6.2 12.5.2017

Discrete Random Variables on the Calculator Last class, somebody asked if we could make our calculators do the mean and standard deviations on our calculators I wanted you to do it by hand on your homework But the answer is yes STAT—EDIT—STAT—1-Var Stats (List: L1, FreqList: L2)

Try it Omaha is a version of poker in which each player is dealt 4 cards. The probability distribution below shows the probability of getting a certain number of spades in one’s hand Calculate the mean and the standard deviation 1 2 3 4 .304 .439 .213 .041 .003

Linear Transformations In chapter 2, we learned about what happens to a distribution when we transform the distribution Add a number to all observations Multiply all observations by a number It works the same way with random variables (probability distributions)

Mean of C: $562.50 (150*3.75), st. dev: $163.50 (150*1.09) Pete offers Jeep tours to tourists. On any given day, he has between 2 and 6 passengers, with the probabilities listed below The mean of X is 3.75, and the standard deviation is 1.09 The random variable C describes the amount of money that Pete collects. If Pete charges $150 per passenger, then: Mean of C: $562.50 (150*3.75), st. dev: $163.50 (150*1.09)

Multiplying/Dividing

Pete’s Jeeps (again) Money Collected: It costs Pete $100 per trip to buy gas and permits. The random variable V describes his profits, once accounting for these costs: What will be the mean of V? What will be the standard deviation of V?

Pete’s Jeeps (again) Money Collected: It costs Pete $100 per trip to buy gas and permits. The random variable V describes his profits, once accounting for these costs: What will be the mean of V? $462.50 What will be the standard deviation of V? $163.50

Adding/Subtracting

Combining Random Variables We learn that Pete’s main competitor is his sister, Erin. The random variable Y represents the number of passengers on a randomly selected trip with Erin. X still represents Pete

Combining Random Variables We want to know how many people go on Jeep adventures COMBINED in any given day. So we need the total (T) of Erin+Pete. T=X+Y How would we figure out the average (mean) number of people that take a jeep tour on any given day?

Combining Random Variables How would we figure out the average (mean) number of people that take a jeep tour on any given day? 3.75+3.1= 6.85 passengers

Combining Random Variables Finding the standard deviation of the combined variable is more difficult

Variance vs. Standard Deviation We cannot add standard deviation. We have to add the variances first.

Finding Standard Deviation We can only add variances…how can we get the variance for each variable?

Finding Standard Deviation We can only add variances…how can we get the variance for each variable? Square them: 1.09^2= 1.1881 0.943^2= .889249

So to get the combined variance: Square them: 1.09^2= 1.1881 0.943^2= .889249 So to get the combined variance: 1.1881+.889249= 2.077349 Last step…how do we get the standard deviation?

So to get the combined variance: Square them: 1.09^2= 1.1881 0.943^2= .889249 So to get the combined variance: 1.1881+.889249= 2.077349 Last step…how do we get the standard deviation? Take the square root 2.077349^.5= 1.4413

Thinking about independence In our example, we assumed that the number of people riding taking Erin’s Jeep tour was independent of the number of people riding Pete’s Jeep tour Can we think of reasons that this might be false?

Thinking about independence In our example, we assumed that the number of people riding taking Erin’s Jeep tour was independent of the number of people riding Pete’s Jeep tour Can we think of reasons that this might be false? Weather Discounts offered by one of them Time of year So maybe they are independent, maybe they are not The book tells us to assume that they are There IS a formula for calculating the variance of combined random variables when they are not independent But it is beyond the scope of AP Statistics—you only need to be able to calculate the combined variance/standard deviation for independent random variables