Probability Continued Chapter 6 Random Variables Probability Continued Chapter 6

Bill and Ted’s Excellent Sports Adventure Bill and Ted play together on the same basketball team. Bill averages 13 points a game with a variance of 5. Ted averages 10 points a game with a variance of 5.

Bill and Ted Team Up How many total points will they average? E(Bill + Ted) = 13 + 10 = 23 points a game. What is the variance of their points? VAR(Bill + Ted) = 5 + 5 = 10 What explains the increase in variance? If Bill and Ted both have good games, variance should increase more than a single player. Same if Bill and Ted both have bad games.

Bill and Ted Compete How many more points will Bill average? E(Bill - Ted) = 13 - 10 = 3 points a game. What is the variance of the difference in points? VAR(Bill - Ted) = 5 - 5 = 0 … right? If we are looking for the difference in scores, would it make sense that there is now NO variation in their scoring? Variance always adds. Always. VAR(Bill - Ted) = 5 + 5 = 10

Bill and Ted’s Conclusions What do we notice from Bill and Ted’s Excellent Sports Adventure? When combining two random variables, the expected value calculation work intuitively. mX + Y = mX + mY mX – Y = mX – mY When combining two random variables, variance always combines. s2X + Y = s2X + s2Y s2X - Y = s2X + s2Y

The “Pythagorean Theorem of Statistics” Remember the steps to solve a Pythagorean Theorem problem. We square the sides of the right triangle. Solve for the missing square. Take the square root to convert back to side length. Working with variation in random variables. We must convert standard deviations to variance. Combine to find the total variance. Take the square root of the total variance to convert back to standard deviation. Next, the mathematics behind what we have learned.

FYI: X+X Please note you will not be responsible for producing this probability distribution. This slide is just to show the mathematics behind the X+X distribution. 1 2 .2 .8

X+X

FYI: X–X Please note you will not be responsible for producing this probability distribution. This slide is just to show the mathematics behind the X-X distribution. 1 2 .2 .8

X – X

Practice Suppose that E(X) = 2.5, Var(X) = .16, E(Y) = 1.2, Var(Y) = .36 What is E(X+Y) = ?, Var(X+Y) = ? E(X+Y) = 2.5 + 1.2 = 3.7 Var(X+Y) = .16 + .36 = .52 E(X – Y) = ?, Var(X – Y) = ? E(X – Y) = 2.5 – 1.2 = 1.3 Var(X – Y) = .16 + .36 = .52

Practice E(X) = 2.5, Var(X) = .16 E(Y) = 1.2, Var(Y) = .36 What is s(X) = ?, s(Y) = ?, s(X+Y) = ? s(X) = .4, s(Y) = .6 Since Var(X+Y) = .52, s(X+Y) = .7211 Cannot add standard deviation directly!! What is E(2X – 4Y) = ?, s(2X-4Y) = ? E(2X–4Y) = 2(2.5) – 4(1.2) = 0.2 Var(2X–4Y) = 22(.16) + 42(.36) = 6.4 So s(2X–4Y) = 2.53