Download "It is the nature of every man to err, but only the fool perseveres in

"It is the nature of every man to err, but only the fool perseveres in error."
Cicero (Roman statesman, 106 - 46B.C.)
7.2 MEANS AND VARIANCES OF RANDOM VARIABLES (Pages 385 -404)
OVERVIEW: If X is a discrete random variable with possible values xi having
probabilities pi, then
n
...mean of x values =  x  x1 p1  x 2 p 2  ...  x n p n   xi pi , and
i 1
...variance of x values =
n
 2 = x1   x 2 p1  x 2   x 2 p 2  ...  x n   x 2 p n   xi   x 2 pi
i 1
This section provides illustrations using these formulas.
Note: the mean is often times referred to as the expected value which describes the
average outcome in the long run. Also keep in mind that the std. deviation or error
is still  , which can be calculated by taking the square root of the variance.
Example: Consider a random variable x that assumes the values 1,2,3 with respective probabilities 60%,
30%, 10%. The following table illustrates use of the formulas in the OVERVIEW.
xi pi
xipi
xi -  (xi -  (xi - pi
SUMS
The formulas shown in the OVERVIEW are consistent with previously established formulas for mean,
standard deviation, and variance. To illustrate this, note that in the example above, one would "expect"
this distribution of the random variable in ten trials: {1,1,1,1,1,1,2,2,2,3}. If you place this set in listL1
and calculate 1-Var Stats, you will find that the mean is 1.5 and the standard deviation is 0.6708203932.
Squaring the standard deviation to obtain variance yields 0.4499999999, or 0.45.
Law of Large Numbers:
Example: When you flip a coin numerous times, you "expect" heads 50% of the time.
RULES FOR MEANS:
The purpose here is to illustrate important rules (page 396) relating to the mean statistic.
Rule 1: IF X is a random variable and a and b area fixed numbers, then
 a bX   Expected Value of a  bX 
Let Sx= {1, 3}. The mean of Sx is _____
Now, add 5 to the product of each element and 7, producing the set S5+7x ={
The mean of S5+7x is
In this situation, mean (S5+7x) =
}.
Rule 2: If X and Y are random variables, then
 X Y 
Now, let Sy = {5, 9, 22}, which has mean =_____
Construct the set Sx+y ={
The mean of Sx+y is
In this situation, mean (Sx+y) =
}={
}.
RULES FOR VARIANCES:
The purpose here is to illustrate important rules (p. 400) relating to the variance statistic.
Rule 1: IF X is a random variable and a and b area fixed numbers, then
 abX 2  Variance a  bX  
*Note: adding a constant to a random variable changes its mean but not the variance.
Let Tx = {10, 14}.
The mean, standard deviation, and variance of Tx are, respectively, __________________
Now, multiply each element by 3 and add 5, obtaining the set T5+3x ={
}.
The mean, standard deviation, and variance of T5+3x are, respectively, _______________
Note that 36 = 32(4). In other words, var(T5+3x) = _______________
*Note also that StDev(T5+3x) = _______________
Rule 2: If X and Y are independent random variables, then
 X Y 2 
Now, let Ty = {6, 9, 12}.
The mean, standard deviation, and variance of Ty are, respectively _________________.
Assume that Tx and Ty are independent sets.
Construct the set Tx-y ={
}={
}.
The mean, standard deviation, and variance for T are, respectively _________________.
Note that
If you construct the set Tx+y, you will find that var(Tx+y) = var(Tx-y) =
In a nutshell, with independence, variances add when sets Tx+y and Tx-y are constructed as
indicated above. Note carefully that the standard deviations do not add.
You should also note mean(Tx-y) = 3 = (12 – 9) = mean(Tx) - mean(Ty).
If you construct Tx+y, you will find that mean(Tx+y) = mean(Tx) + mean(Ty).
Example: You decide to play the 3 digit lotto which pays $500 if you win.
How many three digit numbers are there?
___ ___ ___ = ____
So your chances of winning are
X is the random variable which is the amount you win when playing this game.
Probability Distribution 1
Outcome
Probability
The ordinary average (mean) is $250.00 but does this make sense in this situation?
If you played this game over and over, then in the long run your average winnings would be:
X 
X2 
therefore

So, if you buy a ticket W = X – 1
Rule says  w   X  1
=
Winnings = what you win – 1 dollar spent on ticket
Rule says the variance will stay the same before and after you factor in the dollar you pay to play.
Let’s say you play on two different days. Are these independent events?
So the Expected value in the long run of playing on two days is:  X Y 
The variance of total payoff when playing on two days is:
( X Y ) 2 
Which makes the standard deviation of the total payout:

Which is different than if you just add the standard deviations of X and Y and get __________________
So remember, variances of independent random variables add but standard deviations DO NOT!
1. Let’s play a game. You will roll a six sided fair die and I will pay you $6 if you roll a 6 and
you will pay me $1 if you roll a 1 – 5. Define your random variable X. Each time you play this
game, how much should you expect to win on average in the long run?
2. What would you have to pay when you roll a 1 – 5 to make the game fair?
A new game of dice is being played with two dice on some college campuses. The dice are rolled one
after the other. Instead of adding the numbers on each die, each player subtracts the number on the
second die from the number on the first die. Let S = this difference. Note that for each die,   3.5 and
 2  2.91.
3. Give the values that S consists of in set form.
4. P(S = -2) = _________
5. IF A and B are random variables and A ={10, 11, 12} and B = {1, 2, 3}, then give the set for the new
random variable 3A – 2B.
6. A class in AP Statistics takes a two-part exam. If the mean for the first part was 42 with a standard
deviation of 10 and the mean for the second part was 38 with a standard deviation of 12, find the mean
and standard deviation for the entire exam (if possible). Explain your work.
7. The American Mathematics Contest (AMC-12) has 100 possible points. The American Invitational
Math Exam (AIME) has 15 possible points. Let X be the random variable whose values are the scores
on the AMC-12 test and let Y be the random variable whose values are the scores on the AIME test. An
index score for each student who takes both exams is calculated by the formula:
(AMC-12 score) + 10(AIME score).
Suppose the mean and standard deviation of the students taking both exams are 113.3 and 3.7
respectively on the AMC-12 and suppose that the same values for the AIME test are 6.2 and 2.7
respectively. Find the mean and standard deviation of the index scores for those that have taken both
tests. If this is not possible, explain why it is not.