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 1 2 P(x) 1/4 2/4

Now Graph It……

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 THH HHT THT HTH TTH HTT TTT The probability distribution…. x 1 2 3 P(x) 1/8 3/8

Graph It……

Example…… Construct a probability distribution for rolling a die. The probability distribution…. x 1 2 3 4 5 6 P(x) 1/6

2 Requirements for a Probability Distribution…… 1. Probability has to fall between 0 and 1 for individual probabilities 2. All the P(x)’s must add up to 1.

Is it a probability distribution? Example 1…… 5 10 15 20 P(x) 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 P(x) 0.5 0.3 0.4 NO

Example 4….. NO Individual P(x) does not fall between 0 and 1 x 2 6 2 6 P(x) -1.0 1.5 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 ? 1 – 0.72 = 0.28

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.

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 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 if x = 0, 1, 2 x 1 2 P(x) 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.

Example…… A. Find the mean, variance, and standard deviation. B. Find the probability that C. Find the probability that x P(x) 0.2 1 0.1 2 0.3 3 0.4

Mean: We are going to do this on our graphing calculator. x P(x) 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.

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: You’ll need to add a L4 in your calculator. Set a formula to find the formula above.

The sum of L4 is your variance.

Standard Deviation: The variance was 1.29.

Find the probability that Remember that 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. P(x) x[P(x)] Variance Formula 0.2 0.722 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 Remember that 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. P(x) x[P(x)] Variance Formula 0.2 0.722 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 B. Find the mean, variance, and standard deviation of the probability distribution. C. Find the probability that

Set up the distribution…… x P(x) 1 1/6 2 2/6 3 3/6

Mean: We are going to do this on our graphing calculator. 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……

Variance: 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.

Standard Deviation: The variance was 0.555555555. The standard deviation is 0.75.

Find the probability that Remember that 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. 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