deviation, and Variance of a Probability Distribution Section 5.3 Take Note: Probability distributions may be used to represent theoretical populations, therefore, we will use population parameters and our symbols used will be for population values. mean 2 var iance s tan dard deviation Mean : [ x P( x)] Variance : [( x ) P( x)] 2 St. Deviation : 2 2 Example…… A. Find the mean, variance, and standard deviation. B. Find the probability that x is between 2 C. Find the probability that x is between x P(x) 0 0.2 1 0.1 2 0.3 3 0.4 Mean: [ x P( x)] x P(x) 0 0.2 1 0.1 2 0.3 3 0.4 We are going to do this on our graphing calculator. Put x’s in L1 and P(x) in L2. Set a formula in L3: L1xL2 The sum of this column is your mean. This is what it looks like on the calculator. The mean is 1.9 Another way to sum a list….. 2nd Stat Math Sum L(#) This is actually better because you can store the mean to a location Variance: 2 [x P( x)] You’ll need to add a L4 in your calculator. Set a formula to find the formula above. 2 The sum of L4 is your variance. The var iance is 1.29. Standard Deviation: 2 The variance was 1.29. 2 1.29 1.14 The s tan dard deviation is 1.14. Find the probability that x is between 2 Remember that Remember that 1.9. 1.14 Find the x’s for which you should sum their probabilities: The x’s will be between 1.9 – 2(1.14) = -0.38 1.9 + 2(1.14) = 4.18 Remember: The x values are between -0.38 and 4.18. x P(x) x[P(x)] Variance Formula 0 0.2 0 0.722 1 0.1 0.1 0.081 2 0.3 0.6 0.003 3 0.4 1.2 0.484 All of our x’s fall in between these 2 values. Add the probabilities that goes along with these values. 0.2+0.1+0.3+0.4 = 1 The answer is 1. Find the probability that x is between Remember that Remember that 1.9. 1.14 Find the x’s for which you should sum their probabilities: The x’s will be between 1.9 – 1.14 = 0.76 1.9 + 1.14 = 3.04 Remember: The x values are between 0.76 and 3.04. x P(x) x[P(x)] 0 0.2 0 Variance Formula 0.722 1 0.1 0.1 0.081 2 0.3 0.6 0.003 3 0.4 1.2 0.484 Only 3 of the x values fall between the values listed above. Add the probabilities that go with those 3. 0.1+0.3+0.4 = 0.8 The answer is 0.8. Example…… A. Set up the distribution for x P( x) for x 1, 2, 3 6 B. Find the mean, variance, and standard deviation of the probability distribution. C. Find the probability that x is between Set up the distribution…… x P(x) 1 1/6 2 2/6 3 3/6 Mean: [ x P( x)] x P(x) 1 1/6 2 2/6 3 3/6 We are going to do this on our graphing calculator. Put x’s in L1 and P(x) in L2. Set a formula in L3: L1xL2 The sum of this column is your mean. This is what it looks like on the calculator…… The mean is 2.3 Variance: 2 [x 2 P( x)] You’ll need to add a L4 in your calculator. Set a formula to find the formula above. The sum of list 4 is the variance. The VARIANCE is 0.56. Standard Deviation: The variance was 0.555555555. 2 0.555555555 0.75 The standard deviation is 0.75. 2 Find the probability that x is between Remember that 2.33 Remember that 0.75 Find the x’s for which you should sum their probabilities: The x’s will be between 2.33 – 0.75 = 1.58 2.33 + 0.75 =3.08 Remember: The x values are between 1.58 and 3.08. x P(x) x[P(x)] Variance Formula 1 1/6 0.16667 0.2963 2 2/6 0.66667 0.03704 3 3/6 1.5 0.22222 Only 2 of the x values fall between 1.58 and 3.08. Add the probabilities of those 2 x-values. 2/6 + 3/6 = 5/6 The answer is 5/6. Worksheet